fractals in dimension theory and complex networks

BUDAPEST UNIVERSITY OF TECHNOLOGY AND

ECONOMICS

DOCTORAL THESIS

Fractals in dimension theory and complexnetworks

Author:István KOLOSSVÁRY

Supervisor:Dr. Károly SIMON

Doctoral School of Mathematics and Computer ScienceFaculty of Natural Sciences

2019

http://www.bme.hu/?language=en

http://www.bme.hu/?language=en

http://math.bme.hu/~istvanko/enghome.html

http://math.bme.hu/~simonk

http://doktori.math.bme.hu/english/index-E.html

iii

AcknowledgementsIt is a great pleasure to thank the many people who have had a direct impact on

my academic life in the past years.Foremost, I thank my supervisor Károly Simon. Beyond his deep knowledge of

the field, his enthusiasm for mathematics and the unique way he gives it on has trulyhad a great influence on how I think about math and the world around us;

Secondly, I thank my co-authors Balázs Bárány, Gergely Kiss, Júlia Komjáthy andLajos Vágó. I gained valuable experience from each and every joint project;

I thank the referees of my home defense for their thorough work. In particular,the comments of István Fazekas, which greatly improved the clarity of one of thechapters.

Moreover, I thank the support and generous hospitality of all the colleagues atthe Department of Stochastics. I feel very fortunate to have landed here after mybachelor years, thanks to Doma Szász. The atmosphere is inspiring for research andat the same time very friendly;

and last but not least all the support I get from all my family and friends, whoare in some way part of my life, even though many don’t know much about what Iactually do on a daily basis.

I acknowledge the financial support of different grants and scholarships withoutwhich all my travels, meeting many new colleagues, presenting results at conferencesand learning many interesting topics would not have been possible.

The template used for the thesis can be found at https://www.latextemplates.com/template/masters-doctoral-thesis. I am very satisfied, it saved me fromlots of unnecessary headaches.

https://www.latextemplates.com/template/masters-doctoral-thesis

https://www.latextemplates.com/template/masters-doctoral-thesis

v

Contents

Acknowledgements iii

List of Figures vii

List of Symbols ix

List of Acronyms xi

1 Introduction 11.1 Fractal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Self-affine sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Planar carpets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Fractal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Apollonian networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Informal explanation of contribution . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Dimension of triangular Gatzouras–Lalley-type planar carpets 81.4.2 Pointwise regularity of zipper fractal curves . . . . . . . . . . . 131.4.3 Distances in Random Apollonian Networks . . . . . . . . . . . 15

2 Triangular Gatzouras–Lalley-type planar carpets with overlaps 192.1 Triangular Gatzouras–Lalley-type carpets . . . . . . . . . . . . . . . . . 19

2.1.1 Results of Gatzouras and Lalley . . . . . . . . . . . . . . . . . . 222.1.2 Separation conditions . . . . . . . . . . . . . . . . . . . . . . . . 23

Separation of the cylinder parallelograms . . . . . . . . . . . . . 23Separation of the columns . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.1 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Box dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.3 Diagonally homogeneous carpets . . . . . . . . . . . . . . . . . 29

2.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.1 Symbolic notation . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.2 Atypical parallelograms . . . . . . . . . . . . . . . . . . . . . . . 332.3.3 Ledrappier–Young formula . . . . . . . . . . . . . . . . . . . . . 34

2.4 Upper bound for dimH Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Proof of Theorem 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5.1 The proof of Theorem 2.2.2 assuming Claim 2.5.1 and Proposi-tion 2.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.2 The proof of Claim 2.5.1 . . . . . . . . . . . . . . . . . . . . . . . 402.5.3 Proof of Proposition 2.5.2 . . . . . . . . . . . . . . . . . . . . . . 41

2.6 Proof of results for box dimension . . . . . . . . . . . . . . . . . . . . . 452.6.1 Diagonally homogeneous subsystems . . . . . . . . . . . . . . . 462.6.2 Counting intersections . . . . . . . . . . . . . . . . . . . . . . . . 492.6.3 Proof of Theorem 2.2.7 . . . . . . . . . . . . . . . . . . . . . . . . 51

vi

2.6.4 Proof of Theorem 2.2.8 . . . . . . . . . . . . . . . . . . . . . . . . 522.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.7.1 The self-affine smiley: a non diagonally homogeneous example 532.7.2 Example for dimH Λ = dimB Λ . . . . . . . . . . . . . . . . . . . 532.7.3 Overlapping example . . . . . . . . . . . . . . . . . . . . . . . . 542.7.4 Example "X ≡ X" . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.7.5 Negative entries in the main diagonal . . . . . . . . . . . . . . . 552.7.6 A family of self-affine continuous curves . . . . . . . . . . . . . 56

2.8 Three-dimensional applications . . . . . . . . . . . . . . . . . . . . . . . 58

3 Pointwise regularity of parameterized affine zipper fractal curves 633.1 Self-affine zippers satisfying dominated splitting . . . . . . . . . . . . . 633.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3 Pressure for matrices with dominated splitting of index-1 . . . . . . . . 683.4 Pointwise Hölder exponent for non-degenerate curves . . . . . . . . . 733.5 Zippers with Assumption A . . . . . . . . . . . . . . . . . . . . . . . . . 803.6 An example, de Rham’s curve . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Distances in random and evolving Apollonian networks 894.1 Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2.1 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3 Structure of RANs and EANs . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.1 Tree-like structure of RANs and EANs . . . . . . . . . . . . . . . 934.3.2 Distances in RANs and EANs: the main idea . . . . . . . . . . . 974.3.3 Combinatorial analysis of shortcut edges . . . . . . . . . . . . . 98

4.4 Distances in RANs and EANs . . . . . . . . . . . . . . . . . . . . . . . . 1004.4.1 A continuous time branching process . . . . . . . . . . . . . . . 1004.4.2 Proof of Theorem 4.2.1 and 4.2.5 . . . . . . . . . . . . . . . . . . 1034.4.3 Proof of Theorem 4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . 106

A Basic dimension theoretic definitions 113

B No Dimension Drop is equivalent to Weak Almost Unique Coding 115

Bibliography 119

vii

List of Figures

1.1 A Bedford–McMullen carpet . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 A de Rham curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Linearly parametrized de Rham curve with parameter ω = 1/10. . . . 61.4 An Apollonian gasket and network . . . . . . . . . . . . . . . . . . . . . 71.5 Random Apollonian Networks . . . . . . . . . . . . . . . . . . . . . . . 81.6 A GL and TGL carpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 The "self-affine smiley" . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8 Example of Falconer and Miao together with overlapping version . . . 101.9 TGL carpets with different overlaps . . . . . . . . . . . . . . . . . . . . 101.10 Example "X ≡ X" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.11 A family of self-affine continuous fractal curves . . . . . . . . . . . . . 111.12 Three-dimensional application . . . . . . . . . . . . . . . . . . . . . . . 121.13 Affine zippers in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . 131.14 A RAN after a few steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.15 Impact of shortcut edges on diameter of RAN . . . . . . . . . . . . . . . 17

2.1 The skewness of Ri1 ...in := fi1...in([0, 1]2) . . . . . . . . . . . . . . . . . . . 212.2 The IFS T , where z and (1, z) are identified . . . . . . . . . . . . . . . . 242.3 Intersecting parallelograms Rı and R in the proof of Lemma 2.5.4. . . . 442.4 Intersecting parallelograms Rı and R in the proof of Lemma 2.6.8. . . . 512.5 Orientation reversing maps generally destroy the column structure . . 56

3.1 An affine zipper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2 Local neighbourhood of points in Bn,l,m . . . . . . . . . . . . . . . . . . 743.3 Well ordered property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1 Coding RANs, initial stpes . . . . . . . . . . . . . . . . . . . . . . . . . . 944.2 Coding RANs, induction step . . . . . . . . . . . . . . . . . . . . . . . . 944.3 Tree like structure of RANs . . . . . . . . . . . . . . . . . . . . . . . . . 964.4 Shortest path from u to v . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

ix

List of Symbols

α1(A) ≥ . . . ≥ αd(A) singular values of a d× d matrix Aα(x), αr(x) pointwise and regular Hölder exponent (3.1.3)F , G, H, T Iterated Function SystemsΓ, Λ, Ω attractor of an Iterated Function SystemΣ, ΣH symbolic spacesΠ, ΠH, ΠH, π natural projections from a symbolic spaceHs

δ, Hs (δ-approximate) s-dimensional Hausdorff measure (A.0.1)µd, σ2

d expectation and variance of Yd (4.1.3)p, q, λ probability vectorsp∗ optimal vector for Hausdorff dimensionp optimal vector for box dimensionµp Bernoulli measure on a symbolic spaceνp push forward of µpφs singular value function (A.0.4)

D(p) formula for dimH µp (2.2.1)E(β), Er(β) β-level set of α(x) and αr(x) (3.1.5)Id(x) large deviation rate function of Yd (4.1.4)P(t) matrix pressure function (3.2.1)M(x) cone centered at x (3.1.6)Yd full coupon collector block with d + 1 coupons (4.1.2)

Diam, Flood, Hop diameter, flooding time, hopcount in a graph (4.1.1)dim an unspecified dimensiondimA affinity dimension (A.0.5)dimB, dimB lower and upper box dimension (A.0.3)dimH Hausdorff dimension (A.0.2)dimP packing dimensionprojx orthogonal projection to x-axis

xi

List of Acronyms

i.i.d. independent and identically distributed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

u.a.r. uniformly at random . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

w.h.p. with high probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91

AN Apollonian network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

AUC Almost Uniqe Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

CLT Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

CTBP continuous time branching process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

EAN evolving Apollonian network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

GL Gatzouras–Lalley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

HESC Hochman’s Exponential Separation Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

IFS Iterated Function System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

NDD No Dimension Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

OSC Open Set Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

RAN random Apollonian network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

ROSC Rectangular Open Set Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

SOSC Strong Open Set Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

TGL triangular Gatzouras–Lalley-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

WAUC Weak Almost Uniqe Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1

Chapter 1

Introduction

1.1 Fractal geometry

The word "fractal" comes from the Latin fractus meaning "broken" or "fractured".Benoit B. Mandelbrot coined this term when he wrote in his book Fractals: Form,Chance and Dimension in 1977 that

Many important spatial patterns of Nature are either irregular or frag-mented to such an extreme degree that ... classical geometry ... is hardlyof any help in describing their form. ... I hope to show that it is possible inmany cases to remedy this absence of geometric representation by using afamily of shapes I propose to call fractals – or fractal sets. []

Roughly speaking, any object detailed on arbitrarily small scales that resemblesitself in some way on different magnification scales can be called a fractal. Since the1970s, 80s, fractal geometry has become an important area of mathematics with manyconnections to theory and practice alike. Fractals have found application in geomet-ric measure theory, dimension theory, dynamical systems, number theory, analysis,differential equations or probability theory to name a few. Also there is increasinginterest in more applied areas of mathematics and natural sciences such as networktheory, wavelets, percolation problems all the way to computer graphics, image com-pression, financial markets, fluid turbulence or fractal antenna, etc. The list is boundto expand further in the coming years.

Aim and structure of thesis

The main aim of the Thesis is to demonstrate this diverse applicability of fractals indifferent areas of mathematics. Namely,

1. widen the class of planar self-affine carpets for which we can calculate the dif-ferent dimensions especially in the presence of overlapping cylinders,

2. perform multifractal analysis for the pointwise Hölder exponent of a family ofcontinuous parameterized fractal curves in Rd including deRham’s curve,

3. show how hierarchical structure can be used to determine the asymptotic growthof the distance between two vertices and the diameter of a random graph model,which can be derived from the Apollonian circle packing problem.

Since these topics are not confined to a narrow area of mathematics, a great deal ofeffort has been put into making the presentation accessible to a wider mathematicalaudience.

2 Chapter 1. Introduction

• This Chapter contains a brief (far from exhaustive) introduction to the topicsand is concluded with a section containing informal explanations of the contri-butions made to the different topics, together with the various methods used inthe proofs.

• For those not familiar with the terminologies used, Appendix A contains a briefbackground material with the basic definitions and results.

• Besides the theoretical results, several examples with illustrative pictures areprovided for explanation.

• Plenty of figures assist the Reader through proofs.

Chapters 2, 3 and 4 contain the precise definitions and rigorous formulations of our re-sults, together with the proofs. They are based on the papers [KS18; BKK18; KKV16],respectively.

1.2 Self-affine sets

Let (X, d) be a compact metric space. Usually, we will work on a compact subsetof Euclidean space Rd. A very natural technique to construct fractals is via IteratedFunction Systems (IFSs). An IFS consists of a finite collection F of contracting mapsfi : X → X for i = 1, . . . , N. Huthinson proved in his seminal paper [Hut81], that forevery IFS there exists a unique non-empty compact set Λ, called the attractor, whichsatisfies

Λ =⋃

i∈[N]

fi(Λ),

where [N] = 1, 2, . . . , N. An extensively studied class of IFSs are the self-affine sets,in which case each map of the IFS is an affine transformation, i.e.

fi(x) = Aix + ti,

where Ai ∈ Rd×d is a contracting, invertible matrix and ti ∈ Rd is a translation vector.An important further subclass consists of the self-similar sets, when the matrices canbe written in the form Ai = riOi, where Oi is an orthogonal matrix and 0 < ri < 1 isthe contracting ratio in every direction.

A natural way to depict an IFS is to provide the images fi(R), where R is the small-est rectangle which contains Λ. Without loss of generality we may assume throughoutthis thesis that R = [0, 1]2. The correspondence between the IFS and a figure showingthe collection of images of R will be unique in our study, since the maps do not containany rotations or reflections (except in Subsection 2.7.5). See Figure 1.1 for example.

Perhaps the most fundamental question in fractal geometry is to determine thedimension of a set. Roughly speaking, dimension indicates how much space a setoccupies near to each of its points. Several different types of fractal dimension areused. For basic dimension theoretic definitions such as the Hausdorff, packing and(lower and upper) box dimension of a set and the Hausdorff and local dimension ofmeasures, see Appendix A. Throughout, the Hausdorff, packing, lower and upperbox dimension will be denoted by dimH, dimP, dimB, dimB and dimB, respectively.

The relative position of cylinder sets fi(R) greatly influence the degree of difficultyto calculate the dimension. The simplest case is the Strong Separation Property (SSP)when all sets fi(Λ) are pairwise disjoint. Somewhat weaker is the (Strong) Open

1.2. Self-affine sets 3

Set Condition. The most difficult is when there is heavy overlapping between thecylinders. This will be central in Chapter 2.

Definition 1.2.1. An IFSF with attractor Λ satisfies the Strong Open Set Condition (SOSC),if there exists a non-empty open set U, with Λ ∩U 6= ∅ and such that

⋃

i∈[N]

fi(U) ⊆ U with fi(U) ∩ f j(U) = ∅ for i 6= j. (1.2.1)

In particular, if U can be chosen to be R, then we say that the Rectangular Open Set Condition(ROSC) holds. This will usually be the case.

If U above and Λ can be disjoint, then the Open Set Condition (OSC) holds.

In case of self-similar sets satisfying the OSC, all mentioned dimensions are equalto s, often called the similarity dimension, which is the solution of the Hutchinson–Moran formula

∑i∈[N]

rsi = 1,

see [Fal90, Section 9.2]. Regardless of overlaps, in the self-similar case, the similaritydimension is always an upper bound for the dimensions considered in this thesis. Theanalog upper bound for self-affine sets is the affinity dimension dimAff, introduced byFalconer [Fal88a], which comes from the "most natural" cover of the set, see (A.0.5)for the definition. All self-affine sets satisfy

dimH Λ ≤ dimP Λ ≤ dimBΛ ≤ mindimAff Λ, d.

A central question for the past 30 years has been to determine what can cause thedrop of dimension (from the natural upper bounds). An obvious cause is the presenceof an exact overlap, i.e. there are two distinct sequences i1, . . . , in 6= j1, . . . jk such thatfi1 . . . fin(Λ) = f j1 . . . f jk(Λ). Another cause in higher dimensions can be thehighly regular alignment of cylinder sets, planar carpets are great examples.

The full picture is not completely understood even in the simplest self-similarcase on the real line. The transversality method of Pollicott and Simon [PS95], furtherdeveloped in [Sol95; PS96; PS00] etc., has proven to be a useful tool in determining thedimension of a parametrized family if IFSs for almost all parameter values. Recently,Hochman [Hoc14; Hoc15] made a big breakthrough for the Hausdorff dimension ofself-similar measures, which in particular implies that if an IFS on the real line isdefined by algebraic parameters, then the drop of dimension is equivalent to havingexact overlaps.

In a generic sense, equality of dimensions is typical for self-affine sets. Falconerproved in his seminal paper [Fal88a] that for fixed linear parts A1, . . . , AN if ‖Ai‖ <1/3 and the translations are chosen randomly according to N× d dimensional Lebesguemeasure then all the aforementioned dimensions of the self-affine set are equal. The1/3 bound was later relaxed by Solomyak [Sol98a] to 1/2, which is sharp due toan example of Przytycki and Urbanski [PU89]. Building on the mentioned result ofHochman and results about the Ledrappier–Young formula for self-affine measures[BK17] (see Subsection 2.3.3), very recently Bárány, Hochman and Rapaport [BHR17]greatly improved these results in two dimensions by giving specific, but mild condi-tions on A1, . . . , AN under which the dimensions are equal.

However, in specific cases, which do not fall under these conditions, strict in-equality is possible. Planar carpets form a large class of examples in R2 for whichthis exceptional behavior is typical. The highly regular column and/or row structurecauses the drop of the Hausdorff dimension.


1.2.1 Planar carpets

Independently of each other, Bedford [Bed84] and McMullen [McM84] were the firstto study planar carpets. They split the unit square R into m columns of equal widthand n rows of equal height for some integers n > m ≥ 2 and considered iteratedfunction systems of the form

f(i,j)(x) :=(

1/m 00 1/n

)(xy

)+

(i/mj/n

)

for (i, j) ∈ A ⊆ 0, . . . , m − 1 × 0, . . . , n − 1, see Figure 1.1. They gave explicitformula for the Hausdorff and box-counting dimension of the corresponding attractorΛ. It turns out that dimH Λ = dimB Λ is atypical, namely equality holds if and only ifall non-empty columns have the same number of elements.

FIGURE 1.1: A Bedford–McMullen carpet: IFS on left, attractor onright. In this case dimH Λ < dimB Λ.

Later Gatzouras and Lalley [GL92] generalized the results to IFSs, which kept the(non-overlapping) column structure, however the width of the columns could varyand no row structure is required. They still assumed ROSC (1.2.1) and that the widthof a rectangle is larger than its height, we say that direction-x dominates. The precisedefinition is given in Definition 2.1.3, see also Figure 1.6 for an example.

More recently, Baransky [Bar07] kept the row and column structure, but relaxedthe direction-x dominates assumption by allowing an arbitrary subdivision of the hor-izontal and vertical axis. After appropriately choosing which direction is "dominant",the results resemble that of [GL92]. Diagonal systems assuming only ROSC (1.2.1)and no further restrictions on the translations were studied by Feng–Wang [FW05]and Fraser [Fra12]. Former determined the Lq spectrum of self-affine measures νpand in particular the box dimension of the attractor. In [Fra12] linear isometries whichmap [−1, 1]2 to itself are allowed and the box dimension is determined. Fraser calledthese box-like sets. Observe that in all the mentioned papers the ROSC (1.2.1) wasassumed.

Carpets with overlaps were not studied until the last few years. Fraser and Shmerkin [FS16]shift the columns of Bedford–McMullen carpets to get overlaps, while Pardo-Simón [PS]allows shifts in both directions of Baranski carpets. Relying on a recent breakthroughby Hochman [Hoc14] on the dimension of self-similar measures on the line, both pa-pers show that apart from a small exceptional set of parameters the results in [Bed84;McM84] and [Bar07] remain valid in the overlapping case. This is the type of shiftedcolumns that can be seen in Figure 1.9.

1.2. Self-affine sets 5

In Chapter 2 triangular Gatzouras–Lalley carpets are introduced: the column struc-ture of Gatzouras–Lalley carpets are kept, however instead of diagonal matrices,lower triangular ones are used to define the linear parts of the maps. Hence, theusual rectangular cylinder sets become parallelograms with two vertical sides, seeFigure 1.6. Moreover, we also allow different types of overlaps in the construction, seeFigure 1.9. The results of [GL92] are generalized to this setting. The parallelogramsand overlaps give a much more flexible framework in which existing and many newexamples can be treated in a unified way.

1.2.2 Fractal curves

Fractal curves have already appeared in the 19th century, although at the time the un-usual constructions were considered by many to be mathematical "monsters". Peanocreated the first space-filling curve, Weierstrass presented the first example for a con-tinuous but nowhere differentiable function and von Koch gave a more geometricdefinition of a fractal curve simply referred to today as the von Koch snowflake. Sincethen, fractal curves have found abundant applications in areas such as wavelets, in-terpolation functions or signal processing. The starting point of our work was thecurve introduced and studied by de Rham [Rha47; Rha56; Rha59].

The geometric construction of the curve goes as follows. Starting from the square[0, 1]× [−1, 0], it can be obtained by trisecting each side with ratios ω : (1− 2ω) : ωand "cutting the corners" by connecting each adjacent partitioning point to get anoctagon. Again, each side is divided into three parts with the same ratio and adjacentpartitioning points are connected, and so on. The de Rham curve is the limit curve ofthis procedure. With a more analytic approach, in the language of iterated functionsystems we can say that de Rham’s curve is the attractor Γ of the IFS

f0(x) =[

ω 0ω 1− 2ω

]x−

[0

2ω

]and f1(x) =

[1− 2ω ω

0 ω

]x +

[2ω0

], (1.2.2)

where ω ∈ (0, 1/2) is the parameter. More precisely, the self-affine curve Γ definedby (1.2.2) gives the segment between two midpoints of the original square. Observethat the fixed point of f0 is z0 = [0,−1]T, while for f1 it is z2 = [1, 0]T. Furthermore,f0(z2) = z1 = f1(z0), we say that the cross-condition holds. This ensures that Γ iscontinuous. Figure 1.2 shows both approaches: first four steps of the "corner cutting"on left and the images of f0 and f1 after one (red), two (green) and three (black) levelsof iteration.

FIGURE 1.2: A de Rham curve: geometric construction (left), IFS con-struction (right).


The main goal of Chapter 3 is to analyze the pointwise regularity of such self-affine curves given by a linear parametrization in the much more general frameworkof zipper IFSs as defined in [ATK03], see Definition 3.1.1. In case of de Rham’s curve,a linear parametrization v : [0, 1] 7→ R2 is of the form

v(x) = fi(v(2x− i)), for x ∈[ i

2,

i + 12

), i = 0, 1. (1.2.3)

Figure 1.3 shows a visualisation of a linearly parametrized de Rham curve. Suchlinear parameterizations occur in the study of wavelet functions in a natural way, seefor example Protasov [Pro06], Protasov and Guglielmi [PG15], and Seuret [Seu16].

FIGURE 1.3: Linearly parametrized de Rham curve with parameterω = 1/10. Left: The image of the unit cube w.r.t the IFS generatingthe graph of the de Rham curve. Middle: The second iteration. Right:

The curve itself.

The analysis is done by performing multifractal analysis, i.e. to study the possiblevalues, which occur as (regular) pointwise Hölder exponents, and determine the mag-nitude of the sets (in terms of Hausdorff dimension), where it appears. This propertywas studied for several types of singular functions, for example for wavelets by Barraland Seuret [BS05], Seuret [Seu16], for Weierstrass-type functions by Otani [Ota17], forcomplex analogues of the Takagi function by Jaerisch and Sumi [JS17] or for differentfunctional equations by Coiffard, Melot and Willer [CMT14], by Okamura [Oka16]and by Slimane [BS03] etc. Our results applied to de Rham’s curve give finer resultsthan existing ones in the literature.

1.3 Apollonian networks

Since the seminal work of Erdos and Rényi [ER60], the theory of random graphs hasreceived immense interest in many areas of science. Social networks, the World WideWeb, traffic/shipping routes etc. have made it essential to create models which cap-ture the main features and driving forces of these real-world networks, often also calledcomplex networks. These main features include large clustering coefficient, power lawdecay of degree distribution and small distances between nodes (logarithmic withsize) meaning that networks tend to contain smaller, dense communities; there arefew really large "hubs" and many-many much smaller ones; finally "it’s a small worldafter all".

A particularly successful class of models are the Preferential attachment models [BA99;BR04; Bol+01; FP16]. In these dynamically evolving graphs, the probability that anew node attaches to an existing one is proportional to the degree of the existing one.

1.3. Apollonian networks 7

Thus, nodes with already large degree tend to get new edges more frequently, hencethe term preferential attachment. This simple mechanism gives rise to the power lawdecay. Several variants of the model have been defined, here we focus on the shortestpaths between vertices of Apollonian networks.

The construction of deterministic and random Apollonian networks originatesfrom the problem of Apollonian circle packing: starting with three mutually tangentcircles, we inscribe in the interstice formed by the three initial circles the unique circlethat is tangent to all of them: this fourth circle is known as the inner Soddy-circle. It-eratively, for each new interstice its inner Soddy-circle is drawn, see Figure 1.4. Afterinfinite steps the result is a fractal set, an Apollonian gasket [Boy82; Gra+03].

An Apollonian network (AN) is the resulting graph if we place a vertex in the centerof each circle and connect two vertices if and only if the corresponding circles aretangent, see Figure 1.4. This model was introduced independently by Andrade et al.[And+05] and Doye and Massen [DM05] as a model for real-world networks. Thesenetworks have the important properties mentioned above, in addition, due to theconstruction, Apollonian networks also show hierarchical structure: another propertyvery commonly observed.

FIGURE 1.4: An Apollonian gasket (left), the deterministic networkgenerated by it (right).

It is straightforward to generalize Apollonian packings to Rd for d ≥ 2 withmutually tangent d dimensional hyperspheres. Analogously, if each d-hyperspherecorresponds to a vertex and vertices are connected by an edge if the correspond-ing d-hyperspheres are tangent, then we obtain a d-dimensional AN (see [Zha+08;Zha+06]).

The network arising by this construction is deterministic. Zhou et al. [ZYW05]proposed to randomize the dynamics of the model such that in one step only oneinterstice is picked uniformly at random and filled with a new circle. This construc-tion yields a d dimensional random Apollonian network (RAN) [ZRC06], see Figure 1.5.Using heuristic and rigorous arguments the results in [AM08; CFU14; DS07; Ebr+13;FT12; ZRC06; ZYW05] show that RANs have the above mentioned main features ofreal-world networks.

A different random version of the original Apollonian network was introduced byZhang et al. [ZRZ06], called Evolutionary Apollonian networks (EAN) where in everystep every interstice is picked and filled independently of each other with probabilityq. If an interstice is not filled in a given step, it can be filled in the next step again. Wecall q the occupation parameter. For q = 1 we get back the deterministic AN model. It is


FIGURE 1.5: An array of Random Apollonian Networks in two dimen-sions with n = 50, 250, 750 vertices.

conjectured in [ZRZ06] that an EAN with parameter q, as q→ 0, should show similartopological behaviour to RANs. To make this statement rigorous, instead of lookingat a sequence of evolving EAN-s with decreasing parameters, we slightly modify themodel and investigate the asymptotic behaviour of a single EAN when q might differin each step of the dynamics. That is, we consider a series qn∞

n=1 of occupationparameters so that qn applies for step n of the dynamics, and assume that qn tendsto 0. In this setting, the interesting question is to determine the correct rate for qnthat achieves the observation that EAN shows similar behaviour as RAN when theparameter tends to zero.

The main goal of Chapter 4 is to determine the precise asymptotic growth of theshortest path between two vertices and the diameter of the graph.

1.4 Informal explanation of contribution

The main body of the Thesis is based on three articles [KS18; BKK18; KKV16]. Wenow informally explain the contribution and methods used in each of these papers.

1.4.1 Dimension of triangular Gatzouras–Lalley-type planar carpets

Gatzouras–Lalley (GL) carpets [GL92] are the attractors of self-affine IFSs on the planewhose first level cylinders are aligned into columns using orientation preservingmaps with linear parts given by diagonal matrices. In Chapter 2, we consider a nat-ural generalization of such carpets by replacing the diagonal matrices with lowertriangular ones so that the column structure is preserved, see Definition 2.1.1. Wecall them Triangular Gatzouras–Lalley-type (TGL) planar carpets, indicating that thelinear part of the maps defining the IFS are triangular matrices and it is a naturalgeneralization of the Gatzouras–Lalley construction.

The shaded rectangles and parallelograms in Figure 1.6 show the images of R un-der the maps defining a GL carpet on the left and a TGL carpet on the right. Theseare typical examples which satisfy the ROSC (1.2.1). Furthermore, there is a corre-spondence between the rectangles and parallelograms so that the height and widthof corresponding ones coincide. We call the Gatzouras–Lalley carpet the GL-brother ofthe TGL carpet, see Definition 2.1.4. Even though the ROSC holds, it is not immedi-ate that the dimension of the two attractors should be the same. The parallelogramscan be placed in a way that there is no bi-Lipschitz map between the two attrac-tors. Nevertheless, Baranski essentially shows in [Bar08] that assuming the ROSC the

1.4. Informal explanation of contribution 9

Hausdorff and box dimension of a TGL carpet is equal to the respective dimension ofits GL brother.

FIGURE 1.6: The IFS defining a Gatzouras–Lalley carpet on left andtriangular Gatzouras–Lalley-type carpet on right, which are brothers.

The IFS in Figure 1.6 is an example for which dimH Λ < dimB Λ < dimAff Λ. If theorthogonal projection of Λ to the x-axis is the whole [0, 1] interval, then the box- andaffinity dimensions are equal. Figure 1.7 shows such an example, where the outlinesof fi(R) are shown together with the attractor, which we call the "self-affine smiley".

A special class of examples consists of affine IFSs in which all matrices have thesame main diagonal. We call them diagonally homogeneous carpets. Such a particu-lar TGL carpet (see the left hand side of Figure 1.8) was introduced by Falconer andMiao [FM07, Figure 1 (a)], where the box dimension of the attractor was calculated.Later, Bárány [B15, Subsection 4.3] showed that for this example the box and theHausdorff dimensions are the same. This is due to the fact that all columns have thesame number of maps. The well-known Bedford–McMullen carpets [Bed84; McM84]form a proper subclass of these TGL carpets. In all these examples only the boundaryof the cylinder sets fi(R) could intersect.

However, the main contribution of Chapter 2 is to continue the works [Bar08;GL92] to allow different types of overlaps in the construction. Figure 1.9 illustratesthe overlaps we consider. All three examples are brothers of the GL carpet in Fig-ure 1.6. On the left, the columns are shifted in a way that the IFS on the x-axis gen-erated by the columns, denoted by H, satisfies Hochman’s Exponential SeparationCondition (HESC), see Definition 2.1.9. This type of shifted columns were considered

FIGURE 1.7: The "self-affine smiley", whose dimH = 1.20665 . . . <1.21340 . . . = dimB = dimA, see Subsection 2.7.1.


FIGURE 1.8: The attractor Λa from Subsection 2.7.2 with parametera = 3/10 (left) and Subsection 2.7.3 with parameter a = 3/20 (right),

shown together with the outlines of the images of fi(R).

by Fraser and Shmerkin [FS16] and Pardo-Simón [PS] on different carpets. In the cen-ter, columns do not overlap, however, parallelograms within a column may do so ifa certain transversality like condition holds. The one on the right on Figure 1.9 hasboth types of overlaps.

FIGURE 1.9: Triangular Gatzouras–Lalley-type carpets with differentoverlaps. Left: shifted columns satisfying Hochman’s exponential sep-aration condition. Center: non-overlapping columns, transversality

condition. Right: mixture of both.

By modifying the translation vectors in the example on the left hand side of Fig-ure 1.8, we got a brother with overlaps seen on the right hand side, for which weshow in Subsection 2.7.3 that transversality holds. Another concrete overlapping ex-ample satisfying transversality is "X ≡ X" in Figure 1.10, for which there is strictinequality between the Hausdorff, box and affinity dimensions. If instead the con-struction would be "X = X", then the Hausdorff and box dimensions would be equal.Moreover, if there were no empty columns in this example, then the box and affinitydimensions would coincide.

Results

Section 2.2 contains the formal statements of all the main results. Roughly speaking,we show that for any TGL carpet Λ

dim Λ ≤ dim Λ,


FIGURE 1.10: Example "X ≡ X" from Subsection 2.7.4 for whichdimH Λ = 1.13259 . . . < dimB Λ = 1.13626 . . . < dimA Λ = 1.2170 . . . .

where Λ is the GL brother of Λ and dim means either box or Hausdorff dimension,see Theorems 2.2.1 and 2.2.4. When ROSC holds and the IFS H generated by thecolumns satisfies Hochman’s condition, then equality can be deduced from recentworks [BK17; Fra12]. Also, equivalent conditions are given for the equality of thedifferent dimensions.

Our main contribution is that in the presence of overlaps described above, wegive sufficient conditions under which dim Λ does not drop below dim Λ, see Theo-rems 2.2.2 and 2.2.7. In particular, for the Hausdorff dimension we allow both typesof overlaps simultaneously (like the third figure in Figure 1.9), however for the boxdimension it is either one or the other type (like the first two in Figure 1.9).

For a discussion on generalizing towards orientation reversing maps, see Subsec-tion 2.7.5. In particular, we calculate the dimension of a family of self-affine continu-ous curves Λa, which is generated by an IFS Fa containing a map that reflects on they-axis, see Figure 1.11. They are examples for the type of fractal curves we study froma different perspective in Chapter 3.

The formal treatment of all the mentioned examples is done in Section 2.7.

FIGURE 1.11: Left: first (red), second (green) and third (black) levelcylinders of Fa. Right: Λa rotated 90 degrees from Subsection 2.7.6

with parameter a = 0.2 (black), 0.12 (red) and 0.08 (green).

One motivation to study self-affine fractals of overlapping construction is that


sometimes the dimension of a higher dimensional fractal of non-overlapping con-struction coincides with its lower dimensional orthogonal projection which can be aself-affine fractal of overlapping construction. We obtain such a set in 3D by startingfrom a TGL carpet with overlaps on the xy-plane and then "lift" it to 3D so that the in-teriors of the first level cylinders are disjoint. Figure 1.12 shows such an example withthe first level cylinders (left), the attractor (center) and the projection of the cylindersand attractor to the xy-plane (right). Section 2.8 contains the formal treatment of thistype of construction.

FIGURE 1.12: A three-dimensional fractal whose dimension is equal tothe dimension of its orthogonal projection to the xy-plane.

Methods used to handle overlaps

For each type of overlap and dimension we used different methods:

• The upper bounds on the Hausdorff and box dimensions (after some simple ob-servations) follow from proper adaptations of the results of Gatzouras–Lalley [GL92]and Fraser [Fra12], respectively.

• To estimate the Hausdorff dimension from below we use the Ledrappier-Youngformula of Bárány and Käenmäki [BK17] (cited in Theorem 2.3.4) for self-affinemeasures. We show that this lower bound equals the upper bound

– in case of overlapping like on the second figure of Figure 1.9 by an argu-ment inspired by the transversality method introduced in [BRS];

– in case of overlapping like on the third figure of Figure 1.9, we introduce anew separation condition for the self-similar IFS obtained as the projectionof the TGL carpet under consideration to the horizontal line. This sepa-ration condition is a non-trivial consequence of Hochman’s ExponentialSeparation Condition [Hoc14]. We prove this in Appendix B since it couldbe of separate interest.

• To estimate the box dimension from below we could not simply use the Haus-dorff dimension of the attractor, because, in our case, it is (typically) strictlysmaller than the box dimension. Therefore,

– in case of overlapping like on the first figure of Figure 1.9, we used themethod of Fraser and Shmerkin [FS16]: the main idea is to pass to a spe-cial subsystem of a higher iterate of the IFS which has non-overlappingcolumns;


– in case of overlapping like on the second figure of Figure 1.9, we intro-duced a new argument to count overlapping boxes. It uses transversalityand a result of Lalley [Lal88] based on renewal theory, which gives the pre-cise asymptotics of the number of boxes needed to cover the projection ofthe attractor to the horizontal line.

1.4.2 Pointwise regularity of zipper fractal curves

We postpone the general definition of zipper fractal curves, as given in [ATK03], un-til Chapter 3. For now it is enough to consider a self-affine IFS F = f0, . . . , fN−1in which all matrices have strictly positive entries and map [0, 1]2 into itself. Fur-thermore, the fixed points of f0 and fN−1 are [0, 0]T and [1, 1]T, respectively, andfi([1, 1]T) = fi+1([0, 0]T). Figure 1.13 shows such examples in the plane with thefirst (red), second (green) and third (black) level cylinders visible.

FIGURE 1.13: Affine zippers in the plane defined by matrices withstrictly positive entries.

Let λ = (λ0, . . . , λN−1) be a probability vector, which defines a subdivision of[0, 1]. Let gi be the affine function mapping the unit interval [0, 1] to the ith subintervalof the division (indexing from 0 to N− 1). Then a linear parametrization of the attractorΓ of F is a function v : [0, 1]→ Rd defined by the functional equation

v(x) = fi

(v(g−1

i (x)))

if x ∈ gi([0, 1]).

In comparison, we call F : Rd 7→ R a self-similar function if there exists a boundedopen set U ⊂ Rd, and contracting similarities g1, . . . , gk of Rd such that gi(U) ∩gj(U) = ∅ and gi(U) ⊂ U for every i 6= j, and a smooth function g : Rd 7→ R,and real numbers |ρi| < 1 for i = 1, . . . , k such that

F(x) =k

∑i=1

ρiF(g−1i (x)) + g(x), (1.4.1)

see [Jaf97b, Definition 2.1]. The graph of F over the attractor of g1, . . . , gk can bewritten as the unique, non-empty, compact invariant set of the family of functionsS1, . . . , Sk in Rd+1, where

Si(x, y) = (gi(x), ρiy + g(gi(x))).


The multifractal formalism of the pointwise Hölder exponent of self-similar func-tions was studied in several aspects, see for example Aouidi and Slimane [AS04],Slimane [BS01; BS03; BS12] and Saka [Sak05].

The main difference between the self-similar function F defined in (1.4.1) and vdefined in (3.1.2) is the contraction part. Namely, while F is a real valued functionrescaled by only a real number, the function v is Rd valued and a strict affine trans-formation is acting on it. This makes the study of such functions more difficult.

When studying the pointwise Hölder exponent

α(x) = lim infy→x

log ‖v(x)− v(y)‖log |x− y| ,

we need a good control over the distance ‖v(x) − v(y)‖ as y → x. This is muchsimpler for the self-similar function F, since F was scaled only by a constant. Roughlyspeaking

‖F(gi1,...,in(x))− F(gi1,...,in(y))‖ ≈ |ρi1 · · · ρin |‖F(x)− F(y)‖.

Whereas, in the case of self-affine systems, this is not true anymore. That is,

‖v(gi1,...,in(x))− v(gi1,...,in(y))‖ ≈ ‖Ai1 · · · Ain(v(x)− v(y))‖.

However, in general ‖Ai1 · · · Ain(v(x)− v(y))‖ 6≈ ‖Ai1 · · · Ain‖‖v(x)− v(y)‖. In orderto be able to compare the distance ‖v(gi1,...,in(x))− v(gi1,...,in(y))‖with the norm of theproduct of matrices, we need an extra assumption on the family of matrices. Thisnotion is called dominated splitting, see Definition 3.1.2. The example of matrices withstrictly positive entries satisfies this condition.

Results

The key technical tool is the matrix pressure function P(t) defined in (3.2.1). Undera mild non-degeneracy condition (3.1.7) and dominated splitting we show that α(x)is equal to a constant α = P′(0) for Lebesgue almost all x ∈ [0, 1]. Furthermore, themultifractal formalism holds for the Hausdorff dimension of the level sets E(β) =x ∈ [0, 1] : α(x) = β of a linear parametrization of an affine zipper in a smallinterval of possible β values. In other words dimH E(β) is equal to the Legendre-transform of P(t), see Theorem 3.2.1.

Moreover, an important contribution was to find an additional rather natural as-sumption, see Assumption A (3.2), which is equivalent for the lim inf in the definitionof α(x) to exist as a limit for Lebesgue almost every point, see Theorem 3.2.3. We callthe limit the regular Hölder exponent and denote it αr(x). Under these conditionsTheorem 3.2.2 states that the multifractal formalism for αr(x) can be extended to thefull spectrum of possible β values. For example, de Rham’s curve and matrices withstrictly positive entries satisfy Assumption A, hence these results can be applied toget stronger results than the ones in the literature.

In general, our results do not imply differentiability of the linear parametrizationv, because the lower bound we obtain for α can be strictly smaller than 1. However,in the particular case of de Rham’s curve we show in Proposition 3.6.2 that v is differ-entiable for Lebesgue-almost every x with derivative vector equal to zero and the setwhere it is not differentiable has strictly positive Hausdorff dimension.


Unfortunately, even in the simplest case of positive matrices, there is not muchhope for the computation of the precise values of P(t). Only fast approximation algo-rithms exist, see works of Pollicott and Vytnova [PV15] and Morris [Mor18].

Methods

The most important technical contribution was to generalize the results of Feng [Fen03],and Feng and Lau [FL02] for the pressure function of infinite products of positive ma-trices to the more general dominated splitting setting. This is based on results byBochi and Gourmelon [BG09] characterizing systems with dominated splitting andgeneral ergodic theoretic machinery of Bowen [Bow08].

The proof of the main results use the properties of the pressure function and relyon handling the points which are far away symbolically but close on the self-affinecurve, which is done by a fine study of the underlying dynamical system associatedwith the matrices Ai.

1.4.3 Distances in Random Apollonian Networks

We postpone the precise definition of RANs and EANs in arbitrary dimensions untilChapter 4, instead just focus on RANs in two dimensions.

There is a natural representation of RANs as evolving triangulations in two di-mensions: take a planar embedding of the complete graph on four vertices as in Fig-ure 1.14 and in each step pick a face of the graph uniformly at random, insert a vertexand connect it with the vertices of the chosen triangle (face). The result is a maxi-mal planar graph. Hence, a (RAN2(n))n∈N is equivalent to an increasing family oftriangulations by successive addition of faces in the plane, called stack-triangulations.Stack-triangulations were investigated in [AM08] where the authors also consideredtypical properties under different weighted measures, (e.g. ones picked u.a.r. havingn faces). Under a certain measure stack-triangulations with n faces are an equivalentformulation of RAN2(n), see [AM08] and references therein.

FIGURE 1.14: A RAN2(n) after n = 0, 2, 8 steps

Let u and v be two vertices of a RAN with n vertices. To study distances we denoteby Hop(n, u, v) the hopcount between the vertices, i.e., the number of edges on (oneof) the shortest paths between u and v. The flooding time Flood(n, u) is the maximalhopcount from u, while the diameter Diam(n) is the maximal flooding time (4.1.1).The terminology comes from the intuitive picture that if we were to open a sourceof fluid at u that spreads with unit speed along the edges, then the flooding time isexactly the time when the fluid reaches all other vertices of the graph.


Results

We determine the exact asymptotic growth of Hop, Flood and Diam as the numberof vertices tend to infinity. The results are valid in all dimensions d and all constantsare explicit, only depending on d. All three quantities scale as c log n for differentconstants, which confirms the small-world property.

More precisely, we show that for both RANs and EANs the hopcount obeys a cen-tral limit theorem (CLT), when the two vertices are chosen randomly according to awell-defined probability distribution. The centralizing constant is of order log n whilethe normalizing one is of order

√log n, see Theorems 4.2.1 and 4.2.5. Furthermore,

for RANs the flooding time and diameter divided by log n tend in probability to welldefined constants, see Theorem 4.2.3. The constants that arise have very intuitivemeaning once the structure of these graphs is understood.

Understanding the hierarchical structure using the fractal viewpoint is perhapsthe most important contribution of this work. It gave the opportunity to analyzeall the quantities simultaneously, independently of the dimension in a unified way.The CLT for the hopcount is novel, it is especially interesting, because to the best ofour knowledge in all previous cases, when a CLT was proved, the underlying graphmodel had random edgeweights, while RANs and EANs do not. Determining thediameter is always a greater challenge than the hopcount. The flooding time is veryrarely studied, in particular for RANs it was not known before. The exact relation ofour results compared to existing ones in the literature is surveyed in Subsection 4.2.1.

We remark that the article [KKV16] where these results appeared also contain re-sults about the degree distribution and clustering coefficient, however they are notpresented in this thesis.

Methods

The proofs rely on a few key observations. The first is to realize that RANs have anice tree-like structure: each vertex can be assigned a unique finite code and all edgescan be grouped into two categories, either tree edges or shortcut edges. The tree edgesgo "down" in the hierarchy of the graph, while a shortcut edge goes back "upward"toward the initial graph along a branch of the tree. See Figure 4.3 for an illustrationof the graph in Figure 1.14. All neighbors of a vertex higher in the hierarchy canbe determined simply by looking at its unique code. This allows a combinatorialanalysis and the well-known coupon collector problem comes up naturally. The ideaof using codes can be attributed to my co-author Lajos Vágó, which was then furtherdeveloped.

Secondly, there is a natural embedding of the evolution of a RAN into the evolu-tion of a continuous time branching process (CTBP). This allows for the application ofresults for CTBPs [AN04; BD06; Büh71] to determine the average and maximal depthof the tree created by the tree edges. The CLT for the hopcount can be derived fromhere.

However, for the flooding time and diameter an extra important observation isneeded. Due to the shortcut edges, the deepest branch of the tree created by the treeedges is not necessarily the farthest from the initial graph in graph distance. Therecan be many branches slightly shorter, however farther from the initial graph, becausethe shortcut edges climb to the top slower. Figure 1.15 illustrates this phenomena. Todetermine the exact constants, we achieve the maximal distance by an entropy vsenergy argument.


branch farthest in graph distance

branch with most edges

other branches

shortcut edge

FIGURE 1.15: Impact of shortcut edges on diameter of RAN: branchwith most edges is not the one farthest in graph distance.

The precise proofs use techniques from renewal theory, CTBP theory, large devia-tions, second moment method, symbolic dynamics and combinatorics.

19

Chapter 2

Triangular Gatzouras–Lalley-typeplanar carpets with overlaps

This chapter is based on [KS18] written jointly with Károly Simon. Recall Figure 1.6.

2.1 Triangular Gatzouras–Lalley-type carpets

Denote the closed unit square by R = [0, 1]× [0, 1]. LetA = A1, . . . , AN be a familyof 2 × 2 invertible, strictly contractive, real-valued lower triangular matrices. Thecorresponding self-affine IFS is the collection of affine maps

F = fi(x) := Aix + tiNi=1, where Ai =

(bi 0di ai

)and ti =

(ti,1ti,2

), (2.1.1)

for translation vectors ti, with ti,1, ti,2 ≥ 0. We assume that ai, bi ∈ (0, 1).Orthogonal projection of F to the horizontal x-axis, denoted projx, generates an

important self-similar IFS on the line

H = hi(x) := bix + ti,1Ni=1. (2.1.2)

We denote the attractor of F and H by Λ = ΛF and ΛH respectively.

Definition 2.1.1. We say that an IFS of the form (2.1.1) is triangular Gatzouras–Lalley-type (TGL) and we call its attractor Λ a TGL planar carpet if the following conditions hold:

(a) direction-x dominates, i.e.

0 < ai < bi < 1 for all i ∈ [N] := 1, 2, . . . , N, (2.1.3)

(b) column structure: there exists a partition of [N] into M > 1 sets I1, . . . , IM with cardi-nality |Iı| = Nı > 0 so that

I1 = 1, . . . , N1 and Iı = N1 + . . . + Nı−1 + 1, . . . , N1 + . . . + Nı (2.1.4)

for ı = 2, . . . , M. Assume that for two distinct indices k and ` ∈ 1, . . . , N

if there exists ı ∈1, . . . , M such that k, `∈Iı, then

bk = b` =: rı,tk,1 = t`,1 =: uı.

(2.1.5)

We also introduceH = hı(x) := rıx + uıM

ı=1, (2.1.6)

and we observe that the attractor ΛH ofH is identical with ΛH.

20 Chapter 2. Triangular Gatzouras–Lalley-type planar carpets with overlaps

(c) we assume that ∑j∈Iıaj ≤ 1 holds for every ı ∈ 1, . . . , M and the non-overlapping

column structure

uı + rı ≤ uı+1 for ı = 1, . . . , M− 1 and uM + rM ≤ 1. (2.1.7)

(d) Without loss of generality we always assume in this paper that

(A1) fi(R) ⊂ R for all i ∈ [N] and

(A2) The smallest and the largest fixed points of the functions of H are 0 and 1 respec-tively.

Observe that the definition allows overlaps within columns (like the second figure in Fig-ure 1.9), but columns do not overlap.

We say that Λ is a shifted TGL carpet if we drop the assumption (2.1.7), that is non-overlapping column structure is NOT assumed, we require only that ∑M

ı=1 rı ≤ 1 (like thefirst figure in Figure 1.9).

We often consider the following special cases:

Definition 2.1.2. We say that a shifted TGL carpet Λ has uniform vertical fibres if

∑j∈Iı

as−sHj = 1 for every ı ∈ [M], (2.1.8)

where s = dimB Λ and sH = dimB ΛH.Furthermore, we call Λ a diagonally homogeneous shifted TGL carpet if

bi ≡ b and ai ≡ a for every i ∈ [N].

In particular, a diagonally homogeneous carpet has uniform vertical fibres if N/M ∈ N andNı = N/M for every ı ∈ 1, . . . , M.

The special case when Nı = 1 for all ı = 1, . . . , M is treated in the paper of Bárány,Rams and Simon [BRS, Lemma 3.1]. GL carpets are just special cases of TGL carpets.

Definition 2.1.3. A self-affine IFS F is a Gatzouras–Lalley (GL) IFS and its attractor Λ isa GL carpet if F is a TGL IFS as in Definition 2.1.1 with the additional assumptions thatall off-diagonal elements di = 0 and the rectangular open set condition (ROSC) holds, recallDefinition 1.2.1.

Definition 2.1.4. Let Λ be a shifted TGL carpet generated from the IFS F of the form (2.1.1).We say that the Gatzouras–Lalley IFS

F = fi(x) := Aix + tiNi=1, where Ai =

(bi 00 ai

)and ti =

(ti,1ti,2

),

and its attractor Λ is the GL brother of F and Λ, respectively, if ai = ai and bi = bi forevery i ∈ [N], furthermore, F has the same column structure (2.1.10) as F . If the shiftedTGL carpet Λ is actually a TGL (that is Λ has non-overlapping column structure) then wealso require that ti,1 = ti,1 holds for all i ∈ [N].

There always exists such a brother since we assume Definition 2.1.1 (c) and ∑Mı=1 rı ≤ 1.

Throughout, the GL brother of Λ will always be denoted with the extra tilde Λ.

2.1. Triangular Gatzouras–Lalley-type carpets 21

Some notation

The map fi is indexed by i ∈ [N]. To indicate which column i belongs to in thepartition (2.1.4) we use the function

φ : 1, 2, . . . , N → 1, 2, . . . , M, φ(i) := ı if i ∈ Iı. (2.1.9)

With this notation we can formulate the column structure (2.1.5) as

if φ(k) = φ(`) = ı, then bk = b` =: rı and tk,1 = t`,1 =: uı. (2.1.10)

Throughout, i is an index from [N], while ı with the hat is an index corresponding to acolumn from 1, . . . , M. We use analogous notation for infinite sequences i = i1i2 . . .and ı = ı1 ı2 . . ., see Subsection 2.3.1 for details.

For compositions of maps we use the standard notation fi1...in := fi1 fi2 . . . fin ,where ij ∈ 1, . . . , N. Similarly, for products of matrices we write

Ai1 ...in := Ai1 · . . . · Ain :=(

bi1...in 0di1 ...in ai1...in

).

Immediate calculations give bi1...in = bi1 · . . . · bin , ai1 ...in = ai1 · . . . · ain and

di1 ...in =n

∑`=1

di` ·∏k<`

aik ·n

∏r=`+1

bir , (2.1.11)

where by definition ∏k<1

aik := 1 andn∏

r=n+1bir := 1. The image Ri1...in := fi1...in(R) is a

parallelogram with two vertical sides, see Figure 2.1. We refer to bi1 ...in as the width,ai1...in as the height and γi1...in as the angle of the longer side of the parallelogram Ri1...in ,in other words

tan γi1 ...in :=di1 ...in

bi1 ...in

. (2.1.12)

bi1...in

a i1.

..in

|di 1

...i n|

R i1...in

γi1...in

FIGURE 2.1: The skewness of Ri1 ...in := fi1 ...in([0, 1]2)

Since direction-x dominates, Ri1 ...in is extremely long and thin for large n. A simpleargument gives that | tan γi1...in | remains uniformly bounded away from +∞.


Lemma 2.1.5. There exists a non-negative constant K0 < ∞ such that for every n and everyfinite length word i1 . . . in ∣∣∣∣

di1...in

bi1 ...in

∣∣∣∣ ≤ K0.

Proof. Since direction-x dominates, maxiai/bi < 1, hence using (2.1.11)

∣∣∣∣di1...in

bi1 ...in

∣∣∣∣ ≤|di1 |bi1

+n

∑k=2

|dik |bik

k−1

∏j=1

aij

bij

≤ maxi|di|/bi1−maxiai/bi

< ∞.

2.1.1 Results of Gatzouras and Lalley

A standard technique to give a lower bound for the Hausdorff dimension of the at-tractor Λ =

⋃i∈[N] fi(Λ) is to study self-affine measures νp, i.e. compactly supported

measures with support Λ satisfying

νp =N

∑i=1

piνp f−1i ,

for some probability vector p = (p1, . . . , pN). Let P be the set of all probability distri-butions on the set [N] and P0 be the subset when all pi > 0. By definition

supp∈P

dimH νp ≤ dimH Λ.

Gatzouras and Lalley proved that there always exists a p∗ for which the supremumis attained, furthermore p∗ ∈ P0. Let

α∗ := dimH νp∗ = supp∈P

dimH νp.

They explicitly calculated

dimH νp =∑N

i=1 pi log pi

∑Ni=1 pi log ai

+

(1− ∑N

i=1 pi log bi

∑Ni=1 pi log ai

)∑M

ı=1 qı log qı

∑Ni=1 pi log bi

, (2.1.13)

where qı = ∑j∈Iıpj. This formula is a special case of the Ledrappier–Young formula,

see Subsection 2.3.3 for details and references. For Bedford–McMullen carpets theoptimal p∗ can be given by routine use of the Lagrange multipliers method. Themain result of [GL92] is that for a GL carpet the α∗ bound is sharp, i.e.

α∗ = dimH Λ.

In [GL92] Gatzouras and Lalley also gave an implicit formula to calculate the boxdimension of their carpet. Let sx be the unique real such that ∑M

ı=1 rsxı = 1 (rı was

defined in (2.1.5)). Then dimB Λ = s is the unique real such that

N

∑i=1

bsxi as−sx

i = 1.


Again, equality of dimH Λ and dimB Λ is highly atypical. It holds if and only if the α∗-dimensional Hausdorff measure of Λ, denoted Hα∗(Λ), is positive and finite, whichis equivalent to the condition

∑j∈Iı

aα∗−sxj = 1, for every ı = 1, . . . , M.

For Bedford–McMullen carpets Peres showed in [Per94] that Hα∗(Λ) = ∞ whendimH Λ < dimB Λ.

2.1.2 Separation conditions

In our main results, we assume different extents of separation for the parallelogramsfi(R), recall Figures 1.6 and 1.9. This will be considered in Subsection 2.1.2. In Sub-section 2.1.2 we consider separation conditions for H which are actually conditionsabout the extent of separation of the column structure.

Separation of the cylinder parallelograms

Definition 2.1.6 (Separation conditions for a shifted TGL IFS F ). We say that

• F satisfies the rectangular open set condition (ROSC): recall Definition 1.2.1.

• each column independently satisfies the ROSC if for every ı ∈ [M] andk, ` ∈ Iı we have fk(U) ∩ f`(U) = ∅. In other words, if the interior of two first levelcylinders intersects, then they are from different columns.

• F satisfies the transversality condition if there exists a K1 > 0 such that forevery n and words (i1 . . . in), (j1 . . . jn) ∈ 1, . . . , Nn with φ(ik) = φ(jk) for k =1, . . . , n and i1 6= j1 (φ was defined in (2.1.9)), we have

|projx(int(Ri1 ...in) ∩ int(Rj1 ...jn))| < K1 ·maxai1 · . . . · ain , aj1 · . . . · ajn. (2.1.14)

Given two finite words i1 . . . in and j1 . . . jn, i1 6= j1, the angle of the two corre-sponding parallelograms Ri1 ...in and Rj1 ...jn can be defined as the angle between theirnon-vertical sides. The transversality condition ensures that any such pair of paral-lelograms in the same column have either disjoint interior or have an angle uniformlyseparated from zero.

Observe that this definition of transversality coincides in the diagonally homoge-neous case with the one in [BRS]. In [BRS, Section 1.5] a sufficient condition for thetransversality condition was given. Namely, the authors introduced a self-affine IFSS in R3 which is (in our setup)

S :=

Si(x, z) := ( fi(x), Ti(z))N

i=1, (x, z) ∈ [0, 1]2 ×R,

where fiNi=1 was defined in (2.1.1) and Ti : R→ R is given by

T :=

Ti(z) :=ai

bi· z + bi

di

N

i=1.

The relevance of the IFS T is that


ξ

(1, 0)

`

A i(`)

z

Ti(z) Ti

Ti(z) :=aibi· z + di

bi

FIGURE 2.2: The IFS T , where z and (1, z) are identified

tan γi1...in = Ti1...in(0). (2.1.15)

Indeed, from the definition (2.1.12) of tan γi1 ...in and formula (2.1.11) it immediatelyfollows that

tan γi1 ...in =di1bi1

+n

∑`=2

di`bi`·`−1

∏k=1

aik

bik

= Ti1...in(0).

Using the same argument as in the proof of [BRS, Lemma 1.2] we obtain that

Lemma 2.1.7. If S satisfies the Strong Separation Property (that is Si(Λ) ∩ Sj(Λ) = ∅ ifi 6= j and Λ is the attractor of the IFS S) then the transversality condition holds.

The next lemma gives a different, easy-to-check sufficient condition for transver-sality.

Lemma 2.1.8. Let P :=(k, `) : k, ` ∈ I , k 6= `, Rk ∩ R` 6= ∅

, where A denotes

the interior of a set A. Moreover, we introduce

sk :=dk

bk, rk :=

ak

bk, r∗ := max

1≤k≤Nrk, bmin := min

1≤k≤Nbk and s∗ := min

1≤ ≤MP 6=∅

min(k,`)∈P

|sk − s`|.

Assume thats∗ > 2

1bmin

· r∗

1− r∗or equivalently

s∗bmin

2 + s∗bmin> r∗.

Then the transversality condition holds.In particular, in the diagonally homogeneous case transversality holds if

ab<

d∗2 + d∗

, (2.1.16)

where d∗ := min1≤ ≤MP 6=∅

min(k,`)∈P

|dk − d`|.

Proof. Using that Rk ⊂ [0, 1]2 we obtain that |dk| < 1. Hence

|sk| ≤1

bmin. (2.1.17)


For an m ∈ 1, . . . , M let Σm := j ∈ Σ, j1 ∈ Im. The transversality condition holdsif there exists c > 0 such that for every n, for all m ≤ M with Pm 6= ∅ and

for all i, j ∈ Σ with (i1, j1) ∈ Pm, we have:∣∣∣γi|n − γj|n

∣∣∣ > c. (2.1.18)

It follows from (2.1.15) that (2.1.18) holds whenever for all such pair of i, j and for alln

|si1 − sj1 | −n

∑`=2

(si` ·

`−1

∏k=1

rik − sj` ·`−1

∏k=1

rjk

)

is greater than the same positive constant uniformly. However by (2.1.17) this holdsif

s∗ > 21

bmin· r∗

1− r∗.

Separation of the columns

We will also need some separation conditions for the column structure which arerepresented by separation properties ofH, recall (2.1.6).

The symbolic spaces for F andH are

Σ := 1, . . . , NN and ΣH := 1, . . . , MN .

The natural projections form Σ → Λ and ΣH → ΛH are Π and ΠH respectively,see Subsection 2.3.1 for details. Whenever we are given a probability vector p on1, . . . , N , we always associate to it another probability vector q on 1, . . . , M suchthat

qı := ∑j∈Iı

pj. (2.1.19)

Slightly abusing the notation we write P0 for both the set of the probability vectorsof positive components on 1, . . . , N and 1, . . . , M. The Bernoulli measure pN onΣ is denoted µp and its push forward is νp = Π∗µp = µp Π−1. Analogously for µqand νq.

Definition 2.1.9 (Separation conditions forH). We say thatH satisfies

• Hochman’s Exponential Separation Condition (HESC) (see [Hoc14, p. 775]) ifthere exist an ε > 0 and nk ↑ ∞ such that for

∆n := minı,∈1...Mn

ı 6=

|hı(0)− h(0)|, if h′ı(0) = h′(0);∞, otherwise.

we have ∆nk > e−ε·nk . Here h′ denotes the derivative of the function h.

• Weak Almost Uniqe Coding (WAUC) if for all Bernoulli measures µq there existsBH ⊂ ΣH (may depend on q) for which

µq(BH) = 0 and for every ı ∈ ΣH \ BH : #(Π−1H ΠH(ı) \ BH) = 1.

Almost Uniqe Coding (AUC) holds if for every Bernoulli measure µq and for µq-a.e.ı ∈ ΣH : #Π−1

H ΠH(ı) = 1.


• No Dimension Drop (NDD) if for all push forward measures νq = (ΠH)∗µq

dimH νq =−∑M

ı=1 qı log qı

−∑Mı=1 qı log rı

.

The following implications hold between these conditions

HESC =⇒ NDD⇐⇒WAUC. (2.1.20)

HESC =⇒ NDD follows from Hochman’s work [Hoc14, Theorem 1.1]. AUC im-plies NDD from Feng–Hu [FH09, Theorem 2.8 and Corollary 4.16], but we do notknow if the reverse direction NDD =⇒ AUC holds or not. Feng informed us [Fen19,Corollary 4.7] that he can prove the equivalence NDD ⇐⇒ WAUC for ergodic mea-sures. This result just appeared on the arXiv. However, we use it only for Bernoullimeasures. For completeness, we give our own complete (much simpler) proof ofNDD⇐⇒WAUC for Bernoulli measures in Appendix B.

The set U of translations (u1, . . . , uM) defining H for which HESC does not holdis small. It is stated in [PS, Proposition 2.7] that it essentially follows from [Hoc15,Theorem 1.10] that the Hausdorff and packing dimension of U is M− 1, in particularU has 0 M-dimensional Lebesgue measure. Moreover, [Hoc14, Theorem 1.5] statesthat if the parameters (r1, . . . , rM, u1, . . . , uM) definingH are all algebraic, then HESCdoes not hold if and only if there is an exact overlap, i.e. ∆n = 0 for some n.

2.2 Main results

We now state our main results for the Hausdorff dimension of shifted TGL carpets inSubsection 2.2.1, the box dimension in Subsection 2.2.2 and discuss diagonally homo-geneous carpets in Subsection 2.2.3. For a discussion on generalizing towards nega-tive entries in the main diagonal, see Subsection 2.7.5.

2.2.1 Hausdorff dimension

For any vector c = (c1, . . . , cK) with strictly positive entries and a probability vectorp = (p1, . . . , pK) we write

〈c〉p :=K

∏i=1

cpii .

When no confusion is made, we suppress p and write 〈c〉 = 〈c〉p. Throughout, weuse this notation for the vectors a = (a1, . . . , aN), b = (b1, . . . , bN), p = (p1, . . . , pN),N = (N1, . . . , NM) and q = (q1, . . . , qM), where q is derived from p via (2.1.19). Usingthis notation let us denote the function on the right-hand side of (2.1.13) by

D(p) :=log〈p〉plog〈a〉p

+

(1− log〈b〉p

log〈a〉p

)log〈q〉qlog〈b〉p

=log〈q〉log〈b〉 +

log〈p〉 − log〈q〉log〈a〉 . (2.2.1)

Theorem 2.2.1 (Upper bound). Regardless of overlaps, for any shifted triangular Gatzouras–Lalley-type planar carpet Λ

dimH Λ ≤ supp∈P

D(p) =: α∗.

Furthermore, there always exists a p∗ ∈ P0 for which D(p∗) = α∗.

2.2. Main results 27

The proof is given in Section 2.4. Throughout, let q∗ denote the vector q∗ı =

∑j∈Iıp∗j . The next theorem states sufficient conditions under which the Hausdorff

dimension of a self-affine measure νp on Λ is equal to D(p).

Theorem 2.2.2. Let p ∈ P0, µp := pN and νp := Π∗µp. For a shifted triangularGatzouras–Lalley-type planar carpet Λ we have

dimH νp = D(p)

if the horizontal IFS H satisfies Hochman’s Exponential Separation Condition (in particular,always holds for non-overlapping columns) and

(i) either each column independently satisfies the ROSC or

(ii) Λ satisfies transversality (see Definition 2.1.6) and the following inequality holds:

log〈a〉plog〈b〉p

> 1 +log〈N〉q− log〈q〉q

. (2.2.2)

We remark that Proposition 2.2.10 provides a simple way to check condition (2.2.2)in the diagonally homogeneous case. Section 2.5 is devoted to the proof of this theo-rem. As an immediate corollary, we get

Corollary 2.2.3 (Sufficient conditions). Whenever a shifted TGL carpet Λ satisfies the con-ditions of Theorem 2.2.2 with replacing p and q in (2.2.2) by p∗ and q∗, then

dimH Λ = α∗.

2.2.2 Box dimension

Recall the IFSs H (2.1.2) andH (2.1.6) obtained by projecting F to the x-axis. Recall sxwas defined so that ∑M

ı=1 rsxı = 1 and let sx be the unique real such that ∑N

i=1 bsxi = 1.

Furthermore, introducesH := dimB ΛH = dimB ΛH.

Since ΛH is a self-similar set, sH is well defined. If Λ is a TGL carpet then sH = sx,otherwise sH ≤ sx. The affinity dimension dimAff of Λ can be deduced from theresult of Falconer–Miao [FM07, Corollary 2.6] together with the description in [BRS,Subsection 1.3] and the fact that direction-x dominates: dimAff Λ = sA is the uniquereal such that

N

∑i=1

bminsx ,1i asA−minsx ,1

i = 1. (2.2.3)

In particular, if sx < 1 then sA = sx, otherwise sA solves ∑Ni=1 bia

sA−1i = 1. So sA only

depends on the main diagonals (bi, ai), but not on the off-diagonal elements di. So,the affinity dimension of a shifted TGL carpet Λ is and its GL brother coincide.

The following theorem gives an upper bound for dimB Λ, which can be strictlysmaller than sA. It was proved for diagonal iterated function systems by Feng–Wangin [FW05, Corollary 1] and also follows from Fraser’s work [Fra12, Theorem 2.4,Corollary 2.7]. Here we extend its scope to triangular IFSs. In a different context,Hu [Hu98] studied a related problem, where a version of Bowen’s formula deter-mines the box dimension.


Theorem 2.2.4 (Upper bound). Regardless of overlaps, for any shifted triangular Gatzouras–Lalley-type planar carpet Λ

dimP Λ = dimBΛ ≤ s ≤ sA,

where s is the unique solution of the equation

N

∑i=1

bsHi as−sH

i = 1. (2.2.4)

In particular, if Λ satisfies the ROSC, then dimP Λ = dimB Λ = s.

Corollary 2.2.5 (Equality of box- and affinity dimension). For any shifted TGL carpet

s = sA ⇐⇒ sH = minsx, 1.

Proof. Follows immediately from comparing equations (2.2.3) and (2.2.4) defining sAand s, respectively, together with the fact that ai < 1 and bi/ai > 1 for every i =1. . . . , N.

Remark 2.2.6.

a) The proof of Fraser [Fra12] does not make use of any column structure (2.1.10). Hence,Theorem 2.2.4 immediately extends to an IFS F of the form (2.1.1) as long as direction-xdominates (0 < ai < bi < 1) and the ROSC holds.

b) Since Λ is compact and every open set intersecting Λ contains a bi-Lipschitz image of Λ,we get that dimP Λ = dimBΛ, see [Fal90, Corollary 3.9].

Handling overlaps to calculate the box dimension is a greater challenge, sincetypically dimH Λ < dimB Λ and thus the usual technique of giving a lower boundby bounding the Hausdorff dimension from below does not suffice. Hence, a newcounting argument was necessary.

Theorem 2.2.7 (Box dimension with overlaps). For a shifted TGL carpet Λ we havedimBΛ ≥ s, hence dimB Λ = s, if either of the following hold:

(i) H satisfies HESC and each column independently satisfies the ROSC or

(ii) Λ is a TGL carpet, satisfies transversality and the following inequality:

− log〈p〉p + log〈q〉q < sH(log〈b〉p − log〈a〉p), (2.2.5)

where p := ( p1, . . . , pN) and q := (q1, . . . , qM) are defined by equation (2.2.4):

pi = bsHi as−sH

i and qı = ∑j∈Iı

bsHj as−sH

j . (2.2.6)

The analogue of the following sufficient and necessary condition for the equalityof the box- and Hausdorff dimensions was proved in [GL92, Theorem 4.6].

Theorem 2.2.8 (Equality of box- and Hausdorff dimension). Assume the shifted TGLcarpet Λ satisfies ROSC and H satisfies No Dimension Drop. Then the following three con-ditions are equivalent,

dimH Λ = dimB Λ ⇐⇒ sH = dimH νq ⇐⇒ ∑j∈Iı

as−sHj = 1 for every ı ∈ [M]. (2.2.7)

All results for box dimension are proved in Section 2.6.


2.2.3 Diagonally homogeneous carpets

We show how the conditions and formulas of our main results simplify in the diag-onally homogeneous case. Recall the easy-to-check sufficient condition (2.1.16) fortransversality in Lemma 2.1.8. Moreover, observe that the vector p becomes the uni-form vector pi = 1/N and thus qı = Nı/N. A routine use of the Lagrange multipliersmethod gives the optimal p∗

p∗k = Nlog blog a−1

ı ·( M

∑=1

Nlog blog a

)−1if k ∈ Iı. (2.2.8)

Thus, conditions (2.2.2) and (2.2.5) become

log〈p∗〉p∗log〈q∗〉q∗

<log alog b

andlog Nlog M

+ 1 +log〈q〉qlog M

<log alog b

, (2.2.9)

respectively. If in addition, the system has uniform vertical fibres, then pi = p∗i =1/N also qı = q∗ı = 1/M. Hence, both conditions (2.2.2) and (2.2.5) become

log Nlog M

<log alog b

. (2.2.10)

Next, we give an equivalent explicit formulation of condition (2.2.2). Let ϕ(y) :=y log y and for x ∈ (0, 1) define

R(x) := x + (r(x)− 1)−1 , where r(x) =ϕ(

∑Mı=1 Nx

ı)

∑M=1 ϕ(Nx

).

Lemma 2.2.9. R(x) is a continuous, strictly monotone increasing function.

Proof. Continuity is obvious. It is enough to show that r(x) is strictly monotone de-creasing. Let r′ denote the derivative. Then

x ·( M

∑=1

ϕ(Nx ))2

︸︷︷︸>0

·r′(x) =

=:A︷︸︸︷( M

∑=1

ϕ(Nx ))2

+

=:B︷︸︸︷

log( M

∑ı=1

Nxı

)·( M

∑=1

ϕ(Nx ))2

− ϕ( M

∑ı=1

Nxı

)·

M

∑=1

ϕ(Nx )

︸︷︷︸=:C

− ϕ( M

∑ı=1

Nxı

) M

∑=1

ϕ(Nx ) · log Nx

︸︷︷︸=:D

.

We claim that C > A and D ≥ B, which will conclude the proof of the lemma. Forbrevity, write yı := Nx

ı . yı = 1⇔ Nı = 1, otherwise yı > 1.To show that C > A, it is enough to prove that for 1 ≤ u ≤ v

ϕ(u) + ϕ(v) < ϕ(u + v). (2.2.11)


Then a simple induction implies that ∑ ϕ(yı) < ϕ(∑ yı). Recall ϕ(1) = 0. The meanvalue theorem implies that

ϕ(u + v)− ϕ(v) = u · ϕ′(ξ), for some ξ ∈ (v, u + v)ϕ(u)− ϕ(1) = (u− 1) · ϕ′(ζ), for some ζ ∈ (1, u).

Since the derivative ϕ′(y) = 1+ log y is strictly increasing and ζ < ξ, we have ϕ′(ζ) <ϕ′(ξ). This implies (2.2.11). To prove the other inequality

D− B =( M

∑ı=1

yı

)· log

( M

∑ı=1

yı

) M

∑=1

ϕ(y ) · log y − log( M

∑ı=1

yı

)·( M

∑=1

ϕ(y ))2

.

We can pull out log(

∑ yı)> 0 and divide by it. This gives

D− Blog(

∑ yı) =

( M

∑ı=1

yı

) M

∑=1

ϕ(y ) ·y log y

y −

M

∑=1

ϕ2(y )− 2 ∑ı<

ϕ(yı)ϕ(y )

=M

∑=1

∑Mı=1 yı

y · ϕ2(y )−

M

∑=1

ϕ2(y )− 2 ∑ı<

ϕ(yı)ϕ(y )

=M

∑=1

∑i 6=j yı

y · ϕ2(y )− 2 ∑

ı<

ϕ(yı)ϕ(y )

= ∑ı<

(yı

y · ϕ2(y ) +

y

yı· ϕ2(yı)− 2ϕ(yı)ϕ(y )

)

= ∑ı<

(√yı

y ϕ(y )−

√y

yıϕ(yı)

)2≥ 0.

Proposition 2.2.10. The solution of the equation R(x) = 1 is unique. Let x0 denote thissolution. Then in the diagonally homogeneous case

(2.2.2) holds ⇐⇒ log blog a

< x0.

Remark 2.2.11. Observe that all the conditions for transversality, (2.2.2), (2.2.5) are satisfiedif the heights of the parallelograms Ri are "small enough" compared to their width. See theexamples with overlaps in Section 2.7 for some explicit calculations.

Proof of Proposition 2.2.10. Let x := log b/ log a < 1. In the diagonally homogeneouscase (2.2.2) simplifies to

log alog b

=1x> 1 +

∑Mı=1 q∗ı log Nı

−∑Mı=1 q∗ı log q∗ı

,

where q∗ı = Nxı / ∑ Nx

. Multiplying each side by x we get

1 > x +∑M

ı=1 q∗ı log Nxı

−∑Mı=1 q∗ı log q∗ı

. (2.2.12)

2.3. Preliminaries 31

It is straightforward to check that for any y1, . . . , yM ∈ R and qı := eyı / ∑ ey

−M

∑ı=1

qı log qı +M

∑ı=1

qı · yı = logM

∑ı=1

eyı .

Applying this with yı = log Nxı (then qı = q∗ı ) in the denominator of (2.2.12) we get

that (2.2.2) is equivalent to

1 > x +∑M

ı=1 q∗ı log Nxı

log ∑Mı=1 Nx

ı −∑Mı=1 q∗ı log Nx

ı= R(x).

For x small enough (2.2.2) holds, since 1/x tends to infinity while the right hand sideremains finite. On the other hand for x = 1 it does not hold. Hence, R(x) < 1 forsmall enough x, while R(1) ≥ 1. Thus, Lemma 2.2.9 implies that there exists a uniquex0 ∈ (0, 1) such that R(x0) = 1. So any x < x0 satisfies (2.2.2).

Finally, in the diagonally homogeneous case, the dimension formulas agree withthe ones for Bedford–McMullen carpets.

Corollary 2.2.12. If a diagonally homogeneous shifted TGL carpet Λ satisfies the conditionsof Theorems 2.2.2 and 2.2.7, then

dimH Λ =1

− log blog

M

∑=1

Nlog blog a and dimB Λ =

log N− log a

+

(1− log b

log a

)log M− log b

.

In particular, dimH Λ = dimB Λ if and only if Λ has uniform vertical fibres.

Proof. For diagonally homogeneous shifted TGL carpets the expression (2.2.1) forD(p) simplifies to

D(p) =log〈p〉p

log a+

(1− log b

log a

)log〈q〉q

log b.

Applying this for p∗ from (2.2.8) gives the result dimH Λ = D(p∗).The equation for the box dimension s = dimB Λ, recall (2.2.4), simplifies to

N · bsH · as−sH = 1. (2.2.13)

SinceH has No Dimension Drop (recall Definition 2.1.9), we have sH = log M/(− log b).Substituting this back into (2.2.13) and expressing s from the equation gives the de-sired formula for dimB Λ.

Comparing the formula for dimB Λ with the one for D(p), we immediately getthat equality holds if and only if Nı = N/M for every ı ∈ 1, . . . , M.

2.3 Preliminaries

In this section, we collect important notation, definitions, preliminary lemmas andcite results used in the proofs of the subsequent sections.

2.3.1 Symbolic notation

Throughout, we work simultaneously with the IFSs F , H andH, which were definedin (2.1.1), (2.1.2) and (2.1.6) respectively. Their attractors are Λ, ΛH = ΛH respectively.


We define the symbolic spaces

Σ = 1, 2, . . . , NN and ΣH = 1, 2, . . . , MN

with elements i = i1i2 . . . ∈ Σ and ı = ı1 ı2 . . . ∈ ΣH. The function φ : 1, 2, . . . , N →1, 2, . . . , M, recall (2.1.9), naturally defines the map Φ : Σ→ ΣH

Φ(i) := ı = φ(i1)φ(i2) . . . . (2.3.1)

Finite words of length n are either denoted with a ’bar’ like ı = i1 . . . in ∈ Σn or asa truncation i|n = i1 . . . in of an infinite word i, the length is denoted | · |. The set ofall finite length words is denoted by Σ∗ =

⋃n Σn and analogously Σ∗H. The left shift

operator on Σ and ΣH is σ, i.e. σ(i) = i2i3 . . . and σ(ı) = ı2 ı3 . . ..The longest common prefix of i and j is denoted i ∧ j, i.e. its length is |i ∧ j| =

mink : ik 6= jk − 1. This is also valid if one of them has or both have finite length.The nth level cylinder set of i ∈ Σ is [i|n] := j ∈ Σ : |i∧ j| ≥ n. Similarly for ı ∈ Σnand ı ∈ ΣH. Recall that R = [0, 1]2. We use the standard notation Ai|n = Ai1 · . . . · Ain

and fi|n = fi1 fi2 . . . fin to write

Λi|n := fi|n(Λ) and Ri|n := fi|n(R)

for the nth level cylinder corresponding to i. The sets Ri|n∞n=1 form a nested se-

quence of compact sets with diameter tending to zero, hence their intersection is aunique point x ∈ Λ. This defines the natural projection Π : Σ→ Λ

Π(i) := limn→∞

∞⋂

n=1

Ri|n = limn→∞

fi|n(0) = ti1 +∞

∑n=2

Ai|n−1 · tin . (2.3.2)

The natural projections generated by H andH are

ΠH(i) := limn→∞

hi|n(0), i ∈ Σ; and ΠH(ı) := limn→∞

hı|n(0), ı ∈ ΣH.

The following commutative diagram summarizes these notations:

Σ Φ //

Π

ΠH

ΣH

ΠH

Λprojx// ΛH

(2.3.3)

We also introduce the measurable partitions α and β of Σ whose classes containing ani ∈ Σ are defined

α(i) := Π−1Π(i) and β(i) := Φ−1Φ(i). (2.3.4)

The fact that these partitions are measurable are immediate consequences of the def-inition of measurability of a partition. Alternatively, this also follows from [Sim12,Theorem 2.2]. Thus, α(i) contains those j ∈ Σ which get mapped to the same pointon the attractor, Π(i) = Π(j) ∈ Λ, and β(i) corresponds to the ’symbolic column’ ofi, i.e. for j ∈ β(i) we have ΠH(i) = ΠH(j). These partitions play an important rolewhen handling overlaps.

2.3. Preliminaries 33

Bernoulli measures on Σ are key in obtaining the lower bound for dimH Λ. Recallthe set

P := p = (p1, . . . , pN) : pi ≥ 0,N

∑i=1

pi = 1

of all probability distributions on the set 1, 2, . . . , N and let P0 denote the subsetwhen all pi > 0. The Bernoulli measure on Σ corresponding to p ∈ P is the productmeasure µp = pN, i.e. the measure of a cylinder set is µp([i|n]) = pi1 · . . . · pin . AllBernoulli measures can be uniquely disintegrated according to the family of condi-tional measures µp,α(i) = µα(i) generated by the measurable partition α. That is for allBorel sets U ⊂ Σ

µp(U) =∫

µα(i)(U)dµp(i). (2.3.5)

The entropy of a Bernoulli measure µp is

hµp = −N

∑i=1

pi log pi = − log〈p〉p. (2.3.6)

The push forward νp := Π∗µp is the self-affine measure on Λ defined by νp = µp Π−1 or equivalently

νp =N

∑i=1

piνp f−1i .

Recall that a p ∈ P defines another distribution q = (q1, . . . , qM) via (2.1.19). Thenµq = qN is a Bernoulli measure on ΣH. Moreover, the self-similar measure on ΛH isνq = (ΠH)∗µp = (projx)∗νp. Our convention is that µ always denotes a measure on(some) symbolic space, while ν is supported on (a part of) R.

2.3.2 Atypical parallelograms

The exponential rate of growth of the size of nth level parallelograms, the numberof parallelograms in a column and the column’s measure can vary a lot for differenti ∈ Σ. However, in measure-theoretic sense those i which behave atypically form asmall set. Define the function

X : Σ→ R+, X(i) := ci1 ,

where c = (c1, . . . , cN) is an arbitrary vector with strictly positive elements. Let

Xn(i) :=n−1

∏j=0

X(σji) =n

∏j=1

cij .

In particular, if c = a := (a1, . . . , aN) or b := (b1, . . . , bN), then Xn(i) is the height andwidth of the parallelogram Ri|n. If c = N := (Nφ(1), . . . , Nφ(N)) or q := (qφ(1), . . . , qφ(N)),then Xn(i) gives the number of parallelograms in and the measure of the columnΦ(i)|n.

Fix an arbitrary p ∈ P . Recall the notation 〈c〉p := ∏Nj=1 c

pjj . When no confusion is

made, we suppress p and write 〈c〉 = 〈c〉p. In the rest of the subsection δ > 0 is fixed.Define

Badδ,n(c) :=

i ∈ Σ : Xn(i) < 〈c〉(1−δ)n or Xn(i) > 〈c〉(1+δ)n, if 〈c〉 > 1,i ∈ Σ : Xn(i) < 〈c〉(1+δ)n or Xn(i) > 〈c〉(1−δ)n, if 〈c〉 < 1.

(2.3.7)


The definition can be extended to a positive real t, by setting Badδ,t(c) := Badδ,btc(c).Let µp be the Bernoulli measure on Σ defined by p ∈ P .

Lemma 2.3.1. IF 〈c〉p 6= 1 then there exists a constant C and an r ∈ (0, 1) such that

µp(Badδ,n(c)) < C · rn for every n ≥ 1.

Hence, the Borel-Cantelli lemma immediately implies that

µp(i ∈ Badδ,n(c) for infinitely many n) = 0.

Proof. Assume 〈c〉 > 1. Let Sn(X) := 1n ∑n−1

j=0 log X(σji). Then

µp(Xn(i) < 〈c〉(1−δ)n) = µp(Sn(X) < (1− δ) log〈c〉).

The log X(σji)j are independent and identically distributed with expectation

E(log X) =N

∑j=1

pj log cj = log〈c〉.

Hence, Cramér’s large deviation theorem [DZ10, Theorem 2.1.24.] implies that µp(Xn(i) <〈c〉(1−δ)n) decays exponentially fast in n. The argument for Xn(i) > 〈c〉(1+δ)n is ex-actly the same. The proof is analogous when 〈c〉 < 1.

2.3.3 Ledrappier–Young formula

Let 0 < α2(A) ≤ α1(A) < 1 denote the two singular values of a 2× 2 contractive,non-singular matrix A. Namely, αi(A) is the positive square root of the ith largesteigenvalue of AT A, where AT is the transpose of A. The geometric interpretation ofthe singular values is that the linear map x 7→ Ax maps the unit disk to an ellipsewith principal semi-axes of length α2(A) and α1(A). The singular values can also beexpressed with the matrix norm: α1(A) = ‖A‖ and α2(A) = ‖A−1‖−1. For a familyof matrices A = A1, . . . , AN, the asymptotic exponential growth rate of the semi-axes of the ellipses determined by the maps x 7→ Ai1...in x is given by the Oseledetstheorem.

Theorem 2.3.2 (Oseledets [Ose68]). LetA = A1, . . . , AN be a set of non-singular 2× 2matrices with ‖Ai‖ < 1 for i ∈ 1, . . . , N. Then for any ergodic σ-invariant measure µ onΣ there exist constants 0 < χ1

µ ≤ χ2µ such that for µ-almost every i

limn→∞

1n

log α1(Ai1 ...in) = limn→∞

1n

log ‖Ai1 ...in‖ = −χ1µ,

limn→∞

1n

log α2(Ai1 ...in) = limn→∞

1n

log ‖(Ai1 ...in)−1‖−1 = −χ2

µ.

The numbers χ1µ and χ2

µ are called the Lyapunov-exponents of ν = Π∗µ. If χ1µ 6= χ2

µ then wesay that µ has simple Lyapunov spectrum.

It is an easy exercise to calculate the Lyapunov exponents of Bernoulli measuresµp for a family of lower triangular matrices for which direction-x dominates. Forgreater generality see Falconer–Miao [FM07].

Lemma 2.3.3. Fix any p ∈ P and a family of lower triangular matricesA = A1, . . . , ANfor which direction-x dominates. Then the Lyapunov spectrum of the Bernoulli measure µp is

2.4. Upper bound for dimH Λ 35

simple and the exponents equal

χ1νp

= −N

∑i=1

pi log bi = − log〈b〉p and χ2νp

= −N

∑i=1

pi log ai = − log〈a〉p.

Sketch of proof. Both the singular values or the norm of Ai1 ...in can be calculated di-rectly. Since direction-x dominates, the off-diagonal element does not play a role. Anapplication of Oseledets theorem and the strong law of large numbers concludes theproof.

The Ledrappier–Young formula originates from the seminal work of Ledrappierand Young [LY85a; LY85b] on determining the Hausdorff dimension of invariant mea-sures of diffeomorphisms on compact manifolds. Through a succession of papers byPrzytycki–Urbanski [PU89], Feng–Hu [FH09], Bárány [B15] and Bárány–Käenmäki [BK17]the formula was proved for the Hausdorff dimension of wider and wider classes ofself-affine measures. In fact, Feng [Fen19] recently announced that the Hausdorffdimension of the push-forward of a shift-invariant, ergodic measure µ satisfies aLedrappier–Young type formula in full generality for any self-affine IFS on Rd whichis contracting on average with respect to µ. Also observe that the formulas proved inthe earlier works of [Bar07; Bar08; Bed84; GL92; McM84] are all special cases of theLedrappier–Young formula. The main result of [BK17, Theorem 2.4, Corollary 2.8]can be stated in a simpler form in our context when direction-x dominates.

Theorem 2.3.4 ([BK17], direction-x dominates). Let F be a shifted TGL-type IFS of theform (2.1.1). Furthermore, using the notation from Subsection 2.3.1, let µp be any Bernoullimeasure on Σ, νp = Π∗µp its push forward and νq = (projx)∗νp. Then, regardless ofoverlaps, νp is exact dimensional and satisfies the Ledrappier–Young formula

dimH νp =hµp − H

χ2νp

+

(1−

χ1νp

χ2νp

)dimH νq, (2.3.8)

where H = −∫

log µp,α(i)([i1])dµp(i). Recall µp,α(i) is the family of conditional measuresof µp defined by the measurable partition α(i) = Π−1(Π(i)).

Moreover, if the IFS satisfies the ROSC and p ∈ P0, then H = 0.

2.4 Upper bound for dimH Λ

Consider a shifted triangular Gatzouras–Lalley-type planar carpet Λ without any sep-aration condition. To prove Theorem 2.2.1 we essentially lift the original argument in[GL92], formulated on the attractor Λ, to the symbolic space Σ. This can be done be-cause the method in [GL92] is completely symbolic in nature. Therefore, we only givea short sketch.

The first step is to define a proper metric on Σ, which captures the distance be-tween points on the attractor. Observe that for two points i, j ∈ Σ the distance|Π(i)−Π(j)| (recall (2.3.2)) can be small even if |i ∧ j| is small. This occurs if |Φ(i) ∧Φ(j)| = |ı ∧ | (recall (2.3.1)) is much larger than |i ∧ j|, i.e. the corresponding cylin-ders belong to the same column for a long time.

Lemma 2.4.1. (Σ, d) is a metric space, where the distance between i, j ∈ Σ is defined

d(i, j) :=|ı∧ |∏k=1

bik +|i∧j|∏k=1

aik .


Proof. The fact that d is non-negative and symmetric is trivial. Need to check thetriangle inequality, for all i, j, k ∈ Σ : d(i, j) ≤ d(i, k) + d(k, j). If

• |i ∧ k| ≤ |i ∧ j| then ∏|i∧k|`=1 ai` ≥ ∏

|i∧j|`=1 ai` ,

• |i ∧ k| > |i ∧ j| then |k ∧ j| = |i ∧ j|, thus ∏|k∧j|`=1 ai` = ∏

|i∧j|`=1 ai` .

Analogously, if

• |ı ∧ k| ≤ |ı ∧ | then ∏|ı∧k|`=1 bi` ≥ ∏

|ı∧ |`=1 bi` ,

• |ı ∧ k| > |ı ∧ | then |k ∧ | = |ı ∧ |, thus ∏|k∧ |`=1 bi` = ∏

|ı∧ |`=1 bi` .

The triangle inequality now follows.

Remark 2.4.2. Without difficulty, one can show the stronger assertion that d(i, j) is an ul-trametric on Σ, i.e. d(i, j) ≤ maxd(i, k) , d(k, j) for every i, j, k ∈ Σ.

The next step is to prove that the natural projection with this metric is Lipschitz.

Lemma 2.4.3. For any shifted triangular Gatzouras–Lalley-type planar carpet

dimH Λ ≤ dimH(Σ, d).

Proof. It is enough to show that there exists C > 0 such that |Π(i) − Π(j)| ≤ C ·d(i, j), i.e. Π : Σ → Λ is a Lipschitz-function, which can not increase the Hausdorffdimension.

For i, j ∈ Σ let k := |i ∧ j|, ` := |ı ∧ | and(

xy

):= Π(i)−Π(j) =

∞

∑n=1

(ai|n−1tin,1 − aj|n−1tjn,1

di|n−1tin,1 − dj|n−1tjn,1 + bi|n−1tin,2 − bj|n−1tjn,2

).

The first k terms coincide in both coordinates and bin = bjn for n = 1, . . . , `. Thus,

x2 = a2i|k ·

( ∞

∑n=k+1

(aσki|n−k−1tin+k,1 − aσkj|n−k−1tjn+k,1))2≤ a2

i|k ,

y2 ≤ 2[( ∞

∑n=1

di|n−1tin,1

)2+( ∞

∑n=1

dj|n−1tjn,1

)2+( ∞

∑n=1

(bi|n−1tin,2 − bj|n−1tjn,2))2]

.

In the first two sums using Lemma 2.1.5 we can bound di|n−1 ≤ K0 · bi|n−1 and dj|n−1 ≤K0 · bj|n−1. Now we can pull out bi|` from all three sums. The remaining sums are alluniformly bounded in i, j by some constant c. This gives

y2 ≤ 2K20 · c · b2

i|`, thus |Π(i)−Π(j)| ≤√

a2i|k + 2K2

0 · c · b2i|` ≤ C · d(i, j).

It remains to show that the value α∗ maximizing the expression for D(p) in (2.2.1)is an upper bound for the Hausdorff dimension of (Σ, d).

Proposition 2.4.4. For any choice of parameters defining a shifted triangular Gatzouras–Lalley-type triangular carpet

dimH(Σ, d) ≤ α∗.

2.4. Upper bound for dimH Λ 37

Proof of Theorem 2.2.1. The upper bound is a corollary of Lemma 2.4.3 and Proposition2.4.4. The compactness of P and the continuity of D(p) implies that supp D(p) isattained for some p∗ ∈ P . Moreover, it is easy to check that p∗ ∈ P0, see [GL92,Proposition 3.4].

Proposition 2.4.4 is essentially proved in [GL92, Section 5] formulated on the at-tractor Λ. For completeness we sketch the main steps adapted to (Σ, d) and cite[GL92] when necessary. Most of the notation we bring over from [GL92].

The balls in (Σ, d) are exactly the "approximate squares" defined in [GL92, eq.(1.2)]

Bk(i) := j ∈ Σ : |i ∧ j| ≥ Lk(i) and |ı ∧ | ≥ k, where (2.4.1)

Lk(i) := max

n ≥ 0 :k

∏j=1

bij ≤n

∏j=1

aij

.

Note, k > Lk(i) for every i and k, since ai < bi for every i. The j ∈ Bk(i) for which|i ∧ j| = Lk(i) and |ı ∧ | = k are the ones for which d(i, j) is maximal. The definitionof Lk(i) implies that

1 ≤∏Lk(i)

j=1 aij

∏kj=1 bij

≤ maxi

a−1i for every k, i. (2.4.2)

Hence, diamBk(i) ≤ C ·∏kj=1 bij for some C independent of i.

The main ingredient is a form of the mass distribution principle adapted to (Σ, d).

Lemma 2.4.5. Let µ be a probability measure on Σ and assume

lim infk→∞

log µ(Bk(i))log ∏k

j=1 bij

≤ α for every i ∈ Σ,

where Bk(i) is the approximate square defined in (2.4.1). Then

dimH(Σ, d) ≤ α.

Proof. The assumption states that for every ε, δ > 0 and i ∈ Σ there exists a k(i) suchthat

k(i)

∏j=1

bij < δ and( k(i)

∏j=1

bij

)α+ε≤ µ(Bk(i)(i)).

The collection Bk(i)(i)i∈Σ is a δ-cover of Σ, thus the Vitali- or 5r-covering lemma[Fal86] implies that there exists a (perhaps uncountable) sub-collection J ⊂ Σ of dis-joint balls Bk(i)(i) giving a 5δ-cover of Σ, i.e.

Σ ⊆⊔

i∈J

5Bk(i)(i) and Bk(i)(i) ∩ Bk(j)(j) = ∅ for every i 6= j ∈ J.

Hence, we can bound the α + ε-dimensional Hausdorff measure

Hα+ε5δ (Σ) ≤ (5c)α+ε ∑

i∈J

( k(i)

∏j=1

bij

)α+ε≤ (5c)α+ε ∑

i∈Jµ(Bk(i)(i)) ≤ (5c)α+ε · µ(Σ)


independent of δ and therefore Hα+ε(Σ) ≤ (5c)α+ε < ∞ for every ε > 0. Thus,dimH(Σ, d) ≤ α.

The lemma implies that to prove Proposition 2.4.4 it is enough to find a measureµ satisfying the condition of the lemma with the value α∗. This can be achieved usingthe family of Gatzouras–Lalley Bernoulli measures introduced in [GL92, eq. (5.2)].Let ϑ ∈ R, λ ∈ R and ρ ∈ (0, 1). Define the probability vector p = (p1, . . . , pN) by

pi = pi(ϑ, λ, ρ) := C(ϑ, λ, ρ)aϑi bλ−ϑ

i (γi(ϑ))ρ−1, where γi(ϑ) = ∑

j∈Iφ(i)

aϑj (2.4.3)

and C(ϑ, λ, ρ) normalizes so that ∑i pi = 1. In fact [GL92, Lemma 5.1] shows thatthere exists a real-valued continuous function ϑ(ρ), ρ ∈ (0, 1), such that for everyρ ∈ (0, 1)

C(ϑ(ρ), α∗, ρ) = 1.

From now we work with such p.

Lemma 2.4.6. The Bernoulli-measure µ := pN on Σ satisfies the condition of Lemma 2.4.5with the optimal value α∗, i.e.

lim infk→∞

µ(Bk(i))log ∏k

j=1 bij

≤ α∗ for every i ∈ Σ.

Sketch of proof. By definition of Bk(i)

µ(Bk(i)) =Lk(i)

∏j=1

pij ·k

∏j=Lk(i)+1

qφ(ij) =( k

∏j=1

bij

)α∗

·∏Lk(i)

j=1 aϑij

∏kj=1 bϑ

ij

·(

∏kj=1 γij(ϑ)

)ρ

∏Lk(i)j=1 γij(ϑ)

,

where qφ(ij) = ∑`∈Iφ(ij)p`. Taking logarithm and dividing by log ∏k

j=1 bij gives

log µ(Bk(i))log ∏k

j=1 bij

= α∗ +ϑ log(∏Lk(i)

j=1 aij / ∏kj=1 bij)

log ∏kj=1 bij

+

log

(∏k

j=1 γij (ϑ))ρ

∏Lk(i)j=1 γij (ϑ)

log ∏kj=1 bij

.

Due to (2.4.2), the second term tends to zero as k→ ∞. We can increase the third termby replacing the denominator with k · log mini bi. Hence, it is enough to prove thatthere exists ρ ∈ (0, 1) such that for ϑ = ϑ(ρ)

lim supk→∞

ρ

k

k

∑j=1

log γij(ϑ)−1k

Lk(i)

∑j=1

log γij(ϑ) ≥ 0.

This is exactly the statement in [GL92, eq. (5.10)]. For details see [GL92, pg. 565-566].

Proof of Proposition 2.4.4. The Proposition is a direct corollary of Lemmas 2.4.5 and2.4.6.

2.5. Proof of Theorem 2.2.2 39

2.5 Proof of Theorem 2.2.2

Our goal is to show that the Ledrappier–Young formula (2.3.8) of [BK17] for dimH νp,cited in Theorem 2.3.4, always equals the formula for D(p) in (2.2.1) under the con-ditions of Theorem 2.2.2. For the rest of this proof, we fix a p ∈ P0 and assume Hsatisfies Hochman’s Exponential Separation Condition and either each column inde-pendently satisfies the ROSC or Λ satisfies transversality and (2.2.2).

The entropy of the system is hµp = − log〈p〉p (recall (2.3.6)), the Lyapunov-exponentsfrom Lemma 2.3.3 are χ2

νp= − log〈a〉p and χ1

νp= − log〈b〉p . Hochman’s Exponential

Separation Condition for H implies No Dimension Drop for νq, recall (2.1.20), hencedimH νq = log〈q〉q/ log〈b〉p . As a result, to prove the theorem it is enough to showthat the integral

H = −∫

log µα(i)([i1])dµp(i) = 0,

where µα(i) is the family of conditional measures of µp defined by the measurablepartition α(i) = Π−1(Π(i)), recall (2.3.5). Since − log µα(i)([i1]) ≥ 0, we have thatH = 0 if and only if

µα(i)([i1]) = 1 for µp-a.a. i. (2.5.1)

Thus, it suffices to show that µα(i) is concentrated on i for µp-typical i. Overlapsarising from the translations of columns or from intersections within a column canin theory cause problems. However, the next two results ensure that there is a fullmeasure subset of Σ for which µα(i) is a point mass distribution.

Recall from (2.3.4) that β(i) = Φ−1Φ(i) is the ’symbolic column’ if i. The firstclaim ensures that there is a full measure subset Σ1 ⊂ Σ where the translations ofcomplete columns have no effect.

Claim 2.5.1. Assume Weak Almost Unique Coding holds for ΣH, recall Definition 2.1.9.Then there exists a full measure subset Σ1 ⊂ Σ such that for all i ∈ Σ1 and for all ( 1, . . . , n) 6=(ı1, . . . , ın)

µα(i)([j1, . . . , jn]) = 0, (2.5.2)

where φ(ik) = ık and φ(jk) = k for k = 1, . . . , n, recall (2.1.9) for the definition of φ.Consequently, for every i ∈ Σ1 we have

µα(i)(β(i)c) = 0. (2.5.3)

The second claim defines the full-measure set Σ2 ⊂ Σ where intersections withincolumns have no effect.

Proposition 2.5.2. Assume that the conditions of Theorem 2.2.2 hold. Then there exists aΣ2 ⊂ Σ, with µp(Σ2) = 1 such that for every i ∈ Σ2 and k ∈ Iφ(i1) \ i1

µα(i) (β(i) ∩ α(i) ∩ [k]) = 0.

Theorem 2.2.2 is a corollary of these two results. Sometimes we use the followingnotation:

Definition 2.5.3. Let F ⊂ Σ be a subset of full measure. Then we define

F :=

i ∈ F : µα(i)(Σ \ F) = 0

.

Since µp(F) = 1, the disintegration formula (2.3.5) implies that µp(F) = 1.


2.5.1 The proof of Theorem 2.2.2 assuming Claim 2.5.1 and Proposition2.5.2

Proof of Theorem 2.2.2 assuming Claim 2.5.1 and Proposition 2.5.2. As we established abovein (2.5.1) that to prove the theorem it is enough to check that

µα(i) (α(i) ∩ [i1]c) = 0, for µ-a.a. i. (2.5.4)

Clearly,

α(i) ∩ [i1]c ⊂(

⋃

k 6∈Iφ(i1)

(α(i) ∩ [k])

)∪(

⋃

k∈Iφ(i1)\i1

(α(i) ∩ [k])

).

It follows from (2.5.2) that for every i ∈ Σ1

µα(i)

(⋃

k 6∈Iφ(i1)

(α(i) ∩ [k])

)= 0, (2.5.5)

where Σ1 is defined in Claim 2.5.1. Thus, to prove the theorem we only need to verifythat

µα(i)

(⋃

k∈Iφ(i1)\i1

(α(i) ∩ [k])

)= 0 for µ-a.a. i. (2.5.6)

We can write

⋃

k∈Iφ(i1)\i1

(α(i) ∩ [k]) ⊂ Σc1 ∪ Σc

2

∪(

⋃

k∈Iφ(i1)\i1

(α(i) ∩ β(i) ∩ [k])

)

︸︷︷︸U

∪(

⋃

k∈Iφ(i1)\i1

(α(i) ∩ β(i)c ∩ [k])

)

︸︷︷︸V

.

It follows from Proposition 2.5.2 that µα(i)(U) = 0 for all i ∈ Σ2 and it followsfrom Claim 2.5.1 that µα(i)(V) = 0 for all i ∈ Σ1. So, for all i ∈ Σ1 ∩ Σ2 (2.5.6)holds, which together with (2.5.5) yields that (2.5.4) holds. This completes the proofof Theorem 2.2.2 assuming Claim 2.5.1 and Proposition 2.5.2.

2.5.2 The proof of Claim 2.5.1

Proof of Claim 2.5.1. In the definition of Weak Almost Unique Coding, recall Defini-tion 2.1.9, there is a set BH ⊂ ΣH defined in such a way that for Σ′H := ΣH \ BH wehave µq(Σ′H) = 1 and

ı ∈ Σ′H ⇐⇒ Σ′H ∩(Π−1H ΠH(ı)

)= ı ,

where ΠH is the natural projection from ΣH to ΛH. Let

B := Φ−1(BH) and Σ′ := Φ−1 (Σ′H)

.


Since µq(Σ′H) = 1 we can define Σ1 := Σ′ (recall the notation˜ from Definition 2.5.3)so that µp(Σ1) = 1 and

µα(i)(B) = 0 for all i ∈ Σ1. (2.5.7)

Recall ΠH is the natural projection from Σ to ΛH. Observe that by definition

i ∈ Σ′ =⇒ Σ′ ∩(

Π−1H ΠH(i)

)= β(i). (2.5.8)

Since α(i) ⊂ Π−1H ΠH(i), we get from (2.5.8) that i ∈ Σ′ =⇒ Σ′ ∩ α(i) ⊂ β(i). Equiva-

lently,i ∈ Σ′ =⇒ α(i) ⊂ β(i) ∪ B.

By definition

[j1, . . . , jn] ∩ β(i) = ∅ iff ( 1, . . . , n) 6= (ı1, . . . , ın).

That is for i ∈ Σ1 whenever ( 1, . . . , n) 6= (ı1, . . . , ın) then [j1, . . . , jn] ∩ α(i) ⊂ B. So,(2.5.7) implies that (2.5.2) holds.

To obtain (2.5.3) from (2.5.2), we write β(i)c as a countable union

β(i)c =∞⋃

`=0

j ∈ Σ : |ı ∧ | = ` =∞⋃

`=0

⋃

r=ır ,r≤`j`+1 6=i`+1

[j1, . . . j`+1] .

By (2.5.2) the measure of each cylinder of the right hand side is

µα(i)([j1, . . . j`+1]) = 0 if [ 1, . . . `+1] 6= [ı1, . . . , ı`+1] , i ∈ Σ1.

2.5.3 Proof of Proposition 2.5.2

If the columns independently satisfy ROSC, then the proof of [BK17, Corollary 2.8]can be repeated in this setting, therefore we omit it. In the remainder we assume theshifted TGL carpet Λ satisfies transversality and (2.2.2):


> 1 +log〈N〉q− log〈q〉q

.

Throughout this proof we fix δ > 0 small enough such that

1 + δ +(1 + δ) log〈N〉q

δ log〈b〉p − log〈q〉q< (1− δ)


. (2.5.9)

This can be achieved since the expression is continuous in δ and we assume (2.2.2).The reason that we require this is that for such a δ and

u := (1− δ)log〈a〉plog〈b〉p

− (1 + δ), (2.5.10)

the inequality in (2.5.9) is equivalent to

〈N〉(1+δ) · 〈q〉u · 〈b〉−δu < 1. (2.5.11)


At the very end of this proof we will need this. The importance of the u defined abovecomes from the fact that for an arbitrary ` and k = u · `,

〈b〉kp =〈a〉(1−δ)`

p

〈b〉(1+δ)`p

. (2.5.12)

Recall α(i) = Π−1Π(i), β(i) = Φ−1Φ(i), that ΠH is the natural projection from Σ toΛH and that in (2.3.7) we define Badδ,n(c) for a c = (c1, . . . , cN) with 〈c〉p 6= 1.

Further notation

Recall that Hochman’s Exponential Separation Condition implies that for the self-similar measure νq on ΛH we have dimH νq = log〈q〉q/ log〈b〉p. Feng and Hu [FH09]proved that νq is exact dimensional. That is for K1 defined in (2.1.14) and

Sn0 :=

i ∈ Σ : ∀n ≥ n0,

νq

((ΠH(i)− 3K1〈b〉n, ΠH(i) + 3K1〈b〉n

))∈(〈q〉n〈b〉δn, 〈q〉n〈b〉−δn

) (2.5.13)

we have

µp

(∞⋃

n0=1

Sn0

)= lim

n0→∞µp (Sn0) = 1. (2.5.14)

We define the set of symbols which are "good" from level m on:

Goodm :=⋂

n≥m

(Badδ,n(a) ∪ Badδ,n(b) ∪ Badδ,n(N) ∪ Badδ,n(q)

)c.

Note that it follows from Lemma 2.3.1 that for

Good :=∞⋃

m=1

Goodm, we have µp(Good) = 1. (2.5.15)

To measure vertical distance and neighborhood on Λ we define

disty((x0, y0), (x, y)) :=

|y− y0|, if x = x0;∞. otherwise,

For every m ≥ 1 the function Lm : Good → [0, 1] is defined as follows: if there existsno j ∈ Goodm with j1 6= i1 and Φ(i) = Φ(j) then Lm(i) := 1. Otherwise we define

Lm(i) := infdisty(Π(i), Π(j)) : j ∈ Goodm ∩ β(i) such that j1 6= i1.

Let

Vm` := i ∈ Good : Lm(i) < 〈a〉`

=

i ∈ Good : ∃j ∈ β(i) ∩ [i1]c ∩Goodm, disty (Π(i), Π(j)) < 〈a〉`

.

Also define

Bm2 :=i ∈ Good : Lm(i) = 0 = i ∈ Good : α(i) ∩ β(i) ∩ [i1]c ∩Goodm 6= ∅ .

(2.5.16)


Clearly, Bm2 ⊂ Bm+1

2 since Bm2 = ∩`≥mVm

` . The key lemma states the following.

Lemma 2.5.4. For arbitrary m ≥ 1 we have µ(Bm2 ) = 0.

Proof of Proposition 2.5.2 assuming Lemma 2.5.4. Let

B2 :=∞⋃

m=1

Bm2 = i ∈ Good : α(i) ∩ β(i) ∩ [i1]c ∩Good 6= ∅ .

By Lemma 2.5.4, µ(B2) = 0. That is, if i ∈ Σ2 := Good ∩ Bc2 then on the one hand

µα(i)(Goodc) = 0, on the other hand α(i) ∩ β(i) ∩ [i1]c ∩Good = ∅. This implies thatµα(i) (α(i) ∩ β(i) ∩ [i1]c) = 0, which completes the proof of Proposition 2.5.2.

It remains to show Lemma 2.5.4. The method of the proof was inspired by [BRS,Lemma 4.7], however there are significant differences. On the one hand, in [BRS] themeasure corresponding to νq is absolutely continuous with Lq density and in [BRS]the diagonal part of all the linear parts of all the mappings are identical. These differ-ences required a much more subtle argument in this situation.

Proof of Lemma 2.5.4. Recall that we fixed an m. Let ` ≥ m. All sets and numbers fromnow on in this proof can be dependent of m but m is fixed so we omit it from notation.

We cover Vm` by the union of the Π−1 pre-images of the parallelograms like the

blue one (Rı,) on the right hand side of Figure 2.3. These are parallelograms slightlybigger than the intersection of Rı and the 〈a〉` neighborhood of R for ı, ∈ Σ` withΦ(ı) = Φ().

To control the size of `th level parallelograms and the number of parallelogramsin any given `-th level column set

Bad1δ,` := Badδ,`(a) ∪ Badδ,`(b) ∪ Badδ,`(N) and Bad1,∗

δ,` := i|` : i ∈ Bad1δ,`,

where Badδ,n(c) was defined in (2.3.7). Observe that Bad1δ,` is the union of complete

`-cylinders. That isBad1

δ,` =⋃

ωωω∈Bad1,∗δ,`

[ωωω].

The level `-cylinders of the symbolic spaces excluding these bad cylinders are:

Good∗` := 1, . . . , N` \Bad1,∗δ,` and Good∗`,ı := ∈ Good∗` : j1 6= i1, Φ() = Φ(ı) \Bad1,∗

δ,` .

For H ⊂ [0, 1]2 let

Uy(H, r) :=⋃

(x0,y0)∈H

(x, y) : x = x0 and |y− y0| < r.

Choose ı ∈ Good∗` , ∈ Good∗`,ı and define

Iı, := projx(Rı ∩Uy(R, 〈a〉(1−δ)`),Rı, := (Iı, × [0, 1]) ∩ Rı ,

Rı, := ([ı] ∩Π−1(Rı,)).


Rı, consists of those elements of Rı which are physically "too close" to R, see Figure2.3. As a result we get a cover of Vm

` :

Vm` ⊂ Bad1

δ,` ∪⋃

ı∈Good∗`

⋃

∈Good∗`,ı

Rı,. (2.5.17)

Namely, if i ∈ Vm` then either i ∈ Bad1

δ,` or ı := i|` ∈ Good∗` . In the second case, thereis a j ∈ β(i) ∩ [i1]c ∩Goodm with disty(Π(i), Π(j)) < 〈a〉`. Hence, := j|` ∈ Good∗`,ı.As a result, with these notations, we have i ∈ Rı,.

Π(H

ı,)

Jı,(0, 0)

(1, 1)

aı

〈a〉(1−δ)`a〈a〉(1−δ)`

Iı,

RıR

Rı,

f−1ı (Rı,)

FIGURE 2.3: Intersecting parallelograms Rı and R in the proof ofLemma 2.5.4.

If Iı, 6= ∅, then there exists a non-empty interval Jı, such that

f−1ı (Rı,) = Jı, × [0, 1].

With symbolic notation Hı, := Π−1( f−1ı (Rı,)) we can represent Rı, as the concatena-

tionRı, = ıHı,. (2.5.18)

On the other hand, Hı, ⊂ Π−1H(

Jı,). Hence,

µp(

Hı,)≤ µp

(Π−1H (Jı,)

)= νq

(Jı,)

. (2.5.19)

To continue we give an upper bound for νq(

Jı,).

Claim 2.5.5. Let ı ∈ Good∗` and ∈ Good∗`,ı and let k := u · `, where u was defined in(2.5.10). If Π−1

H (Jı,) ∩ Sk 6= ∅ (recall (2.5.13) for the definition of Sk), then

νq(

Jı,)≤ 〈q〉k · 〈b〉−kδ.

Proof of Claim 2.5.5. If Iı, 6= ∅ then transversality (recall Definition 2.1.6) implies that

|Iı,| < 3K1 · 〈a〉(1−δ)`.


This is the very important point where use that neither ı nor are contained in Bad1,∗δ,` .

Furthermore, f−1ı expands along the x axis by a factor between 〈b〉−(1−δ)` and 〈b〉−(1+δ)`,

hence

|Jı,| < 3K1

( 〈a〉1−δ

〈b〉1+δ

)`

. (2.5.20)

If we set k as in Claim 2.5.5 then as we mentioned in (2.5.12) the right hand side of(2.5.20) is less than 3K1 · 〈b〉k :

|Jı,| < 3K1 · 〈b〉k.

Now assume that Π−1H (Jı,) ∩ Sk 6= ∅. Pick an arbitrary ωωω ∈ Π−1

H (Jı,) ∩ Sk. Then

Jı, ⊂(

ΠH(ωωω)− 3K1〈b〉k, ΠH(ωωω) + 3K1〈b〉k)

.

Using that ωωω ∈ Sk, we get that νq(ΠH(ωωω) − 3K1〈b〉k, ΠH(ωωω) + 3K1〈b〉k

)≤ 〈q〉k ·

〈b〉−kδ.

Now we conclude the proof of Lemma 2.5.4. From the cover (2.5.17) of Vm` to-

gether with (2.5.18) we obtain that for ` ≥ m

µp(Vm` ) ≤ µp(Bad1

δ,`) + ∑ı∈Good∗`

µp([ı])µp

(⋃

∈Good∗`,ı

Hı,

). (2.5.21)

To further bound (2.5.21), first observe that

#Good∗`,ı ≤ 〈N〉(1+δ)` whenever ı ∈ Good∗` .

Moreover, using (2.5.19) and Claim 2.5.5, for an arbitrary ı ∈ Good∗` we have

µp

(⋃

∈Good∗`,ı

Hı,

)≤ µp (Sc

k) + ∑∈Good∗

`,ıΠ−1H (Jı,)∩Sk 6=∅

µp(

Hı,)≤ µp (Sc

k) + ∑∈Good∗

`,ıΠ−1H (Jı,)∩Sk 6=∅

νq(Jı,)

≤ µp (Sck) + ∑

∈Good∗`,ı

Π−1H (Jı,)∩Sk 6=∅

〈q〉k · 〈b〉−kδ

≤ µp (Sck) +

(〈N〉(1+δ) · 〈q〉u · 〈b〉−uδ

)`.

Pluggung this back into (2.5.21), we deduce from Lemma 2.3.1, (2.5.14) and (2.5.11)that for every m,

lim`→∞

µp(Vm` ) = 0.

By (2.5.16), this implies that µp (Bm2 ) = 0.


2.6 Proof of results for box dimension

We begin the section by briefly commenting on how the upper bound for dimBΛ,recall Theorem 2.2.4, follows directly from the work of Fraser [Fra12] and then proveTheorem 2.2.7 in Subsection 2.6.3.

Recall the notation from Subsection 2.2.2. For δ > 0 and a bounded set F ⊂ R2

let Nδ(F) denote the minimal number of closed axes parallel rectangles for which thevertical sides are not shorter than the horizontal sides but the vertical sides are notlonger than (K0 + 1)-times the horizontal sides, where K0 was defined in Lemma 2.1.5.Then

dimBF = lim infδ→0

log Nδ(F)− log δ

and dimBF = lim supδ→0


.

In particular, it is enough to consider δ → 0 through the sequence δk = ck for some0 < c < 1, see [Fal90, Section 3.1].

For t ≥ 0 and any finite length word ı ∈ Σ∗, Fraser defined the modified singularvalue function ψt, which in our context is

ψt( fı) := bsHı · at−sH

ı ,

where sH = dimB ΛH. He showed that the unique solution s of the equation

limn→∞

(∑

ı∈Σn

ψs( fı))1/n

= 1

is an upper bound for dimBΛ and equals dimB Λ if Λ satisfies the ROSC. In our con-text this equation simply becomes (2.2.4): ∑N

i=1 bsHi as−sH

i = 1. The slight modificationof the GL brother Λ ensures that the solution of (2.2.4) for Λ and Λ is the same.

For any TGL carpet Λ, it follows from Lemma 2.1.5 that the longer side of anyparallelogram Rı is at most (K0 + 1) · bı. This implies that there exists a constant C(independent of δ) such that Nδ(Λ) ≤ C · Nδ(Λ). Hence, dimB(Λ) ≤ dimB(Λ) ≤ s.Furthermore, when the ROSC is assumed, it is clear that the reversed inequalities alsohold. This implies dimB(Λ) ≥ dimB(Λ) = s. This proves Theorem 2.2.4.

In the presence of overlaps, one must be more careful when counting the intersec-tions. The next subsection shows how to select a diagonally homogeneous subsystemfrom a higher iterate of F .

2.6.1 Diagonally homogeneous subsystems

Recall for a p ∈ P0

hp = −N

∑i=1

pi log pi = − log〈p〉p,

χ1p = −

N

∑i=1

pi log bi = − log〈b〉p and χ2p = −

N

∑i=1

pi log ai = − log〈a〉p.

The following is a Ledrappier–Young like formula for the solution s of (2.2.4). It gen-eralizes the formula in Corollary 2.2.12 for the diagonally homogeneous case. A sim-ilar result for Bedford-McMullen like systems in arbitrary dimension was proved in[FH09, Theorem 2.15].

2.6. Proof of results for box dimension 47

Claim 2.6.1. For p := ( p1, . . . , pN) defined by pi = bsHi as−sH

i , recall (2.2.6), we have

s =hp

χ2p+

(1−

χ1p

χ2p

)sH,

where sH = dimB ΛH.

Proof. Immediately follows from the observation that hp = sHχ1p + (s− sH)χ2

p.

The following line of thought is an adaptation of [FS16, Section 6] in order toextract from an arbitrary shifted TGL IFS F = fiN

i=1 with M columns a subsystemof a high enough iterate of F k, which has some nice properties required to prove thetheorem.

Let F k := fi1 ...ik : i1, . . . , ik ∈ Σk. The first step is to pass from F to a diagonallyhomogeneous subsystem of F k. Analogous arguments appear for example in [BRS16,Lemma 5.2], [FS16, Lemma 6.2] or [PS, Lemma 4.9].

Definition 2.6.2. A subsystem G(k) ⊂ F k is called a diagonally homogeneous subsystem ifthere exists a(k) and b(k) for which

g(k)i (x) =(

b(k) 0di a(k)

)x + ti, for every g(k)i ∈ G(k).

Fix an arbitrary vector v = (v1, . . . , vN), where vi ∈ N. Let V := ∑i∈I vi, V :=

(V1, . . . , VM), V = V1 + . . . + VM and define

Mv := (i1, . . . , iV) ∈ ΣV : #` ≤ V : i` = r = vr for every r = 1, . . . , N. (2.6.1)

Claim 2.6.3. The subsystem Gv = fi1...iV : (i1, . . . , iV) ∈ Mv ⊂ FV with Mv columns

(i) is a diagonally homogeneous subsystem with a(v) = ∏Nr=1 avr

r and b(v) = ∏Nr=1 bvr

r ,

(ii) has uniform vertical fibres with ∏Mı=1

Vı !∏r∈Iı

vr ! maps in each column and

(iii) for the probability vectors v = v/V and V = V/V

−N log(V + 1) + V · hv ≤ log #Mv ≤ V · hv,−N log(V + 1) + V · hV ≤ log Mv ≤ V · hV.

Proof. Parts (i) and (ii) are immediate. Part (iii) follows directly from [DZ10, Lemma 2.1.8].

Lemma 2.6.4. Let F = fiNi=1 be a shifted TGL IFS with M columns. For every k choose

vk = (v1,k, . . . , vN,k) such that

vi,k = bkpic for every i = 1, . . . , N, (2.6.2)

where pi was defined in (2.2.6). Let V(k) := ∑i∈I

vi,k, V(k) = ∑Mı=1 V(k)

ı and define thesubsystem

G(k) = Gvk = fi1...iV(k)

: (i1, . . . , iV(k)) ∈ Mvk ⊂ FV(k),

whereMvk is defined by (2.6.1). Then G(k) satisfies the assertions of Claim 2.6.3 with vk. Forbrevity we write a(vk) = a(k) and b(vk) = b(k). Let

N(k) = #Mvk and M(k) = #(Φ(Mvk)

)


denote the number of maps and columns in G(k).Moreover, lim

k→∞s(k) = s, where s(k) is the solution of N(k)(b(k))sH(a(k))s(k)−sH = 1, i.e.

s(k) =log N(k)

− log a(k)+

(1− log b(k)

log a(k)

)sH.

Remark 2.6.5. The box dimension of the attractor of the IFS G(k) is NOT equal to s(k), becausesH is not the box dimension s(k)H of the attractor of the IFS generated by the columns of G(k).The problem is that s(k)H 6→ sH as k→ ∞ (except when dimH Λ = dimB Λ).

Proof. It follows from (2.6.2) that k− N ≤ V(k) ≤ k and

k log〈a〉p −N

∑i=1

log ai ≤ log a(k) ≤ k log〈a〉p.

Same holds for log b(k). Furthermore, for the probability vector vk = vk/V(k)

pi −1k≤ vi,k

V(k)≤ pi +

piNk− N

.

Thus limk→∞ hvk = hp. We can use Claim 2.6.3 (iii) to bound log #Mvk . Hence, puttingtogether all the above we get

limk→∞

s(k) =hp

− log〈a〉p+

(1− log〈b〉p

log〈a〉p

)sH,

which is equal to s due to Claim 2.6.1.

If G(k) already has non-overlapping columns, then the rest of the construction isnot necessary. Otherwise, we can pass further to a subsystem G(k,`) ⊂

(G(k)

)` by

throwing away "not too many" columns of(G(k)

)` in order to ensure that G(k,`) hasnon-overlapping columns.

Projecting G(k) to the x-axis gives a subsystem ofHV(k)

G(k)H := hı1...ıV(k) : there exists (i1, . . . , iV(k)) ∈ Mvk s. t. Φ(i1 . . . iV(k)) = ı1 . . . ıV(k),

which has a total of M(k) maps, each with contracting ratio b(k). Observe that G(k)Halso satisfies Hochman’s Exponential Separation Condition, because this condition isassumed for H and this property passes on to any subsystem. Hence, the Hausdorffand box dimension of G(k)H satisfies

s(k)H =log M(k)

− log b(k). (2.6.3)

It follows from the definition of box dimension that for every ε > 0 there exists asubset of the columns of

(G(k)

)`, which are non-overlapping and have cardinality

M(k,`) ≥ Cε

((b(k))`

)−(s(k)H −ε) (2.6.3)= Cε ·

(M(k))`(b(k)

)`ε. (2.6.4)


This is the subsystem G(k,`) which we will use in the proof of Theorem 2.2.7 undercondition (i). When condition (ii) of Theorem 2.2.7 is assumed we use G(k,`) =

(G(k)

)`since in this case non-overlapping columns are assumed for Λ. Next, we present ourargument to count the number of intersections within a column when Λ has non-overlapping columns.

2.6.2 Counting intersections

Let F be an arbitrary TGL IFS and G(k,`) =(G(k)

)` be the subsystem defined inthe previous subsection. Then G(k,`) is diagonally homogeneous with main diagonal((b(k))`, (a(k))`), has uniform vertical fibres with (N(k)/M(k))` maps in each columnand the columns are non-overlapping. For every fı ∈ G(k,`), ı can be written

ı = ı1ı2 . . . ı`, where ıj ∈ Mvk for j = 1, . . . , `.

Let Σ(k,`) := ı : fı ∈ G(k,`) and for the rest of the subsection fix such an ı ∈ Σ(k,`).Let

Σ∼ı := = 1 . . . ` ∈ Σ(k,`) : Φ() = Φ(ı) and 6= ı,i.e. Σ∼ı collects those which belong to the symbolic column of ı. Recall Λı =fı(Λ), Rı = fı([0, 1]2). Let

Rı :=(projx( fı(Λ))× [0, 1]

)∩ Rı and δ

(k)` := (a(k))`.

Our aim is to give a uniform upper bound for Nδ(k)`

(Rı ∩ (∪∈Σ∼ı R)

). Observe that for

every ∈ Σ∼ı

Nδ(k)`

(Rı ∩ R

)= N

δ(k)`

(projx(Λı ∩Λ)

)= N

δ(k)` /(b(k))`

(h−1

ı (projx(Λı ∩Λ))). (2.6.5)

We state a result of Lalley [Lal88, Theorem 1], which gives the precise asymptoticof Nδ(ΛH). A set r1, . . . , rM of positive reals is τ-arithmetic, if τ > 0 is the greatestnumber such that each ri is an integer multiple of τ, and non-arithmetic if no such τexists. We use the notation f (δ) ∼ g(δ) to denote that limδ→0 f (δ)/g(δ) = 1. Let F bea self-similar set on [0, 1] with contracting ratios r1, . . . , rM. Assume F satisfies thestrong OSC and let dimH F = dimB F = t, where t is the solution of ∑M

ı=1 rtı = 1.

Proposition 2.6.6. [Lal88, Theorem 1] If log r−11 , . . . , log r−1

M is a non-arithmetic set, thenfor some K > 0

Nδ(F) ∼ Kδ−t as δ→ 0.

On the other hand, if log r−11 , . . . , log r−1

M is τ-arithmetic, then for the subsequence δn =e−nτ there exists a constant K′ > 0 such that

Nδn(F) ∼ K′δ−tn as n→ ∞.

Remark 2.6.7. The reason why we can not handle both types of overlaps simultaneously forthe box dimension is that we are unaware of an analogous result in the case that SOSC is notassumed. This question could be of independent interest.

We use the proposition for the self-similar set ΛH with contracting ratios (r1, . . . , rM).If log r−1

1 , . . . , log r−1M is τ-arithmetic, then we can choose ` = `(n) so that

mine−τ, 1 · e−τn < δ(k)` ≤ maxe−τ, 1 · e−τn,


which implies that

limn→∞

`(n)n

=τ

− log a(k)and lim

n→∞

Ne−τn(ΛH)N

δ(k)`

(ΛH)= c

for some universal constant c. Thus the proposition implies that

Nδ(k)`

(Rı)= N

δ(k)` /(b(k))`

(ΛH) = (C + o(1))(b(k)/a(k))`sH , (2.6.6)

where the constant C only depends on whether log r−11 , . . . , log r−1

M is τ-arithmeticor not and the o(1)→ 0 as `→ ∞. The next lemma ensures that a positive proportionof these boxes do not get covered by boxes coming from the cover of R for some ∈ Σ∼ı .

Lemma 2.6.8. If F satisfies transversality and

N(k)

M(k)

(1 + KsH

1

)<

(b(k)

a(k)

)sH

, (2.6.7)

then there exists K3 < 1 such that for ` large enough and every ı ∈ Σ(k,`) we have

Nδ(k)`

(Rı ∩

⋃

∈Σ∼ı

R

)≤ K3N

δ(k)`

(Rı).

Proof. Fix ∈ Σ(k,`) such that | ∧ ı| = z, where we count ım, m ∈ Mvk as one sym-bol. Thus, z ∈ 0, 1, . . . , `− 1. Since F satisfies transversality, then so do all of itssubsystems, in particular G(k,`) as well. Hence,

|projx(

Rı ∩ R

)| ≤ K1

(b(k))z(a(k)

)`−z,

see Figure 2.4. This together with (2.6.5) and Proposition 2.6.6 yields that

Nδ(k)`

(Rı ∩ R) ≤ (C + o(1))KsH1

(b(k)

a(k)

)zsH

.

Since G(k,`) has uniform vertical fibres, it follows that # ∈ Σ∼ı : | ∧ ı| = z ≤(N(k)/M(k))`−z. Thus from a simple union bound we get

Nδ(k)`

(Rı ∩

⋃

∈Σ∼ı

R

)≤

`−1

∑z=0

(N(k)

M(k)

)`−z

(C + o(1))KsH1

(b(k)

a(k)

)zsH

=KsH

1M(k)

N(k)

( b(k)a(k))sH − 1

︸︷︷︸=:K3

(C + o(1))

(

b(k)

a(k)

)`sH

−(

M(k)

N(k)

)`≤ K3N

δ(k)`

(Rı),

where the last inequality holds if N(k)/M(k) ≤ (b(k)/a(k))sH . This holds, because(2.6.7) is an even stronger assumption. Furthermore, simple arithmetic shows thatK3 < 1 if and only if (2.6.7) holds.


(b(k))`

(a(k))`

(a(k))`Rı

R

(a(k))`

|projx(Rı ∩ R

)| ≤ K1

(b(k))z(

a(k))`−z

FIGURE 2.4: Intersecting parallelograms Rı and R in the proof ofLemma 2.6.8.

2.6.3 Proof of Theorem 2.2.7

Throughout the proof, s is the target box dimension defined as the solution of (2.2.4):∑N

i=1 bsHi as−sH

i = 1. Fix ε > 0. We work with the subsystem G(k,`) defined in Subsec-tion 2.6.1. It will be enough to cover the subset

⋃

ı∈G(k,`)

fı(Λ) ⊆ Λ,

with boxes of size δ(k)` := (a(k))`. Recall Rı = (projx( fı(Λ))× [0, 1]) ∩ ( fı([0, 1]2)).

Conclusion of proof assuming condition (i) of Theorem 2.2.7. AssumeF generates a shiftedTGL carpet Λ for whichH satisfies Hochman’s Exponential Separation Condition andthe columns independently satisfy ROSC. In this case it is enough to use the defini-tion of box dimension to bound N

δ(k)`

(Rı)≥ Cε(b(k)/a(k))`(sH−ε) for some constant Cε

depending only on ε. Recall from Lemma 2.6.4 that s(k) → s. We choose k so largethat s(k) ≥ s− ε and we bound

lim inf`→∞

log Nδ(k)`

(Λ)

− log δ(k)`

≥ lim inf`→∞

log(

M(k,`)(N(k)/M(k))`(b(k)/a(k))`(sH−ε))

−` log a(k)(2.6.8)

≥ log N(k)

− log a(k)+

(1− log b(k)

log a(k)

)sH

︸︷︷︸=s(k)≥s−ε

−ε ≥ s− 2ε,

where for the second inequality we substituted the lower bound for M(k,`) from (2.6.4).Letting ε 0 yields dimBΛ ≥ s as claimed.

Conclusion of proof assuming condition (ii) of Theorem 2.2.7. For the remainder we assumethat F has non-overlapping columns, satisfies transversality and (2.2.5):

hp − hq < sH(log〈b〉p − log〈a〉p), (2.6.9)

where hp = − log〈p〉p and pi = bsHi as−sH

i . We need to check that condition (2.6.7)of Lemma 2.6.8 is satisfied, since it ensures that a positive proportion of the boxesneeded to cover fı(Λ) are not intersected by any boxes covering f (Λ) for 6= ı.


Claim 2.6.9. For all k large enough, assumption (2.6.9) implies condition

N(k)

M(k)

(1 + KsH

1

)<

(b(k)

a(k)

)sH

of Lemma 2.6.8.

Proof of Claim 2.6.9. We know from Subsection 2.6.1 that

log a(k) = k log〈a〉p + O(1), log N(k) = khp + o(k),

log b(k) = k log〈b〉p + O(1), log M(k) = khq + o(k).

Taking the logarithm of each side of (2.6.7), substituting these values and dividing byk gives

hp − hq +1k

log(1 + KsH1 ) < sH(log〈b〉p − log〈a〉p),

with an error of o(1) as k → ∞ on either side. The second term on the left hand sidealso tends to zero as k → ∞, thus (2.6.9) indeed implies the condition of Lemma 2.6.8for large k.

The conclusion of the proof of Theorem 2.2.7 is now analogous to the calculationof (2.6.8) with the exception that we need the precise value of N

δ(k)`

(Rı)

from (2.6.6)

and we can use G(k,`) = (G(k))`, so the number of columns M(k,`) = (M(k))`. Choose kso large that s(k) ≥ s− ε and condition (2.6.7) hold simultaneously. Using Lemma 2.6.8we can basically repeat the calculation of (2.6.8)

lim inf`→∞

log Nδ(k)`

(Λ)

− log δ(k)`

≥ lim inf`→∞

log((N(k))`(1− K3)(C + o(1))(b(k)/a(k))`sH

)

−` log a(k)= s(k).

This concludes the proof of Theorem 2.2.7.


The theorem claims that for a shifted TGL carpet Λ

(i) dimH Λ = dimB Λ (ii) sH = dimH νq (iii) ∑j∈Iı

as−sHj = 1 for every ı ∈ [M]

are equivalent, provided ROSC and No Dimension Drop (NDD, recall Definition 2.1.9)hold. We show that (i)⇔ (iii), (iii)⇒ (ii) and (ii)⇒ (i).

Proof of (i) ⇔ (iii). Let Λ be the GL brother of Λ, recall Definition 2.1.4. Fora p ∈ P0 let νp denote the push forward of the Bernoulli measure µp on Λ. Wehave dimH νp = dimH νp for every p ∈ P0. Indeed, in the beginning of Section 2.5we proved dimH νp = D(p) assuming ROSC and NDD, furthermore, Gatzouras–Lalley proved dimH νp = D(p) [GL92, Proposition 3.3]. Hence, dimH Λ = dimH Λ.Also, assuming NDD, sH is the unique real which satisfies ∑M

ı=1 rsHı = 1. This implies

dimB Λ = dimB Λ. The analogous claim of (i) ⇔ (iii) for Λ was proved in [GL92,Theorem 4.6]. Thus (i)⇔ (iii) in our setting as well.

Proof of (iii) ⇒ (ii). Condition (iii) implies that the vector q is simply qı = rsHı

for ı ∈ [M], where rı = bj if j ∈ Iı. NDD is assumed, thus dimH νq = hq/χ1q =

sHχ1q/χ1

q = sH.

2.7. Examples 53

Proof of (ii)⇒ (i). We can use Claim 2.6.1 and (2.3.8) to see that

0 ≤ dimB Λ− dimH Λ ≤ dimB Λ− dimH νp =(

1− χ1p/χ2

p

) (sH − dimH νq

).

Clearly, (ii) implies dimH Λ = dimB Λ. This concludes the proof of Theorem 2.2.8.

2.7 Examples

We now treat the examples presented in Subsection 1.4.1 in detail.We do not calculate numerically the exact value of the dimensions for the TGL

carpet of Figure 1.6, rather just comment why dimH Λ < dimB Λ < dimAff Λ. Itsatisfies the ROSC, thus its dimensions are equal to its GL brother. Clearly, the IFSson [0, 1] generated from a vertical line in each of the columns do not have the samedimension. Hence, the third condition of (2.2.7) of Theorem 2.2.8 does not hold. Fur-thermore, dimB ΛH < 1 because there is an empty column. Thus, Corollary 2.2.5implies that dimB Λ < dimAff Λ.

Except for the "X ≡ X" example, all the other ones of Subsection 1.4.1 satisfyΛH = [0, 1], hence Corollary 2.2.5 implies dimB Λ = dimAff Λ.

2.7.1 The self-affine smiley: a non diagonally homogeneous example

The smiley is constructed from the TGL IFS

F =

fi(x) =

(b 0di ai

)x + ti

8

i=1,

where b = 0.2, a1 = . . . = a5 = 0.1, a6 = a7 = a8 = 0.13 and the off-diagonalelements d1 = −0.2, d2 = −0.1, d3 = d7 = d8 = 0, d4 = 0.1, d5 = d6 = 0.2. The trans-lations were chosen so that the mouth is constructed from f1, . . . , f5, the nose from f6and the eyes from f7 and f8. It is non diagonally homogeneous since the mouth isthinner than the nose and eyes. Clearly, Λ does not have uniform vertical fibres, thusTheorem 2.2.8 implies dimH Λ < dimB Λ. The numerical values of the dimensionsgiven in Figure 1.7 were obtained using Wolfram Mathematica 11.2. The box dimen-sion was calculated from ∑N

i=1 bsHi as−sH

i = 1, recall (2.2.4), while the maximization ofD(p) (2.2.1) gave the Hausdorff dimension.

2.7.2 Example for dimH Λ = dimB Λ

Define the matrices

A1 :=(

1/3 00 a

), A2 :=

(1/3 0

1/2− a a

), A3 :=

(1/3 0

a− 1/2 a

).

For a ∈ (0, 1/3) define the IFS Fa consisting of

f1(x) = A1x +

(1/3

0

), f2(x) = A1x +

(1/3

1− a

), f3(x) = A2x +

(0

1/2

),

f4(x) = A2x +

(2/3

0

), f5(x) = A3x +

(0

1/2− a

), f6(x) = A3x +

(2/3

1− a

).

The attractor Λa is shown in Figure 1.8 for a = 3/10. Falconer and Miao showedin [FM07] how to calculate the box dimension and later Bárány in [B15] showed that


the same value is a lower bound for the Hausdorff dimension. Hence, dimH Λa =dimB Λa.

Alternatively, we can now argue that Λa is a diagonally homogeneous TGL carpetfor every a ∈ (0, 1/3) satisfying ROSC with uniform vertical fibres. Hence, our resultsapply. After some basic arithmetic, the dimension formula simplifies to

dimH Λa = dimB Λa = 1− log 2log a

. (2.7.1)

2.7.3 Overlapping example

With a modification of the translation vectors in the previous example, we constructa carpet with overlapping cylinders, see Figure 1.8. Define

f1(x) = A1x +

(1/31/4

), f2(x) = A1x +

(1/3

3/4− a

), f3(x) = A2x +

(0

1/4

),

f4(x) = A2x +

(2/31/4

), f5(x) = A3x +

(0

3/4− a

), f6(x) = A3x +

(2/3

3/4− a

),

where the matrices A1, A2 and A3 are from Subsection 2.7.2. For a ∈ (0, 1/3) theattractor Λa is a diagonally homogeneous TGL carpet with uniform vertical fibresand non-overlapping columns. Transversality must be satisfied in order to apply ourresults. It would suffice to check (2.1.16) in Lemma 2.1.8, but in fact the constant K1in Definition 2.1.6 of transversality can be directly bounded in this example.

Claim 2.7.1. Transversality holds for every a < 1/6 with

K1 <1/9− a/3

(1/2− a)(1/3− 2a).

Proof. For brevity we write d := 1/2 − a and b = 1/3. Let ı and be two wordsof length n such that i1 6= j1 and φ(i1) . . . φ(in) = φ(j1) . . . φ(jn). Since Rı ∩ R 6= ∅and due to the symmetry in the construction, we may assume i1 = 3 and j1 = 5,hence di1 = d. A simple geometric exercise gives that K1 ≤ (minı tan γı)−1, wheretan γı = dı/bı. We need a lower bound for tan γı. From (2.1.11) we get that

tan γi1 ...in =di1 ...in

bi1 ...in

=1a

n

∑`=1

di`

( ab

)`=

1a

(dab

+n

∑`=2

di`

( ab

)`)

.

This is minimal if di` = −d for every ` ≥ 2. Thus, we obtain the lower bound

tan γi1 ...in ≥db

(1−

n

∑`=2

( ab

)`−1)≥ d

b

(1− a/b

1− a/b

)=

d(b− 2a)b(b− a)

.

This remains positive iff a < b/2 = 1/6. Substituting d and b gives the bound forK1.

Corollary 2.7.2. For every a < 1/6 : dimH Λa = dimB Λa = 1− log 2/ log a.

Proof. For uniform vertical fibres both conditions (2.2.2) and (2.2.5) simplify to

log alog b

>log Nlog M

, which is satisfied here iff a ∈ (0, 1/6).

2.7. Examples 55

Thus, Claim 2.7.1 and Corollary 2.2.12 together imply that for every a ∈ (0, 1/6) wehave dimH Λa = dimB Λa = 1− log 2/ log a.

2.7.4 Example "X ≡ X"

This diagonally homogeneous carpet, recall Figure 1.10, is a modification of the pre-vious from Subsection 2.7.3 in order to show an overlapping example for which alldimensions are different. Indeed, clearly it does not have uniform vertical fibres andthere are empty columns.

The main diagonal of each matrix in the TGL IFS is bi ≡ b = 0.28 and ai ≡ a. Theoff-diagonal elements are either di = ±(1/2− a) or 0. The translation vectors werechosen so that Λa is symmetric on both lines x = 1/2 and y = 1/2. In Figure 1.10a = 0.045.

Transversality for the system can be checked the same way as in Claim 2.7.1, toobtain that transversality holds for every a < b/2 = 0.14 with

K1 <0.28(0.28− a)

(1/2− a)(0.28− 2a).

Corollary 2.7.3. We have dimH Λa < dimB Λa < dimAff Λa, where

dimH Λa = 0.78556 · log(

2 · 21.27297− log a + 3

1.27297− log a

), for every a < 0.10405 . . . ,

dimB Λa =0.84730− log a

+ 0.86303, for every a < 0.10254 . . . ,

dimAff Λa = 1 +0.67294− log a

, for every a < 0.28 .

Proof. The formulas are applications of the ones in Corollary 2.2.12 and (2.2.3). Theaffinity dimension is independent of overlaps. The bound for a in case of the Haus-dorff dimension was obtained using Proposition 2.2.10. The value x0 = 0.56255 . . . forwhich R(x0) = 1 was calculated using Wolfram Mathematica 11.2. Then (2.2.2) holdsfor every a < b1/x0 = 0.10405 . . . . The bound on a for the box dimension simplycomes from substituting the parameters into the second inequality in (2.2.9).

2.7.5 Negative entries in the main diagonal

Throughout we assumed that 0 < ai < bi < 1. We now comment on letting ai orbi < 0. For convenience, assume ROSC and non-overlapping columns.

Proposition 2.7.4. The dimension results of Theorems 2.2.2 and 2.2.4 extend to TGL carpetssatisfying the ROSC under the weaker condition that 0 < |ai| < |bi| < 1 and for every fixedı ∈ 1, . . . , M and every k, ` ∈ Iı : bk = b`.

Sketch of proof. All lower triangular matrix can be written(

bi 0di ai

)=

(|bi| 0di |ai|

)· L

where di = di or −di and L is a reflection on one or both of the coordinate axis. SinceL([−1, 1]2) = [−1, 1]2, such compositions fit into the framework of Fraser’s box-likesets [Fra12]. Furthermore, the direction-x dominates property is preserved. Hence,the proof of the box dimension from Section 2.6 immediately extends to this setting.


The lower bound for the Hausdorff dimension follows from Bárány–Käenmäki[BK17] cited in Theorem 2.3.4. Since in any given column all bi have the same signand we have ROSC, the column structure is preserved for every level. Thus, thedimension of the projected measure νq is not affected by the negative ai, bi. For theupper bound, we can modify the metric defined on Σ in Lemma 2.4.1 to be

d(i, j) :=|ı∧ |∏k=1|bik |+

|i∧j|∏k=1|aik |.

One can easily check that d(i, j) is indeed a metric and the natural projection Π : Σ→Λ is Lipschitz. Only the lengths of the sides of a parallelogram are important, itsorientation is not. The Bernoulli measure defined in (2.4.3) can be modified by againputting ai and bi in absolute value. The original proof of Gatzouras and Lalley [GL92]does not use that ai, bi > 0, only that 0 < |ai| < |bi| < 1.

In general, if a column has bi of different signs, then the initial column structurecan easily be destroyed. This is true even if |bi| ≡ b and possibly empty columnsalso have width b, see Figure 2.5. This motivates us to call a TGL carpet symmetricif Nı = NM−ı+1 for ı = 1, . . . , bM/2c (empty columns are allowed) and |bi| ≡ 1/M.For a particular symmetric carpet, in the next subsection, we show that the dimensionformulas hold.

FIGURE 2.5: Orientation reversing maps generally destroy the columnstructure. First and second level cylinders of the horizontal IFS H areshown, arrows indicating the orientation. Left: different |bi|, right:

equal |bi| and gap as well.

2.7.6 A family of self-affine continuous curves

Let a ∈ (0, 1/5] and d = (1− 5a)/4. Define the matrices

A =

(1/3 0

d a

)and A− =

(−1/3 00 a

).

A− is orientation reversing. We introduce the parameterized family of IFSs Fa givenby the functions

f1(x) = Ax, f2(x) = Ax +

(1/3

a + d

), f3(x) = A−x +

(2/3

2(a + d)

),

f4(x) = Ax +

(1/3

3a + 2d

), f5(x) = Ax +

(2/3

4a + 3d

).

The translation vectors are chosen so that f1(0) = 0, f5((1, 1)) = (1, 1) and fi((1, 1)) =fi+1(0). This ensures that Λa is a continuous curve in R2, see Figure 1.11. Curves sat-isfying this property are also called affine zippers in the literature, see for example[ATK03; BKK18]. Clearly, the attractor Λa is a symmetric, diagonally homogeneousTGL carpet satisfying the ROSC for every value of a. For a = 1/5 all cylinders Ri|n are

2.7. Examples 57

rectangles, however it is not a classical Bedford-McMullen carpet, since A− containsa negative element.

Proposition 2.7.5. For every a ∈ (0, 1/5], the Hausdorff and box dimension of Λa are givenby the continuous, strictly increasing functions

1log 3

· log(

2 + 3log 3− log a

)= dimH Λa < dimB Λa = 1 +

log(3/5)log a

.

Proof. A− can be written as the composition of the reflection on the vertical axis withthe diagonal matrix Diag(1/3, a). Hence, the proof of the box dimension carries overwithout difficulty.

The argument for the Hausdorff dimension follows that in Proposition 2.7.4, withan extra argument why the dimension of νq is not affected by A−.

The symbolic space Σ = 1, . . . , 5N codes the IFS Fa on [0, 1]2 and Ha on [0, 1](recall (2.1.2)). Fix a p = (p1, . . . , p5) ∈ P . Due to the symmetry and diagonallyhomogeneous property we may assume that p1 = p5. Let µp be the Bernoulli measureon Σ and νp = Π∗µp its push forward. Define the IFS Ha := hi(x) = x/3 + (i −1)/3, i = 1, 2, 3, which is coded by ΣH = 1′, 2′, 3′N. The map φ : 1, . . . , 5 →1′, 2′, 3′ is defined

φ(1) = 1′, φ(2) = φ(3) = φ(4) = 2′, φ(5) = 3′.

For ı = i1 . . . in ∈ 1′, 2′, 3′n let us denote Jk(ı) := j : ij = k, j ≤ |ı|, #k(ı) := |Jk(ı)|and define νq := (projx)∗νp. We claim that

νq(hı([0, 1])) = p#1′ (ı)+#3′ (ı)1 · (p2 + p3 + p4)

#2′ (ı), (2.7.2)

i.e. νq is the push forward (ΠH)∗µq of the Bernoulli measure µq on ΣH defined by thevector q = (q1, q2, q3) = (p1, p2 + p3 + p4, p5). This implies that

dimH νq =log〈q〉q− log 3

.

To see (2.7.2), choose an arbitrary ı ∈ 1′, 2′, 3′, n. We determine those ı ∈ 1, . . . , 5n

for which projx f ı([0, 1]2) = hı([0, 1]). For indices j ∈ J2′(ı) we can choose 2, 3 or 4 in ı.Let J3

` (ı) := j : ıj = 3, j ≤ ` ≤ |ı| ⊆ J2′(ı). Orientation is reversed at each j ∈ J3` (ı).

|J3` (ı)| uniquely determines ı` if i` = 1′ or 3′. Namely, whenever

|J3` (ı)| is

odd, if i` = 1′ then necessarily ı` = 5 and if i` = 3′ then ı` = 1;even, if i` = 1′ then necessarily ı` = 1 and if i` = 3′ then ı` = 5.

For indices j ∈ J2′(ı) \ J3|ı|(ı) we can freely choose ıj = 2 or 4. These are precisely the ı

for which projx f ı([0, 1]2) = hı([0, 1]). Using that p1 = p5, the measure equals

νq(hı([0, 1])) = p#1(ı)+#5(ı)1

(#2′(ı)#3(ı)

)p#3(ı)

3

(#2′(ı)− #3(ı)

#2(ı)

)p#2(ı)

2 · p#4(ı)4 ,

which after two applications of the binomial theorem yields (2.7.2).Finally, we conclude that dimH Λa < dimB Λa since Λa does not have uniform

vertical fibres.


2.8 Three-dimensional applications

We can compute the Hausdorff dimension of some self-affine carpets in R3. We donot aim for full generality, rather just demonstrate how our results can be applied.Throughout this section we always use the following definitions:

Definition 2.8.1. Let F be a TGL carpet on [0, 1]2 of the form (2.1.1), that is

F = fi(x) := Ai · x + tiNi=1, where Ai =

(bi 0di ai

)and ti =

(ti,1ti,2

), x ∈ [0, 1]2.

Furthermore, let the vectors u = (u1, . . . , uN), v = (v1, . . . , vN), λλλ = (λ1, . . . , pλN) besuch that for every 1 ≤ i ≤ N

ui, vi ∈ R and λi ∈ (−1, 1) \ 0 .

We say that the three dimensional self-affine IFS

F :=

Fi(x) := Ai · x + ti

N

i=1, where Ai =

bi 0 0di ai 0ui vi λi

, ti :=

ti,1ti,2ti,3

on [0, 1]3 is an uplift of F corresponding to (u, v, λλλ) if the following conditions hold:

(C1) For all 1 ≤ i ≤ N we have

0 < |λi| < ai < bi < 1. (2.8.1)

(C2) F satisfies the ROSC (see Definition 2.1.3).

Let Λ and Λ be the attractor of F and F respectively. We write Π and Π for the naturalprojection from Σ := 1, . . . , NN to Λ and Λ respectively. For a probability vector p :=(p1, . . . , pN) we set νp := Π∗(pN) and νp := Π∗(pN)

We obtain as a corollary of [BK17, Theorem 2.3, Proposition 5.8 and Proposition5.9] that

Corollary 2.8.2 (Bárány, Käenmäki). Assume that for an uplift F of F we have u = v = 0and all components of λλλ are equal to the same λ. Moreover, assume that for a probabilityvector p = (p1, . . . , pN) we have hp < χ1

p + χ2p (i.e. the entropy is less than the sum of the

Lyapunov exponents). Then dimH νp = dimH νp.

That is, the computation of the Hausdorff dimension of a Bernoulli measure forthe three-dimensional non-overlapping system F is traced back to the correspondingtwo-dimensional possibly overlapping system F . In this way, if F satisfies the con-ditions of Theorem 2.2.2 then we can determine dimH(νp) for the three-dimensionalsystem.

In general, we cannot approximate the Hausdorff dimension of a self-affine set inR3 by the Hausdorff dimension of self-affine (or even ergodic) measures (see [DS17,Theorem 2.8]). However, this is possible in some special cases.

Theorem 2.8.3. Given a diagonally homogeneous TGL of the form

F = fi(x) := A · x + tiNi=1, where A =

(b 0di a

)and ti =

(ti,1ti,2

), x ∈ [0, 1]2,

we assume that

2.8. Three-dimensional applications 59

(i) F has uniform vertical fibres (i.e. each column has the same number of maps).

(ii) The projection of Λ to the x-axis is the whole interval [0, 1] (this means that 1/b is equalto the number of columns M). We assume this to guarantee that the box and affinitydimensions of Λ coincide (see Corollary 2.2.5).

(iii) Moreover, we assume that the parameter a is sufficiently small so that both conditions(2.2.10), (2.1.16) and the transversality condition hold:

a < min

blog Nlog M ,

bd∗2 + d∗

, (2.8.2)

where d∗ was defined in Lemma 2.1.8 as

d∗ := min1≤ ≤MP 6=∅

min(k,`)∈P

|dk − d`|,

where (k, `) ∈ P if fk([0, 1]2) and f`([0, 1]2) belong to the same column and havedisjoint interior.

We consider the self-affine IFS F which is an uplift of F corresponding to (u, v, λλλ) accordingto Definition 2.8.1. That is (2.8.1) holds and u, v and λλλ are chosen such that

F :=

Fi(x) := Ai · x + tiN

i=1, where Ai =

b 0 0di a 0ui vi λi

, ti :=

ti,1ti,2ti,3

, x ∈ [0, 1]3

satisfies:

• Fi([0, 1]3

)⊂ [0, 1]3 holds for all i ∈ 1, . . . , N and

• the set Fi([0, 1]3

)∩ Fj

([0, 1]3

)has empty interior for all i 6= j ∈ 1, . . . , N.

Let p :=(

1/N, . . . , 1/N︸︷︷︸N

). Using the notation of Definition 2.8.1 we have

dimH νp = dimH Λ = dimB Λ = dimAff Λ = 1 +log(Nb)− log a

. (2.8.3)

To give the upper bound in the proof of this theorem, first we need to extend thescope of Lemma 2.1.5 to R3.

Lemma 2.8.4. There exists Kx, Ky and Kz such that for an arbitrary n and (i1, . . . , in) ∈(1, . . . , N)n we have

Ai1 ...in ≤

bn 0 0Kx · bn an 0Ky · bn Kz · bn λi1...in

,

that is all the elements of the matrix on the right-hand side are greater than or equal to thecorresponding element on the left-hand side.

Proof. For every n and (i1, . . . , in) ∈ (1, . . . , N)n we introduce xi1 ...in , yi1 ...in and zi1...in

such that

Ai1 ...in =

bn 0 0xi1...in · bn an 0yi1 ...in · bn zi1...in · bn λi1 ...in

.


Since the existence of Kx was proved in Lemma 2.1.5, it suffices to prove that yi1 ...in

and zi1...in are uniformly bounded in (i1, . . . , in) ∈ Σ∗. To do so, observe that

zi1 ...in+1 = zi1 ...in

ab+

λi1 ...in

bn

vin+1

b, (2.8.4)

yi1 ...in+1 = yi1 ...in + zi1...in

din+1

b+

λi1 ...in

bn

uin+1

b. (2.8.5)

By (2.8.1) we obtain from (2.8.4) that there is an r ∈ (0, 1) and c > 0 such that

zi1 ...in < c · rn for all n and (i1, . . . , in) ∈ 1, . . . , Nn . (2.8.6)

Namely, we can write down the formula for zi1...in inductively and thus we get that

zi1...in ≤ (a/b)n ·maxi

vi + n maxivi

b

. From here we get that (2.8.6) holds. This

settles the existence of Kz. Substituting (2.8.6) into (2.8.5) and using (2.8.1) again weobtain the existence of Ky. Namely, the second and third summands in (2.8.5) areexponentially small. More precisely,

Ky = max ui+∞

∑n=1

(c · rn · max di

b+

(max |λi|

b

)n

· max uib

),

where all of the maximums are taken for i ∈ 1, . . . , N.

Proof of Theorem 2.8.3.

Lower bound Observe that if condition (2.8.2) holds then it follows from Lemma 2.1.8that the transversality condition holds. Moreover, as we noted in Section 2.2.3,condition (2.8.2) also implies that conditions (2.2.2) and (2.2.5) hold when p ischosen as above to be the uniform vector. In this way the conditions of Theo-rems 2.2.2 and 2.2.7 are satisfied. As an application of these theorems, we obtainthat

dimH νp =log N− log a

+

(1− log b

log a

)log M− log b

= 1 +log(Nb)− log a

.

This implies that

1 +log(Nb)− log a

< dimH νp ≤ dimH νp ≤ dimH Λ.

Upper bound It is enough to prove that

dimAff Λ ≤ 1 +log(Nb)− log a

. (2.8.7)

This follows from Lemma 2.8.4 since the cylinder Fi1,...in

([0, 1]3

)can be covered

by Nn · bn/an axes parallel rectangular box of dimensions an × Kx · an × (Ky +Kz) · an. This immediately implies that (2.8.7) holds.

2.8. Three-dimensional applications 61

Example 1. Recall the attractor in the center of Figure 1.12. It is defined by an IFS

F =

Fi(x) := Ai · x + ti6

i=1, where for 0 < λ < a < 1/3

A1 = A5 =

1/3 0 01− a a 01− λ 0 λ

, A2 = A6 =

1/3 0 0a− 1 a 0

0 0 λ

, A3 = A4 =

1/3 0 00 a 0

λ− 1 0 λ

.

The translations are chosen appropriately so that F satisfies the ROSC and the projection tothe xy-plane looks like the one on the right-hand side of Figure 1.12. If λ < a < 1/6, thenthe conditions of Theorem 2.8.3 hold and we have from (2.8.3) that for p = (1/6, . . . , 1/6)

dimH νp = dimH Λ = dimB Λ = dimAff Λ = 1− log 2log a

.

Open problems

We plan to study the appropriate dimensional Hausdorff measure of planar carpets.This is always more difficult than determining the actual dimension. Determiningwhether it is 0, infinite or positive and finite is the goal. Not much is known aboutthis for planar carpets.

It is known for Gatzouras–Lalley carpets that if the Hausdorff and box dimensionare equal, then the Hausdorff measure is positive and finite [GL92]. Moreover, if theyare not equal, then Peres showed for Bedford–McMullen carpets that the Hausdorffmeasure is infinite and the set is not σ-finite with respect to the Hausdorff measure.Natural questions can be the following

1. Can the proof of Peres be adapted in order to extend the result to the moregeneral (triangular) Gatzouras–Lalley case?

2. When the measure is infinite, we do not expect overlaps to cause the value ofthe measure to drop to a finite value. However, when the Hausdorff and boxdimension are equal, and thus the measure is positive and finite, then can over-laps cause the Hausdorff measure to drop to zero? A similar phenomena wasproved for solenoids by Rams and Simon [RS03]. We plan to first look at simpleexamples and then see to what extent can those findings be generalized.

63

Chapter 3

Pointwise regularity ofparameterized affine zipper fractalcurves

This chapter is based on the article [BKK18] written jointly with Balázs Bárány andGergely Kiss.

3.1 Self-affine zippers satisfying dominated splitting

Let us begin by defining fractal curves generated by zippers. To the best of ourknowledge, the terminology and definition of a zipper in this generality is due toAseev, Tetenov and Kravchenko [ATK03]. However, special cases already appear inthe works of Hutchinson [Hut81] and Barnsley [Bar86].

Definition 3.1.1. A system F = f0, . . . , fN−1 of contracting mappings of Rd to itselfis called a zipper with vertices Z = z0, . . . , zN and signature ε = (ε0, . . . , εN−1), ε i ∈0, 1, if the cross-condition

fi(z0) = zi+εi and fi(zN) = zi+1−εi

holds for every i = 0, . . . , N − 1. We call the system a self-affine zipper if the functions fiare affine contractive mappings of the form

fi(x) = Aix + ti, for every i ∈ 0, 1, . . . , N − 1,

where Ai ∈ Rd×d invertible and ti ∈ Rd.The fractal curve generated from F is the unique non-empty compact set Γ, for which

Γ =N−1⋃

i=0

fi(Γ).

If F is an affine zipper then we call Γ a self-affine curve.

For an illustration see Figure 3.1. It shows the first (red), second (green) and third(black) level cylinders of the image of [0, 1]2. The cross-condition ensures that Γ is acontinuous curve.

The dimension theory of self-affine curves is far from being well understood.The Hausdorff dimension of such curves is known only in a very few cases. The

64 Chapter 3. Pointwise regularity of parameterized affine zipper fractal curves

FIGURE 3.1: An affine zipper with N = 3 maps and signature ε =(0, 1, 0).

usual techniques, like self-affine transversality, see Falconer [Fal88b], Jordan, Polli-cott and Simon [JPS07], destroys the curve structure. Ledrappier [Led92] gave a suf-ficient condition to calculate the Hausdorff dimension of some fractal curves, andSolomyak [Sol98b] applied it to calculate the dimension of the graph of the Takagifunction for typical parameters. Feng and Käenmäki [FK18] characterized self-affinesystems, which have analytic curve attractor. Bandt and Kravchenko [BK11] studiedsome smoothness properties of self-affine curves, especially the tangent lines of pla-nar self-affine curves. In Subsection 2.7.6 we used the framework of the TGL carpetsof Chapter 2 to calculate the different values of the Hausdorff and box dimension ofa whole family of self-affine zippers.

In this chapter we use the same notation as in Chapter 2 for the various dimen-sions of sets and measures. Let us recall the definition of pointwise Hölder exponentof a real valued function g, see for example [Jaf97a, eq. (1.1)]. We say that g ∈ Cβ(x)if there exist a δ > 0, C > 0 and a polynomial P with degree at most bβc such that

|g(y)− P(y− x)| ≤ C|x− y|β for every y ∈ Bδ(x),

where Bδ(x) denotes the ball with radius δ centered at x. Let αp(x) = supβ : g ∈Cβ(x). We call αp(x) the pointwise Hölder exponent of g at the point x.

In this chapter, we study the local regularity of a generalized version of self-similarfunctions F, recall (1.4.1). Namely, let λ = (λ0, . . . , λN−1) be a probability vector. Letus subdivide the interval [0, 1] according to the probability vector λ and signatureε = (ε0, . . . , εN−1), ε i ∈ 0, 1 of the zipper F . Let gi be the affine function mappingthe unit interval [0,1] to the ith subinterval of the division which is order-preserving ororder-reversing according to the signature ε i. That is, the interval [0, 1] is the attractorof the iterated function system

G = gi : x 7→ (−1)εi λix + γiN−1i=0 , (3.1.1)

where γi = ∑i−1j=0 λj + ε iλi. Let S = SiN−1

i=0 be an IFS on Rd+1 such that

Si(x, y) = (gi(x), Aiy + ti).

It is easy to see that if Λ is the attractor of S then for every x ∈ [0, 1] there exists aunique y ∈ Rd such that (x, y) ∈ Λ. Thus, we can define a function v : [0, 1] 7→ Γ ⊂ Rd

3.1. Self-affine zippers satisfying dominated splitting 65

such that v(x) = y if (x, y) ∈ Λ. The function v satisfies the functional equation

v(x) = fi

(v(g−1

i (x)))

if x ∈ gi([0, 1]). (3.1.2)

We note that g−1i (x) = x−γi

(−1)εi λi, and g−1

i (x) ∈ [0, 1] if and only if x ∈ gi([0, 1]). More-

over, if fiN−1i=0 is a self-affine zipper then v is continuous. We call v as the linear

parametrization of Γ. Such a particular example is de Rham’s curve, introduced in(1.2.3), see also Section 3.6 for details.

As a slight abuse of the appellation of the pointwise Hölder exponent, we use anotherexponent α(x) of the function v at a point x ∈ [0, 1]

α(x) = lim infy→x

log ‖v(x)− v(y)‖log |x− y| . (3.1.3)

We note that if αp(x) < 1 or α(x) < 1 then αp(x) = α(x). Otherwise, we have onlyα(x) ≤ αp(x).

When the lim inf in (3.1.3) exists as a limit, then we say that v has a regular pointwiseHölder exponent αr(x) at a point x ∈ [0, 1], i.e.

αr(x) = limy→x

log ‖v(x)− v(y)‖log |x− y| . (3.1.4)

Let us define the level sets of the (regular) pointwise Hölder exponent by

E(β) = x ∈ [0, 1] : α(x) = β and Er(β) = x ∈ [0, 1] : αr(x) = β . (3.1.5)

Our goal is to perform multifractal analysis, i.e. to study the maps

β 7→ dimH E(β) and β 7→ dimH Er(β).

Dominated splitting

Let us denote by Mo the interior and by M the closure of a set M ⊆ PRd−1. For apoint v ∈ Rd, denote by 〈v〉 the equivalence class of v in the projective space PRd−1.Every invertible matrix A defines a natural map on the projective space PRd−1 by〈v〉 7→ 〈Av〉. As a slight abuse of notation, we denote this function by A too.

Definition 3.1.2. We say that a family of matrices A = A0, . . . , AN−1 have dominatedsplitting of index-1 if there exists a non-empty open subset M ⊂ PRd−1 with a finitenumber of connected components with pairwise disjoint closure such that

N−1⋃

i=0

Ai M ⊂ Mo,

and there is a d− 1 dimensional hyperplane that is transverse to all elements of M. We callthe set M a multicone.

We adapted the definition of dominated splitting of index-1 from the paper ofBochi and Gourmelon [BG09]. They showed that the tuple of matrices A satisfies theproperty in Definition 3.1.2 if and only if there exist constants C > 0 and 0 < τ < 1such that

α2(Ai1 · · · Ain)

α1(Ai1 · · · Ain)≤ Cτn


for every n ≥ 1 and i0, . . . , in−1 ∈ 0, . . . , N − 1, where αi(A) denotes the ith largestsingular value of the matrix A. That is, the weakest contracting direction and thestronger contracting directions are strongly separated away (splitted), α1 dominatesα2. This condition makes it easier to handle the growth rate of the norm of matrixproducts, which will be essential in our later studies.

We note that for example a tuple A formed by matrices with strictly positive el-ements, satisfies the dominated splitting of index-1 of M = 〈x ∈ Rd : xi > 0, i =1, . . . , d〉. Throughout the paper we work with affine zippers, where we assume thatthe matrices Ai have dominated splitting of index-1. For more details, see Section 3.3and [BG09].

For a subset M of PRd−1 and a point x ∈ Rd, let

M(x) = y ∈ Rd : 〈y− x〉 ∈ M. (3.1.6)

We call the set M(x) a cone centered at x.We callF a non-degenerate system, if it satisfies the SOSC, recall Definition 1.2.1 and

〈zN − z0〉 /∈∞⋂

k=0

⋂

|ı|=k

A−1ı (Mc), (3.1.7)

where for a finite length word ı = i1 . . . ik, Aı denotes the matrix product Ai1 Ai2 . . . Aik .We note that the non-degenerate condition guarantees that the curve v : [0, 1] 7→ Rd

is not self-intersecting and it is not contained in a strict hyperplane of Rd.

3.2 Main results

The key technical tool for our work is the matrix pressure function. Denote by P(t) thepressure function which is defined as the unique root of the equation

0 = limn→∞

1n

logN−1

∑i1,...,in=0

‖Ai1 · · · Ain‖t (λi1 · · · λin)−P(t) . (3.2.1)

A considerable attention has been paid for pressures, which are defined by ma-trix norms, see for example Käenmäki [K04], Feng and Shmerkin [FS14], and Morris[Mor16; Mor17]. Feng [Fen03] and later Feng and Lau [FL02] studied the properties ofthe pressure P for positive and non-negative matrices. In Section 3.3, we extend theseresults for the dominated splitting of index-1 case. Namely, we will show that thefunction P : R 7→ R is continuous, concave, monotone increasing, and continuouslydifferentiable.

Let d0 > 0 be the unique real number such that P(d0) = 0, i.e.

0 = limn→∞

1n

log ∑|ı|=n‖Aı‖d0 . (3.2.2)

Observe that for every n ≥ 1, fı(U) : |ı| = n defines a cover of Γ. But since Γ isa curve and thus dimH Γ ≥ 1, and since every fı(U) can be covered by a ball withradius ‖Aı‖|U| we have d0 ≥ 1.

Let

αmin = limt→+∞

P(t)t

, αmax = limt→−∞

P(t)t

and α = P′(0). (3.2.3)


The values αmin and αmax correspond to the logarithm of the joint- and the lower-spectral radius defined by Protasov [Pro06].

Now, we state our main theorems on the pointwise Hölder exponents.

Theorem 3.2.1. Let v be a linear parametrization of Γ defined by a non-degenerate system F .Then there exists a constant α, defined in (3.2.3), such that for L-a.e. x ∈ [0, 1], α(x) = α ≥1/d0. In addition, there exists an ε > 0 such that for every β ∈ [α, α + ε]

dimH x ∈ [0, 1] : α(x) = β = inft∈Rtβ− P(t). (3.2.4)

Moreover, (3.2.4) can be extended for every β ∈ [αmin, α + ε] if v satisfies

λ0 = λN−1 and limk→∞

‖Ak0‖

‖AkN−1‖

= 1. (3.2.5)

Furthermore, the functions β 7→ dimH E(β) and β 7→ dimH Er(β) are continuous andconcave on their respective domains.

In the following, we give a sufficient condition to extend the previous result,where (3.2.4) holds to the complete spectrum [αmin, αmax]. As a slight abuse of no-tation for every θ ∈ PRd−1, we say that 0 6= v ∈ θ if 〈v〉 = θ.

Assumption A. For a non-degenerate affine zipper F = fi : x 7→ Aix + tiN−1i=0 with

vertices z0, . . . , zN assume that there exists a convex, simply connected closed cone C ⊂PRd−1 such that

1.⋃N

i=1 AiC ⊂ Co and for every 0 6= v ∈ θ ∈ C, 〈Aiv, v〉 > 0,

2. 〈zN − z0〉 ∈ Co.

Observe that if F satisfies Assumption A then it satisfies the strong open set con-dition with respect to the set U, which is the bounded component of Co(z0)∩ Co(zN).We note that if all the matrices have strictly positive elements and the zipper has sig-nature (0, . . . , 0) then Assumption A holds.

Theorem 3.2.2. Let F be an affine zipper satisfying Assumption A. Then for every β ∈[α, αmax]

dimH x ∈ [0, 1] : α(x) = β = inft∈Rtβ− P(t), (3.2.6)

and for every β ∈ [αmin, αmax]

dimH x ∈ [0, 1] : αr(x) = β = inft∈Rtβ− P(t). (3.2.7)

Moreover, if F satisfies (3.2.5) then (3.2.6) can be extended for every β ∈ [αmin, αmax].The functions β 7→ dimH E(β) and β 7→ dimH Er(β) are continuous and concave on

their respective domains.

Assumption A has another important role. In Theorem 3.2.2, we calculated thespectrum for the regular Hölder exponent, providing that it exists. We show that theexistence of the regular Hölder exponent for Lebesgue typical points is equivalent toAssumption A.

Theorem 3.2.3. Let F be a non degenerate system. Then the regular Hölder exponent existsfor Lebesgue almost every point if and only ifF satisfies Assumption A. In particular, αr(x) =P′(0) for Lebesgue almost every x ∈ [0, 1].


Remark 3.2.4. In the sequel, to keep the notation tractable we assume the signature ε =(0, . . . , 0). The results carry over for general signatures, and the proofs can be easily modifiedfor the general signature case, see Remark 3.6.3.

The organization of the chapter is as follows. In Section 3.3 we prove severalproperties of the pressure function P(t), extending the works of [Fen03; FL02] to thedominated splitting of index-1 case using [BG09]. We prove Theorem 3.2.1 in Sec-tion 3.4. Section 3.5 contains the proofs of Theorems 3.2.2 and 3.2.3 when the zippersatisfies Assumption A. Finally, as an application in Section 3.6, we show that ourresults can be applied to de Rham’s curve, giving finer results than existing ones inthe literature.

3.3 Pressure for matrices with dominated splitting of index-1

In this section, we generalize the result of Feng [Fen03], and Feng and Lau [FL02]. In[FL02] the authors studied the pressure function and multifractal properties of Lya-punov exponents for products of positive matrices. Here, we extend their results for amore general class of matrices by using Bochi and Gourmelon [BG09] for later usage.

Let Σ be the set of one side infinite length words of symbols 0, . . . , N − 1, i.e.Σ = 0, . . . , N − 1N. Let σ denote the left shift on Σ, its n-fold composition byσni = (in+1, in+2, . . .). We use the standard notation i|n for i1, . . . , in and

[i|n] := j ∈ Σ : j1 = i1, . . . , jn = in .

Let us denote the set of finite length words by Σ∗ =⋃∞

n=00, . . . , N − 1n, and foran ı ∈ Σ∗, let us denote the length of ı by |ı|. For a finite word ı ∈ Σ∗ and for a j ∈ Σ,denote ıj the concatenation of the finite word ı with j.

Denote i ∧ j the length of the longest common prefix of i, j ∈ Σ, i.e. i ∧ j =minn− 1 : in 6= jn. Let λ = (λ0, . . . , λN−1) be a probability vector and let d(i, j) bethe distance on Σ with respect to λ. Namely,

d(i, j) =i∧j

∏n=1

λin =: λi|i∧j. (3.3.1)

If i ∧ j = 0 then by definition i|i∧j = ∅ and λ∅ = 1. In the sequel, whenever we useHausdorff dimension in Σ it is with respect to this metric d(i, j). For every r > 0, wedefine a partition Ξr of Σ by

Ξr =[i1, . . . , in] : λi1 · · · λin ≤ r < λi1 · · · λin−1

. (3.3.2)

For a matrix A and a subspace θ, denote ‖A|θ‖ the norm of A restricted to θ, i.e.‖A|θ‖ = supv∈θ ‖Av‖/‖v‖. In particular, if θ has dimension one ‖A|θ‖ = ‖Av‖/‖v‖for any 0 6= v ∈ θ. Denote G(d, k) the Grassmanian manifold of k dimensional sub-spaces of Rd. We define the angle between a 1 dimensional subspace E and a d− 1dimensional subspace F as usual, i.e.

^(E, F) = arccos( 〈v, projFv〉‖projFv‖‖v‖

),

where 0 6= v ∈ E arbitrary and projF denotes the orthogonal projection onto F.The following theorem collects the most relevant properties of a family of matrices

with dominated splitting of index-1.

3.3. Pressure for matrices with dominated splitting of index-1 69

Theorem 3.3.1. [BG09, Theorem A, Theorem B, Claim on p. 228] Suppose that a finite setof matrices A0, . . . , AN−1 satisfies the dominated splitting of index-1 with multicone M.Then there exist Hölder continuous functions E : Σ 7→ PRd−1 and F : Σ 7→ G(d, d − 1)such that

1. E(i) = Ai1 E(σi) for every i ∈ Σ,

2. F(i) = A−1i1

F(σi) for every i ∈ Σ,

3. there exists β > 0 such that ^(E(i), F(j)) > β for every i, j ∈ Σ,

4. there exist constants C ≥ 1 and 0 < τ < 1 such that

α2(Ai|n)

‖Ai|n‖≤ Cτn

for every i ∈ Σ and n ≥ 1,

5. there exists a constant C > 0 such that ‖Ai|n |E(σni)‖ ≥ C‖Ai|n‖ for every i ∈ Σ,

6. there exists a constant C > 0 such that ‖Ai|n |F(in . . . i1j)‖ ≤ Cα2(Ai|n) for everyi, j ∈ Σ.

There are a few simple consequences of Theorem 3.3.1. First, if M is the multiconefrom Definition 3.1.2, then by Theorem 3.3.1(1)

E(i) =∞⋂

n=1

Ai1 · · · Ain(M),

and for every V ∈ M, Ai1 · · · Ain V → E(i) uniformly (independently of V). Hence,by property (5), there exists a constant C′ > 0 such that for every V ∈ M and everyı ∈ Σ∗,

‖Aı|V‖ ≥ C′‖Aı‖. (3.3.3)

So, this gives us a strong control over the growth rate of matrix products on subspacesin M.

Remark 3.3.2. We note if the multicone M in Definition 3.1.2 has only one connected com-ponent then it can be chosen to be simply connected and convex. Indeed, since M is separatedaway from the strong stable subspaces F then cv(M) must be separated away from every d− 1dimensional strong stable subspace, as well, where cv(M) denotes the convex hull of M. ThusAi(cv(M)) ⊂ cv(M)o for every i.

Second, property (1) of Theorem 3.3.1 also implies that

‖Ai|n |E(σni)‖ =n

∏k=1‖Aik |E(σki)‖. (3.3.4)

Indeed, since E(i) is a one dimensional subspace, for every v ∈ E(σni)

‖Ai|n |E(σni)‖ = ‖Ai|n v‖‖v‖ =

n

∏k=1

‖Aik ...in v‖‖Aik+1,...,in v‖ =

n

∏k=1‖Aik |E(σki)‖.

Moreover, since E(i) is Hölder-continuous, the function ψ(i) := log ‖Ai1 |E(σi)‖ isalso Hölder-continuous. That is, there exist C > 0 and 0 < τ < 1 such that

|ψ(i)− ψ(j)| ≤ Cτi∧j. (3.3.5)


It is easy to see that (3.3.5) holds if and only if ψ is Hölder-continuous with respect tothe metric d defined in (3.3.1).

Finally, (3.3.5) implies that for every t, the potential function ϕt : Σ 7→ R definedby

ϕt(i) := log(‖Ai1 |E(σi)‖tλ

−P(t)i1

)= tψ(i)− P(t) log λi1 (3.3.6)

is Hölder-continuous w.r.t the metric d, where P(t) was defined in (3.2.1). Thus, by[Bow08, Theorem 1.4], for every t ∈ R there exists a unique σ-invariant, ergodicprobability measure µt on Σ such that there exists a constant C(t) > 1 such that forevery i ∈ Σ and every n ≥ 1

C(t)−1 ≤ µt([i|n])∏n−1

k=0 eϕt(σki)=

µt([i|n])‖Ai|n |E(σni)‖t · λ−P(t)

i|n

≤ C(t), (3.3.7)

where the equality follows from substituting (3.3.6) and (3.3.4).Moreover, for the Hausdorff dimension w.r.t. the metric d defined in (3.3.1)

dimH µt =hµt

χµt

, (3.3.8)

where

hµt = limn→∞

−1n ∑|ı|=n

µt([ı]) log µt([ı]) = −∫

ϕt(i)dµt(i), (3.3.9)

χµt = limn→∞

−1n ∑|ı|=n

µt([ı]) log λı = −∫

log λi1 dµt(i). (3.3.10)

We call χµt the Lyapunov exponent of µt and hµt the entropy of µt.

Lemma 3.3.3. The map t 7→ P(t) is continuous, concave, monotone increasing on R.

Proof. Since µt is a probability measure on Σ and Ξr is a partition we get

0 =log ∑ı∈Ξr

µt([ı])log r

for every r > 0

and by (3.3.7), Theorem 3.3.1(5) and (3.2.1)

P(t) = limr→0+

log ∑ı∈Ξr‖Aı‖t

log r. (3.3.11)

Using this form it can be easily seen that t 7→ P(t) is continuous, concave and mono-tone increasing.

By Lemma 3.3.3, the potential ϕt depends continuously on t. Moreover, by (3.3.5),|ϕt(i)− ϕt(j)| ≤ Ctτi∧j. Thus, the Perron-Frobenius operator

(Tt(g))(i) =N−1

∑i=0

eϕt(ii)g(ii)

depends continuously on t. Hence, both the unique eigenfunction ht of Tt and theeigenmeasure νt of the dual operator T∗t depend continuously on t. Since dµt = htdνt,see [Bow08, Theorem 1.16], we got that t 7→ µt is continuous in weak*-topology.Hence, by (3.3.9) and (3.3.10), t 7→ hµt and t 7→ χµt are continuous on R.

3.3. Pressure for matrices with dominated splitting of index-1 71

Proposition 3.3.4. The map t 7→ P(t) is continuously differentiable on R. Moreover, forevery t ∈ R

dimH µt = tP′(t)− P(t),

and

limn→∞

log ‖Ai1 · · · Ain‖log λi1 · · · λin

= P′(t) for µt-almost every i ∈ Σ.

Proof. We recall [Heu98, Theorem 3.1]. That is, since µt is a Gibbs measure

τµt(q) = limr→0+

log ∑ı∈Ξrµt([ı])q

log r

is differentiable at q = 1 and τ′µt(1) = dimH µt. On the other hand, by (3.3.7) and

(3.3.11)τµt(q) = P(tq)− P(t)q.

Hence, by taking the derivative at q = 1 we get that P(t) is differentiable for everyt ∈ R \ 0 and

dimH µt = tP′(t)− P(t).

Let us observe that by (3.3.6), (3.3.8) and (3.3.9)

dimH µt = t−∫

log ‖Ai1 |E(σi)‖dµt(i)−∫

log λi1 dµt(i)− P(t).

Thus,

P′(t) =−∫

log ‖Ai1 |E(σi)‖dµt(i)−∫

log λi1 dµt(i)for every t 6= 0.

Since t 7→ µt is continuous in weak*-topology we get that t 7→ P′(t) is continuous onR \ 0. On the other hand, the left and right hand side limits of P′(t) at t = 0 existand are equal. Thus, t 7→ P(t) is continuously differentiable on R.

By Theorem 3.3.1 (5), equation (3.3.4) and ergodicity of µt we get the last assertionof the proposition.

Let us observe that by the definition of pressure function (3.2.1), P(0) = −1 andthus, µ0 corresponds to the Bernoulli measure on Σ with probabilities (λ0, . . . , λN−1).That is,

µ0([i1, . . . , in]) = λi1 · · · λin .

Lemma 3.3.5. For every finite set of matricesA with dominated splitting of index-1, P′(0) ≥1/d0, P′(d0) ≤ 1/d0. Moreover, P′(0) > 1/d0 if and only if P′(d0) < 1/d0 if and only ifµd0 6= µ0.

Proof. By the definition of P(t), (3.2.1), P(d0) = 0, where d0 is defined in (3.2.2). To-gether with P(0) = −1 and the concavity and differentiability of P(t) (by Lemma 3.3.3and Proposition 3.3.4), we get P′(0) ≥ 1/d0, P′(d0) ≤ 1/d0. Moreover, P′(0) > 1/d0if and only if P′(d0) < 1/d0.

On the other hand, by Proposition 3.3.4

dimH µd0 = d0P′(d0) = limn→∞

log ‖Ai|n‖d0

log λi|n=

hµd0

χµd0

for µd0-a.e. i,

where in the last equation we used the definition of µd0 , the entropy and the Lyapunovexponent. Since dimH µ0 = 1, if P′(d0) < 1/d0 then µ0 6= µd0 . Otherwise, by [Bow08,


Theorem 1.22], for every σ-invariant, ergodic measure ν on Σ,

hν

−∫

log λi0 dν(i)≤ 1 and

hν

−∫

log λi0 dν(i)= 1 if and only if ν = µ0.

Therefore, if P′(d0) = 1/d0 thenhµd0χµd0

= 1 and so µd0 = µ0.

Lemma 3.3.6. For every α ∈ [αmin, αmax]

dimH

i ∈ Σ : lim inf

m→∞

log ‖Ai|m‖log λi|m

≤ α≤ inf

t≥0tα− P(t) (3.3.12)

and

dimH

i ∈ Σ : lim sup

m→∞


≥ α≤ inf

t≤0tα− P(t) (3.3.13)

Proof. For simplicity, we use the notations

Gα =

i ∈ Σ : lim inf

m→∞


≤ α

and Gα =

i ∈ Σ : lim sup

m→∞


≥ α

.

Let ε > 0 be arbitrary but fixed and let us define the following sets of cylinders:

Dr(ε) =[ı] ∈ Ξρ : 0 < ρ ≤ r and

log ‖Aı‖log λı

≤ α + ε

and

Dr(ε) =[ı] ∈ Ξρ : 0 < ρ ≤ r and

log ‖Aı‖log λı

≥ α− ε

.

By definition, Dr(ε) is a cover of Gα and respectively, Dr(ε) is a cover of Gα. Now letCr(ε) and Cr(ε) be a disjoint set of cylinders such that

⋃

[ı]∈Dr(ε)

[ı] =⋃

[ı]∈Cr(ε)

[ı] and⋃

[ı]∈Dr(ε)

[ı] =⋃

[ı]∈Cr(ε)

[ı].

Then by (3.3.7) and the definition of Cr(ε), for any t ≥ 0

Hαt−P(t)+(1+t)εr (Gα) ≤ ∑

[ı]∈Cr(ε)

λ(αt−P(t)+(1+t)ε)ı

≤ λ−1minrε ∑

[ı]∈Cr(ε)

‖Aı‖tλ−P(t)ı

≤ Cλ−1minrε ∑

[ı]∈Cr(ε)

µt([ı]) ≤ Cλ−1minrε.

Hence, Hαt−P(t)+(1+t)ε(Gα) = 0 for any t > 0 and any ε > 0, so (3.3.12) follows. Theproof of (3.3.13) is similar by using the cover Cr(ε) of Gα.

We note that by the concavity of P

inft∈Rtα− P(t) = inf

t≤0tα− P(t) ,

3.4. Pointwise Hölder exponent for non-degenerate curves 73

for every and α ∈ [P′(0), αmax],

inft∈Rtα− P(t) = inf

t≥0tα− P(t) ,

for every α ∈ [αmin, P′(0)].

3.4 Pointwise Hölder exponent for non-degenerate curves

First, let us define the natural projections π and Π from the symbolic space Σ to theunit interval [0, 1] and the curve Γ. We recall that we assumed that all the signaturesof the affine zipper Definition 3.1.1 is 0, and all the matrices are invertible. Therefore,

π(i) = limn→∞

gi1 · · · gin(0) =∞

∑n=1

λi|n−1γin (3.4.1)

Π(i) = limn→∞

fi1 · · · fin(0) =∞

∑n=1

Ai|n−1tin . (3.4.2)

Observe that by the definition of the linear parametrization v of Γ, v(π(i)) = Π(i).In the analysis of the pointwise Hölder exponent α, defined in (3.1.3), the points

play important role which are far away symbolically but close on the self-affine curve.To be able to handle such points we introduce the following notation

i ∨ j =

minσi∧j+1i ∧ N − 1, σi∧j+1j ∧ 0, if ii∧j+1 + 1 = ji∧j+1,minσi∧j+1i ∧ 0, σi∧j+1j ∧ N − 1, if ji∧j+1 + 1 = ii∧j+1,0, otherwise,

where 0 denotes the (0, 0, . . . ) and N − 1 denotes the (N − 1, N − 1, . . . ) sequence. Itis easy to see that there exists a constant K > 0 such that

K−1(λi|i∧j+i∨j+ λj|i∧j+i∨j

) ≤ |π(i)− π(j)| ≤ K(λi|i∧j+i∨j+ λj|i∧j+i∨j

). (3.4.3)

Hence, the distance on [0, 1] is not comparable with the distance on the symbolicspace. More precisely, let T be the set of points on the symbolic space, which hastail 0 or N − 1, i.e. i ∈ T if and only if there exists a k ≥ 0 such that σki = 0 orσki = N − 1. So if π(σki) is too close to the set π(T) infinitely often then we lose thesymbolic control over the distance |π(i)− π(in)|, where in is such that π(in)→ π(i)as n→ ∞.

On the other hand, the symbolic control of the set ‖Π(i)−Π(in)‖ is also far non-trivial. In general, ‖Π(i)−Π(j)‖ = ‖Ai|i∧j

(Π(σi∧ji)−Π(σi∧jj))‖ is not comparableto ‖Ai|i∧j

‖ · ‖Π(σi∧ji)−Π(σi∧jj)‖, unless 〈Π(σi∧ji)−Π(σi∧jj)〉 ∈ M, where M is themulticone satisfying the Definition 3.1.2. Thus, in order to handle

lim infn→∞

log ‖Π(i)−Π(in)‖log |π(i)− π(in)|

we need that i is sufficiently far from the tail set T and also that the points Π(in) onΓ can be chosen such that 〈Π(σi∧ji) −Π(σi∧in in)〉 ∈ M. So we introduce a kind of


FIGURE 3.2: Local neighbourhood of points in Bn,l,m.

exceptional set B, where both of these requirements fail. We define B ⊆ Σ such that

B = i ∈ Σ : ∀ n ≥ 1, ∀ l ≥ 1, ∀m ≥ 1, ∃K ≥ 0, ∀ k ≥ K(

M(Π(σki)) \ B1/n(Π(σki)))∩ Γ \ (Γσki|l ∪ Γσki|l−1(ik+l−1)(N−1)m ∪ Γσki|l−1(ik+l+1)0m) = ∅

,

(3.4.4)

where Γı = fı(Γ) for any finite length word ı ∈ Σ∗ and M(Π(i)) is the cone centeredat Π(i). We note that if il = 0 (or il = N − 1) then we define Γσki|l−1(il−1)(N−1)m = ∅(or Γσki|l−1(il+1)0m = ∅ respectively).

In particular, B contains those points i, for which locally the curve Γ will leave thecone M very rapidly. In other words, let

Bn,l,m = i ∈ Σ :

(M(Π(i)) \ B1/n(Π(i))) ∩ Γ \ (Γi|l ∪ Γi|l−1(il−1)(N−1)m ∪ Γi|l−1(il+1)0m) = ∅

.

and

Bn,m,l,K =∞⋂

k=K

σ−kBn,l,m and B =∞⋂

n=1

∞⋂

l=0

∞⋂

m=0

∞⋃

K=0

Bn,l,m,K.

For a visualisation of the local neighbourhood of a point in Bn,l,m, see Figure 3.2.In particular, we are able to handle the pointwise Hölder exponents at π(i) outside ofthe set B and we show that B is small in some sense.

Lemma 3.4.1. Let us assume thatF is non-degenerate. Then there exist n ≥ 1, l ≥ 1, m ≥ 1and finite length word with || = l, such that

Bn,l,m ∩ [] = ∅.

Proof. Our first claim is that there exists a finite sequence ı such that 〈Aı(zN − z0)〉 ∈M. Suppose that this is not the case. That is, for every finite length word 〈Aı(zN −


z0)〉 ∈ Mc. Equivalently, for every finite length word ı, 〈zN − z0〉 ∈ A−1ı (Mc). Thus,

〈zN − z0〉 ∈⋂∞

k=0⋂|ı|=k A−1

ı (Mc), which contradicts to our non-degeneracy assump-tion.

Let us fix an ı such that 〈Aı(zN − z0)〉 ∈ M. Then fı(zN) ∈ M( fı(z0)). By continu-ity, one can choose k ≥ 1 large enough such that for every i ∈ [ı0k],

fı(zN) ∈ M(Π(i))

and

‖ fı(z0)−Π(i)‖ = ‖Aı0k(z0 −Π(σ|ı|+ki))‖ ≤ ‖Aı‖‖Ak0‖diam(Γ) ≤ 1

2‖Aı(zN − z0)‖,

where we used the fact that f0(z0) = z0. Then

‖Π(i)− fı(zN)‖ ≥ ‖Aı(zN − z0)‖ − ‖ fı(z0)−Π(i)‖ > 12‖Aı(zN − z0)‖.

We get that for every i ∈ [ı0k]

fı(zN) ∈(

M(Π(i)) \ B 12 ‖Aı(zN−z0)‖(Π(i))

)∩ Γ \ (Γı0k ∪ Γı0k−110 ∪ Γı||ı|−1(ı|ı|−1)N) 6= ∅.

By fixing := ı0k, l := ||, m := 1 and n :=⌈

2‖Aı(zN−z0)‖

⌉, we see that Bn,l,m ∩ [] =

∅.

Proposition 3.4.2. Let us assume that F is non-degenerate. Then dimP π(B) < 1. More-over, for any ν fully supported ergodic measure on Σ, ν(B) = 0.

Proof. By definition, Bn,l,m ⊇ Bn+1,l,m, Bn,l,m ⊇ Bn,l+1,m and Bn,l,m ⊇ Bn,l,m+1. More-over, Bn,l,m,K = σ−KBn,l,m,0. In particular, σ−1Bn,l,m,0 = Bn,l,m,1 ⊇ Bn,l,m,0. Thus, forevery n ≥ 1

Bn,l,m,0 ⊆⋃

|ı|=q

$ı(Bn,l,m,0), (3.4.5)

where $ı(i) = ıi. Let n0 ≥ 1, l0 ≥ 1, m0 ≥ 1 be natural numbers and be the finitelength word with || = l0 as in Lemma 3.4.1, then

Bn0,l0,m0,0 ∩ [] =∞⋂

k=0

(σ−kBn0,m0,l0 ∩ []

)⊆ Bn0,m0,l0 ∩ [] = ∅.

Thus,Bn0,l0,m0,0 ⊆

⋃

|ı|=l0ı 6=

$ı(Bn0,l0,m0,0). (3.4.6)

Hence, σpi /∈ [] for every i ∈ Bn0,l0,m0,0 and for every p ≥ 1. Indeed, if there existsi ∈ Bn0,l0,m0,0 and p ≥ 1 such that σpi ∈ [] then there exist a finite length word ı with|ı| = p such that B ∩ [ı] 6= ∅. But by equations (3.4.5) and (3.4.6),

Bn0,l0,m0,0 ⊆⋃

|ı1|=p

$ı1(Bn0,l0,m0,0) ⊆⋃

|ı1|=p

⋃

|ı2 |=l0ı2 6=

$ı1($ı2(Bn0,l0,m0,0)) ⊆⋃

|ı1|=p

⋃

|ı2 |=l0ı2 6=

[ı1ı2]


which is a contradiction. But for any fully supported ergodic measure ν, ν([]) > 0and therefore ν(Bn0,l0,m0,0) = 0. The second statement of the lemma follows by

ν(B) ≤ infn,l,m

ν(∞⋃

K=0

Bn,l,m,K) ≤∞

∑K=0

ν(Bn0,l0,m0,K) =∞

∑K=0

ν(Bn0,l0,m0,0) = 0.

To prove the first assertion of the proposition, observe that by equation (3.4.6)

π(Bn0,l0,m0,0) ⊆⋃

|ı|=l0ı 6=

gı(π(Bn0,l0,m0,0)).

Therefore, π(Bn0,l0,m0,0) is contained in the attractor Λ of the IFS gı|ı|=l0ı 6=

, for which

dimB Λ < 1. Hence,

dimP π(B) ≤ infn,l,m

dimBπ(Bn,l,m,0) ≤ dimBπ(Bn0,l0,m0,0) ≤ dimB Λ < 1.

Lemma 3.4.3. Let us assume that F is non-degenerate. Then for every i ∈ Σ \ B

α(π(i)) ≤ lim supn→+∞

log ‖Ai|n‖log λi|n

.

Proof. Let i ∈ Σ \ B. Then there exist n ≥ 1, l ≥ 1, m ≥ 1 and a sequence

kp∞

p=1such that kp → ∞ as p→ ∞ and

(M(Π(σkp i)) \ B1/n(Π(σkp i))

)∩

Γ \ (Γσkp i|l ∪ Γσkp i|l−1(ikp+l−1)(N−1)m ∪ Γσkp i|l−1(ikp+l+1)0m) 6= ∅ (3.4.7)

Hence, there exists a sequence jp such that kp ≤ i ∧ jp ≤ kp + l, i ∨ jp ≤ m,

Π(σkp jp) ∈ M(Π(σkp i)) and ‖Π(σkp jp)−Π(σkp i)‖ > 1n

. (3.4.8)

Thus,

α(π(i)) = lim infπ(j)→π(i)

log ‖Π(i)−Π(j)‖log |π(i)− π(j)| ≤ lim inf

p→+∞

log ‖Π(i)−Π(jp)‖log |π(i)− π(jp)|

= lim infp→+∞

log ‖Ai|kp(Π(σkp i)−Π(σkp jp))‖

log |λi|i∧jp+i∨jp(π(σi∧jp+i∨jp i)− π(σi∧jp+i∨jp jp))|

,

and by (3.3.3), (3.4.8),

lim infp→+∞

log ‖Ai|kp(Π(σkp i)−Π(σkp jp))‖

log |λi|i∧jp+i∨jp(π(σi∧jp+i∨jp i)− π(σi∧jp+i∨jp jp))|

≤

lim infp→+∞

log(C−1/n) + log ‖Ai|kp‖

log λi|kp+ log d′

≤ lim supp→+∞

log ‖Ai|p‖log λi|p

,


where d′ = (maxi λi)m+l .

Lemma 3.4.4. Let us assume that F is non-degenerate. Then for every ergodic, σ-invariant,fully supported measure µ on Σ such that ∑∞

k=0(µ[0k] + µ([Nk]) is finite, then

α(π(i)) = limn→+∞


for µ-a.e. i ∈ Σ.

Proof. By Proposition 3.4.2, we have that µ(B) = 0. Thus, by Lemma 3.4.3, for µ-a.e. i

α(π(i)) ≤ limn→+∞


.

On the other hand, for every i ∈ Σ,


log ‖Π(i)−Π(j)‖log |π(i)− π(j)| ≥ lim inf

j→i

log ‖Ai|i∧j‖

log λi|i∧j+i∨j+ log mini λi

.

Hence, to verify the statement of the lemma, it is enough to show that

limj→i

log λi|i∧j

log λi|i∧j+i∨j

= 1 for µ-a.e. i.

It is easy to see that from limj→ii∨ji∧j = 0 follows the previous equation. Let

Rn =

i : ∃jk s. t. jk → i as k→ ∞ and lim

k→∞

i ∨ jk

i ∧ jk>

1n

In other words,

Rn =∞⋂

K=0

∞⋃

k=K

N⋃

i1,...,ik=0

[i1, . . . , ik,

bk/nc︷︸︸︷0, . . . , 0] ∪ [i1, . . . , ik,

bk/nc︷︸︸︷N, . . . , N]

Therefore, for any µ ergodic σ-invariant measure and for every K ≥ 0

µ(Rn) ≤∞

∑k=K

(µ([0bk/nc]) + µ([Nbk/nc])).

Since by assumption the sum on the right hand side is summable, we get µ(Rn) = 0for every n ≥ 1.

Lemma 3.4.5. Let us assume that F is non-degenerate and satisfies (3.2.5). Then for everyi ∈ Σ

α(π(i)) ≥ lim infn→+∞


.

Proof. Let us observe that by the zipper property fi(Π(0)) = fi−1(Π(N − 1)) for ev-ery 1 ≤ i ≤ N − 1. Moreover, for any i, j with ii∧j+1 = ji∧j+1 + 1,

i ∨ j = minσi∧j+1i ∧ N − 1, σi∧j+1j ∧ 0.


enough to show that for µ0-a.e. i ∈ Σ, α(π(i)) is a constant.But by Proposition 3.3.4, there exists α such that for µ0-a.e. i

α = limn→+∞


.

By definition of Bernoulli measure, ∑∞k=0 µ0([0k]) + µ0([Nk]) = 1

1−λ1+ 1

1−λN. Thus, by

Lemma 3.4.4, α(π(i)) = α for µ0-a.e. i, and by Lemma 3.3.5, we have α ≥ 1/d0.We show now the lower bound for (3.2.4). By Lemma 3.3.3 and Proposition 3.3.4,

the map t 7→ P′(t) is continuous and non-increasing on R. Hence, for every β ∈(αmin, αmax) there exists a t0 ∈ R such that P′(t0) = β. By Proposition 3.3.4, thereexists a µt0 Gibbs measure on Σ such that

limn→+∞


= β for µt0-a.e. i ∈ Σ.

It is easy to see that for any i and n ≥ 1, µt0([i|n]) > 0. Thus, by Lemma 3.4.4,

α(π(i)) = β for µt0-a.e. i ∈ Σ.

Observe that π : Σ 7→ [0, 1] is a finite to one Lipschitz-map. Thus, by [FH09, The-orem 2.8, Corollary 4.16], dimH µt = dimH µt π−1 for every t ∈ R. Therefore, byProposition 3.3.4

dimH x ∈ [0, 1] : α(x) = β ≥ dimH µt0 π−1 = t0P′(t0)− P(t0) =

t0β− P(t0) ≥ inft∈Rtβ− P(t) .

On the other hand, by Lemma 3.4.3

dimH x ∈ [0, 1] : α(x) = β≤ max dimH π(B), dimH i ∈ Σ \ B : α(π(i)) = β

≤ max

dimH π(B), dimH

i ∈ Σ : lim sup

n→+∞


≥ β

≤ max

dimH π(B), inft≤0tβ− P(t)

,

where in the last inequality we used Lemma 3.3.6.By Proposition 3.3.4, the function t 7→ tP′(t)− P(t) is continuous and P(0) = −1.

By Proposition 3.4.2, dimH π(B) < 1, thus, there exists an open neighbourhood oft = 0 such that for every t ∈ (−ρ, ρ), tP′(t) − P(t) > dimP π(B). In other words,there exists an ε > 0 such that P′(t) ∈ (α− ε, α + ε) for every t ∈ (−ρ, ρ). Hence, forevery β ∈ [α, α + ε] there exists a t0 ≤ 0 such that P′(t0) = β and inft≤0 tβ− P(t) =t0P′(t0)− P(t0) > dimH π(B) which completes the proof of (3.2.4).

Finally, if (3.2.5) holds then by Lemma 3.4.5 and Lemma 3.3.6

dimH x ∈ [0, 1] : α(x) = β ≤

dimH

i ∈ Σ : lim inf

n→+∞


≤ β

≤ inf

t≥0tβ− P(t) ,

which completes the proof.


3.5 Zippers with Assumption A

Now, we turn to the case when our affine zipper satisfies the Assumption A. We willshow that in fact in this case the exceptional set B, introduced in (3.4.4) is empty.That is, there are no points, in which local neighbourhood, the curve leaves the conerapidly. First, let us introduce a natural ordering on Σ∗. For any ı, ∈ Σ∗with ı∧ = mand |ı| > m, || > m, let

ı < ⇔ im+1 < jm+1.

Moreover, let Z1 := z0, . . . , zN the endpoints of the curves fi(Γ) and let Zn := fı(zk), |ı| = n, k = 0, . . . , N − 1.

For simplicity, let us denote fı(z0) by zı. Observe that by the Zipper propertyfı(zN) = zı||ı|−1(i|ı|−1).

Proposition 3.5.1. Let us assume that F is non-degenerate and satisfies the Assumption A.Then B = ∅, where the set B is defined in (3.4.4).

Proof. It is enough to show that for every i ∈ Σ

C(Π(i)) ∩ Γ = Γ, (3.5.1)

which is equivalent to show that for every i, j ∈ Σ, 〈Π(i)−Π(j)〉 ∈ C.Since 〈z0− zN〉 ∈ C and C is invariant w.r.t all of the matrices then for every ı ∈ Σ∗,

〈zı − zı||ı|−1(i|ı|−1)〉 = 〈 fı(z0)− fı(zN)〉 = 〈Aı(z0 − zN)〉 ∈ C.Observe by convexity of C, for any three vectors x, y, w ∈ Rd, if 〈x− y〉 ∈ C and

〈y−w〉 ∈ C then 〈x−w〉 ∈ C. Thus, by Assumption A and the convexity of the cone,for every n ≥ 1, and for every ı < ∈ Σ with |ı| = || = n, 〈zı − z〉 ∈ C.

Thus, for every i 6= j ∈ Σ and for every n ≥ 1, 〈 fi|n(z0)− fj|n(z0)〉 = 〈zi|n − zj|n〉 ∈C. Since C is closed, by taking n tends to infinity, we get that 〈Π(i)−Π(j)〉 ∈ C.

Lemma 3.5.2. Let us assume that F is non-degenerate and satisfies the Assumption A. Thenfor any µ fully supported, ergodic, σ-invariant measure on Σ

lim supπ(j)→π(i)

log ‖Π(i)−Π(j)‖log |π(i)− π(j)| ≤ lim

n→+∞


for µ-a.e. i.

Proof. Observe that


log ‖Π(i)−Π(j)‖log |π(i)− π(j)| =


log ‖Ai|i∧j+i∨j

(Π(σi∧j+i∨ji)−Π(0)

)+ Aj|i∧j+i∨j

(Π( f rm[o]−−)−Π(σi∧j+i∨jj)

)‖

|λi|i∧j

(λi|i∨j

(π(σi∧j+i∨ji)− 0) + λj|i∨j(1− π(σi∧j+i∨jj))

)|

.

By (3.5.1), 〈Π(σi∧j+i∨ji)−Π(0)〉, 〈Π(N − 1)−Π(σi∧j+i∨jj)〉 ∈ C, therefore by (3.3.3)


log ‖Π(i)−Π(j)‖log |π(i)− π(j)| ≤ lim sup

j→i

log ‖Ai|i∧j+i∨j‖+ log

(1 + ‖Ai∨j

0 ‖‖Ai∨j

N−1‖

)

log λi|i∧j

.

3.5. Zippers with Assumption A 81

It is easy to see that for any fully supported, ergodic, σ-invariant measure µ, for µ-a.e.i

lim supj→i

i ∨ ji ∧ j + i ∨ j

= 0.

Hence, by the previous inequality, the statement follows similarly as in Lemma 3.4.4.

Proof of Theorem 3.2.2. By Lemma 3.4.4 and Lemma 3.5.2, for every t ∈ R

αr(π(i)) = limn→+∞


for µt-a.e. i ∈ Σ.

Thus, similarly to the proof of Theorem 3.2.1

dimH x ∈ [0, 1] : αr(x) = β ≥ dimH µt0 π−1 = t0P′(t0)− P(t0) =

t0β− P(t0) ≥ inft∈Rtβ− P(t) ,

where t0 is defined such that P′(t0) = β. On the other hand,

dimH x ∈ [0, 1] : αr(x) = β ≤ dimH x ∈ [0, 1] : α(x) = β .

By Proposition 3.5.1, B = ∅, and similarly to the proof of Theorem 3.2.1,

dimH x ∈ [0, 1] : α(x) = β ≤ inft≤0tβ− P(t) ,

for every β ∈ [α, αmax]. If Γ satisfies (3.2.5) then by Lemma 3.4.5 and Lemma 3.3.6

dimH x ∈ [0, 1] : α(x) = β ≤ inft≥0tβ− P(t) ,


Now, we turn to the equivalence of the existence of pointwise regular Hölder ex-ponents and the Assumption A. Before that, we introduce another property and weshow that in fact all of them are equivalent. Denote cv(a, b) open line segment inRd connecting two points a, b. Moreover, let us denote the orthogonal projection to asubspace θ by projθ and for a subspace θ let θ⊥ be the orthogonal complement of θ.For a point x and a subspace θ, let θ(x) = y ∈ Rd : x− y ∈ θ.

Definition 3.5.3. We say that Zn is well ordered on l ∈ G(d, d− 1) if for any ı1 < ı2 < ı3

projl⊥(zı2) ∈ cv(projl⊥(zı1), projl⊥(zı3)

). (3.5.2)

We say that Zn is well ordered if there exists a δ > 0 such that Zn is well ordered for alll ∈ Bδ(F(Σ)).

Let us recall that F : Σ 7→ G(d, d− 1) is the Hölder-continuous function defined inTheorem 3.3.1. So Bδ(F(Σ)) is the δ > 0 neighbourhood of all the possible subspaceson which the growth rate of the matrices is at most the second singular value. For avisualisation of the well ordered property, see Figure 3.3. Roughly speaking, the wellordered property on l ∈ G(d, d− 1) means that the curve is parallel to l⊥. The nextlemma indeed verifies that the curve cannot turn back along l⊥.


FIGURE 3.3: Well ordered property of Zn on l ∈ G(d, d− 1).

Lemma 3.5.4. Zn is well ordered if and only if ∃δ > 0, ∀x ∈ Rd, ∀l ∈ Bδ(F(Σ)) eithercv(zı1 , zı2) ∩ l(x) = ∅ or cv(zı2 , zı3) ∩ l(x) = ∅, for every ı1 < ı2 < ı3.

Proof. Fix ı1 < ı2 < ı3 ∈ 0, . . . , N − 1n. Suppose that Zn is well ordered but forall δ > 0 there exists l ∈ Bδ(F(Σ)) and x ∈ Rd such that cv(zı1 , zı2) ∩ l(x) 6= ∅ andcv(zı2 , zı3) ∩ l(x) 6= ∅. Thus,

projl⊥(x) ∈ cv(projl⊥(zı2), projl⊥(zı1)

)∩ cv

(projl⊥(zı2), projl⊥(zı3)

)

Since the right hand side is open, and non-empty line segment, therefore

projl⊥(zı2) /∈ cv(projl⊥(zı1), projl⊥(zı3)

),

which is a contradiction.On the other hand, suppose that Zn satisfy the assumption of the lemma but not

well ordered. Then for every δ > 0 there exists an l ∈ Bδ(F(Σ)) such that projl⊥(zı2) /∈cv(projl⊥(zı1), projl⊥(zı3)

). Since Bδ(F(Σ)) is open, there exists an l′ ∈ Bδ(F(Σ)) for

whichdist(projl′⊥(zı2), cv

(projl′⊥(zı1), projl′⊥(zı3)

)) > 0.

Thus, there exists x ∈ Rd that cv(zı1 , zı2) ∩ l′(x) 6= ∅ and cv(zı3 , zı2) ∩ l′(x) 6= ∅,which is again a contradiction.

The next lemma gives us a method to check the well ordered property.

Lemma 3.5.5. Z0 is well ordered if and only if for every n ≥ 0 Zn is well ordered.

Proof. The if part is trivial.By definition, Zn =

⋃N−1k=0 fk(Zn−1). By Lemma 3.5.4, if Zn−1 is well ordered

then there exists δ > 0 such that for every l ∈ Bδ(F(Σ)) and for every x ∈ Rd ei-ther cv(zı1 , zı2) ∩ l(x) = ∅ or cv(zı2 , zı3) ∩ l(x) = ∅. Thus, in particular for every

3.5. Zippers with Assumption A 83

l ∈ Bδ(F([k])). By Theorem 3.3.1(2), there exists δ′ > 0 such that AkBδ(F([k])) ⊇Bδ′(F(Σ)). Thus, for every l ∈ Bδ′(F(Σ)) and for every x ∈ Rd,

either cv( fk(zı1), fk(zı2)) ∩ l(x) = ∅ or cv( fk(zı2), fk(zı3)) ∩ l(x) = ∅ (3.5.3)

for every zı1 , zı2 , zı3 ∈ Zn−1 with ı1 < ı2 < ı3.Let us suppose that Zn is not well ordered for some n. Hence, there exists a

minimal n such that Zn−1 is well ordered but Zn is not. By Lemma 3.5.4, for everyδ′ > δ > 0 there exist 1 < 2 < 3 ∈ 0, . . . , N − 1n+1, l′ ∈ Bδ(F(Σ)) and x ∈ Rd

cv(z1, z2

) ∩ l′(x) 6= ∅ and cv(z2, z3

) ∩ l′(x) 6= ∅.

Since (3.5.3) holds for every k = 0, . . . , N− 1, there are k < m such that z1∈ fk(Zn−1)

and z3∈ fm(Zn−1). On the other hand by (3.5.3), one of the endpoints of fk(Zn−1)

(and fm(Zn−1)) must be on the same side of l′(x), where z1(and z3

respectively) is.Denote these endpoints by za′ and zb′ . Observe that za′ 6= zb′ . Indeed, if za′ = zb′

then k = m − 1, and thus either z2∈ fk(Zn−1) or z2

∈ fm(Zn−1). Hence, but z2is

separated from z1, z3

, za′ , zb′ by the plane l′(x), which cannot happen by (3.5.3).Moreover,by (3.5.3), one of the endpoints of f(2)0

(Zn−1) is on the same side ofl′(x) with z2

, denote it by zc′ . But the endpoints of fp(Zn−1) are the elements of Z0,moreover, a′ < c′ < b′, which contradicts to the well ordered property of Z0.

Theorem 3.5.6. Let F be a non-degenerate system. Then the following three statements areequivalent

1. S satisfies Assumption A,

2. for L-a.e. x, αr(x) exists,

3. Z0 satisfies the well-ordered property.

Proof of Theorem 3.5.6(1)⇒Theorem 3.5.6(2). Similarly, to the begining of the proof ofTheorem 3.2.1, π∗µ0 = L|[0,1], where µ0 = λ1, . . . , λNN, which is fully supported,σ-invariant, ergodic measure. Moreover, ∑∞

k=0 µ0([0k]) + µ0([Nk]) = 11−λ1

+ 11−λN

.Thus, by Lemma 3.4.4

lim infπ(j)→π(i)

log ‖Π(i)−Π(j)‖log |π(i)− π(j)| = lim

n→+∞


for µ0-a.e. i ∈ Σ.

But since F satisfies Assumption A, by using Lemma 3.5.2,


log ‖Π(i)−Π(j)‖log |π(i)− π(j)| ≤ lim

n→+∞


for µ0-a.e. i ∈ Σ,


Proof of Theorem 3.5.6(2)⇒Theorem 3.5.6(3). Let us argue by contradiction. Assume thatαr(x) exists for L-a.e. x but there exists Zn, n ≥ 0, which does not satisfy the well-ordered property. By Lemma 3.5.5, Z0 does not satisfy the well ordered property. ByLemma 3.5.4, let l ∈ F(Σ), x ∈ Rd and zi−1, zi, zi+1 ∈ Z0 be such that cv(zi−1, zi) ∩l(x) 6= ∅ and cv(zi, zi+1) ∩ l(x) 6= ∅. By continuity of the curve Γ, there exist i, j ∈ Σsuch that i1 = i − 1 6= i = j1, j2 6= 0 and Π(i) −Π(j) ∈ l(x′) with some x′ ∈ Rd.Hence, 〈Π(i)−Π(j)〉 ⊂ l. By definition, there exists a k ∈ Σ such that F(k) = l. By


using the continuity of F : Σ 7→ G(d, d − 1), Γ and Π : Σ 7→ Γ, one can choose n, msufficiently large, such that for every i′ ∈ [i|n] and k′ ∈ [k|m],

F(k′)(Π(i′)) ∩ Γj1 j2 6= ∅. (3.5.4)

By ergodicity, for µ0-a.e. i, σpi ∈ [km, . . . , k1, i1, . . . , in] for infinitely many p ≥ 0,where k|m = k1, . . . , km. Let us denote this subsequence by pk. Let kk be the sequencesuch that kk ∈ [k1, . . . , km, ipk , . . . , i1].

By (3.5.4), there exists a sequence jk such that jk ∧ i = pk + m, σpk+mjk ∈ [j1 j2],and Π(σpk+mjk) − Π(σpk+mi) ∈ F(kk). By construction, σpk+mjk ∧ σpk+mi = 0 andσpk+mjk ∨ σpk+mi = 0, and hence there exists a constant c > 0 such that‖Π(σpk+mjk)−Π(σpk+mi)‖ > c. Therefore for µ0-a.e. i

αr(π(i)) = limπ(j)→π(i)

log ‖Π(i)−Π(j)‖log |π(i)− π(j)| = lim

k→+∞

log ‖Π(i)−Π(jk)‖log |π(i)− π(jk)|

= limk→+∞

log ‖Ai|pk+m(Π(σpk+mi)−Π(σpk+mjk))‖log |λi|pk+m(π(σpk+mi)− π(σpk+mjk))|

≥ limk→+∞

log ‖Ai|pk+m |F(kk)‖log λi|pk+m

≥ limk→+∞

log α2(Ai|pk+m)

log λi|pk+m

≥ log τ

−χµ0

+ limk→+∞

log ‖Ai|pk+m‖log λi|pk+m

=log τ

−χµ0

+ α(π(i)),

(3.3.10) where Theorem 3.3.1(4), Theorem 3.3.1(6) and Lemma 3.4.4. But− log τ/χµ0 >0, which is a contradiction.

Let us recall that for any 0 6= x ∈ Rd, 〈v〉 denotes the unique 1-dimensionalsubspace in PRd−1 such that v ∈ 〈v〉. Also, any V ∈ G(d, d− 1) can be identified witha d− 2 dimensional, closed submanifold V of PRd−1 such that V = θ ∈ PRd−1 : θ ⊂V. Also, for a subset B ⊂ G(d, d − 1) we can identify it with a subset B of PRd−1

such that B = θ ∈ PRd−1 : θ ⊂ V ∈ B.

Proof of Theorem 3.5.6(3)⇒Theorem 3.5.6(1). Suppose that Z0 satisfies the well orderedproperty. By Lemma 3.5.5, Zn satisfies the well-ordered property for every n ≥ 0

and thus, we may assume that 〈zı − z〉 /∈ F(Σ) for every zı, z ∈ Zn. Indeed, if〈zı − z〉 ∈ F(i) for some i ∈ Σ then one could find zı′1

, zı′2, zı′3∈ Zn+1 such that

ı′1 < ı′2 < ı′3, zı′1= zı, zı′3

= z and

projF(i)⊥(zı′2) /∈ cv(projF(i)⊥(zı′1

), projF(i)⊥(zı′3)).

So, for every ı, ∈ Σ∗ there exists open, connected component Cı, of PRd−1 \ F(Σ)such that 〈zı − z〉 ∈ Cı,.

Then for any ı1 < ı2 < ı3, Cı1,ı2 = Cı1,ı3 = Cı2,ı3 . Indeed, if Cı1,ı2 6= Cı1,ı3 then thereexists F(i), which separates 〈zı2− zı1〉 and 〈zı3− zı1〉. But then, for F(i)⊥, projF(i)⊥(zı2) /∈cv(projF(i)⊥(zı1), projF(i)⊥(zı3)), which cannot happen by definition of well orderedproperty.

Therefore, there exists a unique open,connected component C such that 〈zı− z〉 ∈C for every ı, ∈ Σ∗. But, for any i ∈ Σ, since 〈zN − z0〉 /∈ F(Σ)

limn→+∞

〈Ai|n(zN − z0)〉 = E(i),

3.6. An example, de Rham’s curve 85

hence, E(Σ) ⊂ C. Thus, for any multicone M, for which the dominated splittingcondition of index-1 holds, the cone M ∩ C is invariant, i.e. Ai(M ∩ C) ⊂ Mo ∩ C.

On the other hand, by 〈zN − z0〉 ∈ C, one can extend M ∩ C such that 〈zN − z0〉 ∈M ∩ C and M ∩ C remains invariant.

3.6 An example, de Rham’s curve

Now we show an application for our main theorems. Recall, the well-known de Rham’scurve [Rha47; Rha56; Rha59] in R2 is the attractor of the affine zipper defined by thefunctions f0 and f1 given in (1.2.2), where ω ∈ (0, 1/2) is the parameter of the curve.In the introductory Subsection 1.2.2 a linear parametrization v : [0, 1] 7→ R2 was alsogiven, see (1.2.3) and a visualization of the curve, see Figure 1.3.

Protasov [Pro04; Pro06] proved in a more general context that the set of points x ∈[0, 1] for which α(x) = β has full measure only if β = α, otherwise it has zero measure.Just recently, Okamura [Oka16] bounds α(x) for Lebesgue typical points allowing inthe definition (1.2.3) more than two functions and also non-linear functions undersome conditions.

We show that with a suitable coordinate transform the matrices A0 and A1 satisfyAssumption A and hence, our results are applicable.

Lemma 3.6.1. For every ω ∈ (0, 1/3) ∪ (1/3, 1/2) there exists a coordinate transformD(ω) such that D(ω)−1AiD(ω) has strictly positive entries for i = 0, 1.

Proof. For ε > 0 and 0 < δ < 1 define the coordinate transform matrices

Dε =

[1 εε 1

]and Dδ =

[1 −δ−δ 1

].

Elementary calculations show that the matrices A0 = D−1ε A0Dε and A1 = D−1

ε A1Dε

have strictly positive entries whenever

13− ε

< ω <1

2− ε− ε2 .

The largest possible interval (1/3, 1/2) is attained when ε is arbitrarily small. Verysimilar calculations show that the entries of A0 = D−1

δ A0Dδ and A1 = D−1δ A1Dδ are

strictly positive wheneverδ

1 + 3δ< ω <

13 + δ

,

which gives the open interval (0, 1/3). Also trivial calculations show that ‖Ai‖1 =‖Ai‖1 = ‖Ai‖1, i = 0, 1.

Let us recall that in this case P(t) has the form

P(t) = limn→+∞

−1n log 2

log ∑|ı|=n‖Aı‖t,

and

αmin = limt→+∞

P(t)t

and αmax = limt→−∞

P(t)t

.

Proposition 3.6.2. For every ω ∈ (0, 1/4)∪ (1/4, 1/3)∪ (1/3, 1/2) the de Rham functionv : [0, 1] 7→ R2, defined in (1.2.3), the following are true


1. v is differentiable for Lebesgue-almost every x ∈ [0, 1] with derivative vector equal tozero,

2. LetN be the set of [0, 1] such that v is not differentiable. Then dimHN = τ− P(τ) >0, where τ ∈ R is chosen such that P′(τ) = 1,

3. dimH Π∗µ0 < 1, where µ0 = 1

2 , 12

N equidistributed measure on Σ = 0, 1N andΠ is the natural projection from Σ to v([0, 1]).

4. for every β ∈ [αmin, αmax]

dimH x ∈ [0, 1] : α(x) = β = dimH x ∈ [0, 1] : αr(x) = β= inf

t∈Rtβ− P(t).

For ω = 1/4 the de Rham curve is a smooth curve, namely a parabola arc. Forω = 1/3, the matrices does not satisfy the dominated splitting condition. For thiscase, we refer to the work of Nikitin [Nik04].

Proof. By Lemma 3.6.1, we are able to apply Theorem 3.2.2 and Theorem 3.2.3 forω 6= 1/3. It is easy to see that (1.2.2) satisfies (3.2.5). Thus, by Theorem 3.2.2, thestatement (4) of the proposition follows.

On the other hand, let N be the set, where v is not differentiable. Then

x ∈ [0, 1] : α(x) < 1 ⊆ N ⊆ x ∈ [0, 1] : α(x) ≤ 1.

Thus, dimHN = inft∈Rt− P(t).Now, we prove that there exists τ ∈ R such that P′(τ) = 1. Observe that M =

A0 + A1 is a stochastic matrix with left and right eigenvectors p = (p1, p2)T ande = (1, 1)T, respectively, corresponding to eigenvalue 1 and pi > 0, p1 + p2 = 1.There exists a constant c > 0 such that for every ı ∈ Σ∗

c−1 pT Aıe ≤ ‖Aı‖ ≤ cpT Aıe,

and therefore∑|ı|=n

pT Aıe = pT(A0 + A1)ne.

Thus P(1) = 0, and

µ1([i|n]) = pT Ai|n e = pT Ai1 . . . Ain e and

µ0([i|n]) =12n for every i ∈ Σ.

Simple calculations show that,

µ1([00]) =(1

2,

12

)A0A0(1, 1)T = ω2 +

(1− 2ω)ω

2+

(1− 2ω)2

2,

which is not equal to 1/4 if ω 6= 1/4 or ω 6= 1/2. Thus, for ω 6= 1/4 and ω 6= 1/2,µ0 6= µ1, and by Lemma 3.3.5, P′(1) < 1 < P′(0). Since t 7→ P′(t) is continuous, thereexists τ such that P′(τ) = 1 and therefore dimHN = τ − P(τ) > 0, which completes(2).

3.6. An example, de Rham’s curve 87

On the other hand, by Theorem 3.2.3, αr(x) = P′(0) > 1 for Lebesgue almostevery x ∈ [0, 1] and therefore v is differentiable with derivative vector 0. This implies(1).

Finally, we show statement (3) of the proposition. By using the classical result ofYoung

dimH Π∗µ0 = lim infr→0+

log Π∗µ0(Br(x))log r

for π∗µ0-a.e. x ∈ Γ = v([0, 1]).

For an i ∈ Σ and r ∈ R let n ≥ 1 be such that ‖Ai|n‖ ≤ r < ‖Ai|n−1‖. Hence,

Π([i|n]) ⊆ Br(Π(i)) and

lim infr→0+

log Π∗µ0(Br(Π(i)))log r

≤ lim infn→∞

log P([i|n])log ‖Ai|n‖

Since µ0([i|n]) = 1/2n, by Proposition 3.3.4

lim infn→∞

log P([i|n])log ‖Ai|n‖

=1

P′(0)< 1.

Remark 3.6.3. Finally, we remark that in case of general signature vector, one may modifythe definition of i ∨ j to

i∨ j =

minσi∧j+1i ∧ N − 1, σi∧j+1j ∧ 0, ii∧j+1 + 1 = ji∧j+1 & ε ii∧j+1 = 0 & ε ji∧j+1 = 0,minσi∧j+1i ∧ 0, σi∧j+1j ∧ 0, ii∧j+1 + 1 = ji∧j+1 & ε ii∧j+1 = 1 & ε ji∧j+1 = 0,minσi∧j+1i ∧ 0, σi∧j+1j ∧ N − 1, ii∧j+1 + 1 = ji∧j+1 & ε ii∧j+1 = 1 & ε ji∧j+1 = 1,minσi∧j+1i ∧ N − 1, σi∧j+1j ∧ N − 1, ii∧j+1 + 1 = ji∧j+1 & ε ii∧j+1 = 0 & ε ji∧j+1 = 1,minσi∧j+1j ∧ N − 1, σi∧j+1i ∧ 0, ji∧j+1 + 1 = ii∧j+1 & ε ji∧j+1 = 0 & ε ii∧j+1 = 0,minσi∧j+1j ∧ 0, σi∧j+1i ∧ 0, ji∧j+1 + 1 = ii∧j+1 & ε ji∧j+1 = 1 & ε ii∧j+1 = 0,minσi∧j+1j ∧ 0, σi∧j+1i ∧ N − 1, ji∧j+1 + 1 = ii∧j+1 & ε ji∧j+1 = 1 & ε ii∧j+1 = 1,minσi∧j+1j ∧ N − 1, σi∧j+1i ∧ N − 1, ji∧j+1 + 1 = ii∧j+1 & ε ji∧j+1 = 0 & ε ii∧j+1 = 1,0, otherwise.

Open problems

In this chapter we parametrized the points of zippers, but for zippers like in the ex-ample of Subsection 2.7.6 we can consider the attractor as the graph of a functionf : [0, 1] → [0, 1]. Now we can ask similar questions about f , but the methods re-quired are different.

1. Perform multifractal analysis for the pointwise Hölder exponent of f .

2. Consider the level sets Ex := y : f (y) = x. From our dimension result, wecan determine the Hausdorff dimension of a level set for typical x. However,this is not the same value for all x ∈ [0, 1]. By keeping track of the codes ofvertical columns, the goal is to determine the different values that dimH Ex canattain, and then study the map

α 7→ dimHx ∈ [0, 1] : dimH Ex = α.

89

Chapter 4

Distances in random and evolvingApollonian networks

This chapter is based on the article [KKV16] written jointly with Júlia Komjáthy andLajos Vágó. The paper also contains results about the degree distribution and cluster-ing coefficient. However, I did not participate actively in this part, therefore it is leftout of the thesis.

4.1 Definitions and notations

We now define the models and introduce necessary notation to state the results.

Random Apollonian networks

A random Apollonian network (RAN) in d dimensions, denoted RANd(n), is defined asfollows. The graph at step n = 0 consists of d + 2 vertices, embedded in Rd in sucha way that d + 1 of them form a d-dimensional simplex, and the (d + 2)-th vertex islocated in the interior of this simplex, connected to all of the vertices of the simplex.This vertex in the interior forms d + 1 d-simplices with the other vertices: initially weset the status of these d-simplices ‘active’, and call them active cliques. For n ≥ 1, pickan active clique Cn of RANd(n− 1) uniformly at random (u.a.r.), insert a new vertex vnin the interior of Cn and connect vn with all the vertices of Cn. The newly added vertexvn forms new cliques with each possible choice of d vertices of Cn. Set the status ofthe clique Cn ‘inactive’, and the status of the newly formed d-simplices ‘active’. Theresulting graph is RANd(n). At each step n a RANd(n) has n + d + 2 vertices andnd + d + 1 active cliques.

Evolutionary Apollonian networks

Given a sequence of occupation parameters qn∞n=1, 0 ≤ qn ≤ 1, an evolving Apol-

lonian network (EAN) EANd(n, qn) = EANd(n) in d dimensions can be constructediteratively as follows. The initial graph is the same as for a RANd(0) and there ared + 1 active cliques. For n ≥ 1, choose each active clique of EANd(n − 1) indepen-dently of each other with probability qn. The set of chosen cliques Cn becomes inactive(the non-picked active cliques stay active) and for every clique C ∈ Cn we place a newvertex vn(C) in the interior of C that we connect to all vertices of C. This new vertexvn(C) together with all possible choices of d vertices from C forms d + 1 new cliques:these cliques are added to the set of active cliques for every C ∈ Cn. The result-ing graph is EANd(n). The set of inactive vertices after n steps is denoted V(n) andN(n) = |V(n)|.

90 Chapter 4. Distances in random and evolving Apollonian networks

The case qn ≡ q was studied in [ZRZ06] where it was further suggested that forq → 0 the graph is similar to a RANd(n). We prove their conjecture in [KKV16]by showing that EANs obey the same power law exponent as RANs if qn → 0 and∑∞

n=0 qn = ∞.

Remark 4.1.1. Note that both in the RAN and EAN models, there is a one-to-onecorrespondence between cliques and vertices/future vertices: vertex v corresponds tothe clique C that became inactive when v was placed in the interior of the d-simplexcorresponding to C. In this respect, we call vertices that are already present in thegraph inactive vertices, and we refer to active cliques as active vertices: this notationmeans that these vertices are not yet present in the graph, but might become presentin the next step of the dynamics.

Notation

Fix n and consider two ‘active’ or ‘inactive’ vertices u and v from RANd(n) or EANd(n).Recall that Hopd(n, u, v) is the hopcount between the vertices u and v, i.e., the num-ber of edges on (one of) the shortest paths between u and v. The flooding timeFloodd(n, u) and diameter Diamd(n) are

Floodd(n, u) = maxv

Hopd(n, u, v) and Diamd(n) = maxu,v

Hopd(n, u, v). (4.1.1)

Whenever possible d, u and v are suppressed from the notation. We define Dv(n) asthe degree of vertex v after the n-th step.

Let (Xi)d+1i=1 be a collection of independent geometrically distributed random vari-

ables with success probability id+1 for Xi. Define the sum

Yd :=d+1

∑i=1

Xi. (4.1.2)

Yd is commonly referred to as a full coupon collector block in a coupon collector problemwith d + 1 coupons. Denote the expectation and variance of Yd by

µd := E[Yd] = (d + 1)H(d + 1), σ2d := D2 [Yd] , (4.1.3)

where H(d) = ∑di=1 1/i. The Large Deviation rate function of Yd is given by

Id(x) := supλ∈R

λx− log

(E[eλYd

]). (4.1.4)

The rate function Id(x) has no explicit form. It can be computed numerically from

Id(x) = λ∗(x) · x− log E[eλ∗(x)Yd

],

where λ∗(x) is the unique solution to the equation ∂∂λ log E

[eλYd

]= x and

log E[eλYd

]= log d!− d log(d + 1) + (d + 1)λ−

d

∑i=1

log(

1− id + 1

eλ)

.

The following is needed for the flooding time and diameter. Consider the function

fd(c) := c− d + 1d− c log

( dd + 1

c)

. (4.1.5)


Notice that − fd(c) is the rate function of a Poi( d+1d ) random variable. Thus for c >

d+1d the equation fd(c) = −1 has a unique solution which we denote by cd. Finally we

introduce

g(α, β) := 1 + fd(αcd)− αβcd

µdId

(µd

β

). (4.1.6)

A sequence of events En happens with high probability (w.h.p.) if limn P(En) = 1.Note that ‘with high probability’ is the same as ‘asymptotically almost surely’. Wefurther define for an event A and a σ-algebraF the conditional probability P(A|F ) =E[11A|F ], where 11A is the indicator of the event A, i.e., it takes value 1 if A holds and0 otherwise. We will sometimes replace F by a list of random variables, in this casewe drop the σ-algebra notation and list the random variables in the conditioning, andthis means conditional on the σ-algebra generated by this list of random variables.

Structure of the chapter

In Section 4.2 we state our main results and discuss their relation to other results in thefield. Section 4.3 contains the most important combinatorial observations about thestructure of RANs: we work out an approach of coding the vertices of the graph thatenables us to compare the structure of the RAN to a branching process and further,the distance between any two vertices in the graph is given entirely by the coding ofthese vertices. We also give a short sketch of proofs related to distances in this section.Then we prove rigorously the distance-related theorems in Section 4.4.

4.2 Main results

The first theorem describes the asymptotic behavior of typical distances in RANd(n).

Theorem 4.2.1 (Typical distances in RANs). The hopcount between two active verticeschosen u.a.r. in a RANd(n) satisfies a Central Limit Theorem (CLT) of the form

Hopd(n)− 2µd

d+1d log n

√2 σ2

d+µd

µ3d

d+1d log n

d−→ Z, (4.2.1)

where µd, σ2d as in (4.1.3) and Z is a standard normal random variable.

Further, the same CLT is satisfied for the distance between two inactive vertices that arepicked independently with the size-biased probabilities given by

P(v is chosen |Dv(n)) =(d− 1)Dv(n)− d2 + d + 2

dn + d + 1, (4.2.2)

where Dv(n) is the degree of the inactive vertex v.

Remark 4.2.2. Active vertices are not physically present in the graph. The distanceis defined between them so that we view them as inactive vertices with their initiald + 1 edges present in the graph.

The next theorem describes the asymptotic behaviour of the flooding time and thediameter:

Theorem 4.2.3 (Diameter and flooding time in RANs). Let u denote either an activevertex chosen u.a.r. or an inactive vertex chosen according to the size-biased distribution


given in (4.2.2). Then as n→ ∞

Diamd(n)log n

P−→ 2αβcd

µd,

Floodd(n, u)log n

P−→ 1µd

(d + 1

d+ αβcd

),

(4.2.3)

where (α, β) ∈ (0, 1]× [1, µdd+1 ] is the optimal solution of the maximization problem with the

following constraint:maxαβ : g(α, β) = 0. (4.2.4)

Remark 4.2.4. Observe that the set of (α, β) pairs that satisfy the constraint in (4.2.4) isnon-empty since for α = β = 1 by definition f (cd) = −1 and Id(µd) = 0. The fact thatthe pair (α, β) is unique is proved in Lemma 4.4.9. The maximization problem is alsoequivalent to first defining the g(α(x), β(x)) := supα,βg(α, β) : αβ = x and thenchoosing the unique x with g(α(x), β(x)) = 0, where the existence and uniquenessof such x follows from the fact that g(α(x), β(x)) strictly monoton decreases in x andcontinuity. This is shown in Claim 4.4.8.

We conclude with the asymptotic behavior of the typical distances in EANd(n).

Theorem 4.2.5 (Typical distances in EANs). Assume that the sequence of occupation pa-rameters qn satisfies ∑n∈N qn = ∞ and ∑n∈N qn(1 − qn) = ∞. Then, the hopcountbetween two active vertices chosen u.a.r. in a EANd(n) satisfies a central limit theorem ofthe form

Hopd(n)− 2µd

n∑

i=1qi

√2 σ2

d+µd

µ3d

n∑

i=1qi(1− qi)

d−→ Z, (4.2.5)

where µd, σ2d as in (4.1.3) and Z is a standard normal random variable.

Further, the same CLT is satisfied for the distance between two inactive vertices that arechosen independently with the size-biased probabilities given by

P (v is chosen |Dv(n), N(n)) =(d− 1)Dv(n)− d2 + d + 2d(N(n)− d− 2) + d + 1

. (4.2.6)

Remark 4.2.6. Note that in this theorem qn might or might not tend to 0. The secondcriterion rules out the case when qn → 1 so fast that the graph becomes essentiallydeterministic. Further, the statements of Theorems 4.2.1, 4.2.5 also stay valid if oneof the vertices is an active vertex chosen uniformly at random and the other vertex isinactive chosen according to the distribution given in (4.2.2) and (4.2.6), respectively.

4.2.1 Related literature

The statements of Theorem 4.2.1 are in agreement with previous results. In particular,in [ZRC06] the authors estimate the average path length, i.e., the hopcount averagedover all pairs of vertices, and they show that it scales logarithmically with the size ofthe network.

A more refined claim is obtained by Albenque and Marckert [AM08] concerningthe hopcount in two dimensions. They prove that

Hop(n)6/11 log n

P−→ 1.

4.3. Structure of RANs and EANs 93

The constant 6/11 is the same as 2(d + 1)/(dµd) for d = 2. They use the previouslymentioned notion of stack triangulations to derive the result from a CLT similar tothe one in Theorem 4.2.1. We show an alternative approach using weaker results. TheCLT for distances in RANs and EANs is novel.

Central limit theorems of the form (4.2.1) for the hopcount have been proven withthe addition of exponential or general edge weights for various other random graphmodels, known usually under the name first passage percolation. Janson [Jan99] anal-ysed distances in the complete graphs with independent and identically distributed(i.i.d.) exponential edge weights. In a series of papers Bhamidi, van der Hofstad andHooghiemstra determine typical distances and prove CLT for the hopcount for theErdos-Rényi random graph [BHH11], the stochastic mean-field model [BHH12b], theconfiguration model with finite variance degrees [BHH10] and quite recently for theconfiguration model [BHH12a] with arbitrary i.i.d. edge weights from a continuousdistribution. Inhomogeneous random graphs are handled by Bollobás, Janson and Ri-ordan [BJR07; KK15]. Note that in all these models the edges have random weights,while in RANs and EANs all edge weights are 1. The reason for this similarity is hid-den in the fact that all these models have an underlying branching process approxi-mation, and the CLT valid for the branching process implies CLT for the hopcount onthe graph.

Further, there are some previous bounds known about the diameter of RANs:Frieze and Tsourakakis [FT12] establishes the upper bound 2c2 log n for RAN2(n).They use a result of Broutin and Devroye [BD06] that, combined with the branchingprocess approximation of the structure of RANs we describe in this paper, actuallyimplicitly gives the 2cd log n upper bound for all d.

Just recently and independently from our work other methods were used to deter-mine the diameter. In [Ebr+13] Ebrahimzadeh et al. apply the result of [BD06] in anelaborate way, while Cooper and Frieze in [CFU14] use a more analytical approachsolving recurrence relations. We emphasize that the methods in [CFU14; Ebr+13]and in the present paper are all qualitatively different, moreover [CFU14; Ebr+13] donot give results for the hopcount or flooding time. Numerical solution of the maxi-mization problem (4.2.4) for d = 2 yields the optimal (α, β) pair to be approximately(0.8639, 1.500). The corresponding constant for the diameter is 2c2/µ2 ·0.8639·1.5 =1.668, which perfectly coincides with the one obtained in [CFU14] and [Ebr+13]. Tothe best of our knowledge no result has been proven before for the flooding time.

4.3 Structure of RANs and EANs

4.3.1 Tree-like structure of RANs and EANs

The construction method of RANs and EANs enables us to describe a natural wayto code the vertices and active cliques of the graph parallel to each other. Let Σd :=1, 2, . . . , d+ 1 be the symbols of the alphabet. In a d-dimensional simplex, the spher-ical vertex figure at a vertex v is the subspace of the simplex "cut out" by a small enoughd-dimensional ball centered at v. The coding in two dimensions is illustrated in Fig-ures 4.1 and 4.2. In d-dimensions it is done by induction:

1. Initialization: Each vertex of the initial d-dimensional simplex is given a differ-ent auxiliary code ′i′ for i ∈ Σd. For each vertex ′i′, the spherical vertex figure at′i′ is given the label i. Furthermore, the single active clique (the interior of thesimplex) gets the code O. See left-hand side of Figure 4.1.


′1′′2′

′3′

2 1

3

O

2 1

3

21

3

′1′′2′

′3′

O1

2

3

2 1

3

21

3

21

3

′1′′2′

′3′

O

3

12

31 32

33

FIGURE 4.1: Initial steps of coding RANs. Green codes represent ac-tive cliques, blue codes represent inactive cliques, and the red labels

correspond to spherical vertex figures.

2. Step 0: Active clique O becomes inactive and O gets connected to all the ver-tices ′i′, thus creating the new active d-simplexes. The newly formed sphericalvertex figures at each vertex ′i′ inherit the label i. The d − 1-dimensional hy-perplanes defined by the new edges divide the d-dimensional ball centered atO into d + 1 spherical vertex figures. Each of these spherical vertex figures isgiven a different, unique label from Σd so that in every active clique, the d + 1spherical vertex figures all have different labels from Σd. An active clique isassigned the code i corresponding to the label i of the spherical vertex figureinside the clique at the newly added vertex O. See center of Figure 4.1.

3. Step 1: One of the active cliques becomes inactive (in Figure 4.1 it is 3) and weassign the newly added vertex v the code v of the clique that becomes inactive.The new spherical vertex figures around v are assigned different, unique labelsfrom Σd same way as before. Then the new active cliques are assigned codesvj, j = 1, . . . , d + 1 (here vj means concatenation), where j corresponds to thelabel j of the spherical vertex figure inside the new clique at the vertex v. Seeright-hand side of Figure 4.1.

4. Induction step: Same as Step 1. An active clique u becomes inactive, the spher-ical vertex figures around u are assigned labels from Σd. Based on these labels,the newly added active cliques are given the codes uj, j ∈ Σd. See Figure 4.2.Thus, at the beginning and end of each step, every (in)active vertex has a well-defined code and each spherical vertex figure has a well-defined label.

T1uT2u

T3u

2 1

3

u

2 1

3

21

3

T1uT2u

T3u

uu1

u2

u3

FIGURE 4.2: The general induction step in the coding of RANs whenan active clique u is chosen. The neighbors of u are Tiu for i ∈ Σd.


Hence, each vertex in the graph has a code that is a concatenation of symbols fromΣd. For a vertex u we write u = u1u2 . . . u` for its code for some ` ∈ N, and we callthe length of a code |u| = ` the generation of the vertex u. We define |′1′| = . . . = |′d +1′| = |O| = 0. For two vertices u and v with codes u = u1u2 . . . un and v = v1v2 . . . vm,respectively, we say that u is an ancestor of v if n ≤ m and u1u2 . . . un = v1v2 . . . vn.We denote the latest common ancestor of u and v by u ∧ v and its code by u ∧ v, thus|u ∧ v| = mink : uk+1 6= vk+1. For codes u = u1 . . . un and v = v1 . . . vm we denotethe concatenation u1 . . . unv1 . . . vm by uv and the corresponding vertex by uv.

Furthermore, let (i) denote the index for which u(i) is the last occurrence of thesymbol i ∈ Σd in u. For all i ∈ Σd we introduce the cut-operators

Tiu := u1 . . . u(i)−1 and Piu = u(i) . . . un.

In the special case when u(i) = u1, then Tiu := O. Also, if there is no i in u or u = O,then we define Tiu :=′ i′. For future reference, we also define the operator Tminu,which gives the ancestor of u with length mini |Tiu|. If there is at least one i ∈ Σd forwhich Tiu =′ i′ or O, then Tminu := ∅ and |Tminu| := 0.

Remark 4.3.1. Note that there is a one-to-one correspondence between the codes oflength at most n and vertices of a rooted (d + 1)-ary tree of depth n. As a result,we use the codes u to denote vertices as well, that is, we identify vertices in a RANor EAN with their codes and sometimes refer to u as a vertex. In this respect, theconcept ‘u is an ancestor of v’ precisely means the ‘usual’ notion of being an ancestor:the unique path from v to the root in the (d + 1)-ary tree passes through u.

Apart from these ‘tree’ edges, RANs and EANs have other edges as well. How-ever, we will see below that these extra edges always go upwards (or downwards) ona branch of a tree, hence the crucial tree-like properties of the structure are conserved.We collect the most important combinatorial observations in the following lemma.

Lemma 4.3.2 (Tree-like properties of the coding). The coding of the vertices of a RAN orEAN described above has the following properties:

(a) The d + 1 neighbors of a newly formed vertex with code u have codes Tiu, i ∈ Σd.Furthermore, for any edge with endpoints u and v either ‘u is an ancestor of v’ or viceversa.

(b) Any shortest path between two vertices with codes u and v must go through a vertex withcode w which is an ancestor of u ∧ v or one of the initial vertices ′1′, . . . ,′ d + 1′, O.

(c) For any two vertices with codes u and v,

|Hop(u, v)− (Hop(u, u ∧ v) + Hop(v, u ∧ v)| ≤ 2.

Before the proof, let us interpret Lemma 4.3.2. Part (a) means that edges are onlypresent between vertices along the same ancestral line. In particular, the first d + 1neighbours of a newly added vertex with code u can be determined by cutting off thelast pieces of the code of u, up to the last occurrence of a given symbol i ∈ Σd.

The coding gives a natural grouping of the edges. Edges of the initial graph arenot given any name. An edge is called a forward edge if its endpoints have codes ofthe form u and uj for j ∈ Σd. All other edges are called shortcut edges. So in a RAN ateach step one new forward edge and d shortcut edges are formed.

Figure 4.3 below shows an example in two dimensions. Suppose at step n = 0the ‘left’, ‘right’ and ‘bottom’ triangles were given the symbols 1, 2 and 3 respectively.


Then later each new vertex u with code u in the middle of a triangle gives rise to thenew ‘left’, ‘right’ and ‘bottom’ triangles: to these we have to assign the codes v1, v2and v3 respectively.

On the left hand side is a planar embedding of the graph, while on the right thetree-like structure of the same graph becomes more apparent. Interpreting the initialgraph as the root, the forward edges are the edges of the tree: along them we can godeeper down in the hierarchy of the graph. The shortcut edges only run along a treebranch: between vertices that are in the same line of descent, so we can ‘climb up’ tothe root faster along these edges.

O

v

u

Coding of verticesgrouping of edges

initial graph

forward edges

shortcut edges

u u = 132

v v = 3312

O

u

v

FIGURE 4.3: Tree like structure of a realization of RAN2(8)

Part (b) is a consequence of part (a). It says that if we have two vertices with codeu and v in the tree, then any shortest path between them must intersect a path fromthe initial graph to their latest common ancestor u∧ v. Finally, part (c) says that up toa bounded additive error we can calculate the length of the shortest path by lookingat the path from u ∧ v to u and v separately.

Proof of Lemma 4.3.2. Part (a) is a direct corollary of the following observation: in anactive simplex u, the code of the vertex whose spherical vertex figure is labeled by iis equal to Tiu. This is true for the initial Step 0. Thus it is enough to show that it isalso true after the Induction step, refer to Figure 4.2. By construction, the label of thespherical vertex figure of u inside the active simplex ui is i, and of course Ti(ui) = u.

By construction, the other spherical vertex figures inside the active simplex ui in-herit the labels Σd \ i from the spherical vertex figures inside the simplex of u. Theinduction hypothesis holds for u, thus the vertices of the simplex, in which u be-came inactive, have codes Tju for j ∈ Σd \ i. The observation now follows becauseTj(ui) = Tju for all j ∈ Σd \ i. Finally, part (a) follows because a new inactive vertexalways gets the code of its active simplex.

Part (b) follows from part (a): every vertex is connected to d+ 1 vertices with codelength shorter than |u|, and all these vertices are descendants of each other, i.e., theyare in the path from u to the initial graph. The other vertices u is connected to are itsdescendants, i.e., of the form uw for some w. Hence, if we want to build a path fromvertex u to v, we must go up in the tree to at least u ∧ v.

Part (c). First observe that for all i ∈ Σd and all codes u, x, y

|Ti(uxy)| ≥ |Ti(ux)|, (4.3.1)

i.e. the position of the last occurrence of a symbol in a code can not be earlier thanthat in some prefix of the same code.


We describe the shortest path u → v with the help of Figure 4.4. The blue edgesgive the shortest path from u to the initial graph. This path intersects the ancestralline of u ∧ v at the vertex z∅

u , keeping in mind that possibly z∅u = u ∧ v or z∅

u isalready a vertex of the initial graph. For some non-empty word x, let (u ∧ v)x be thenext vertex on the ∅ → u shortest path below the level of u ∧ v. We define (u ∧ v)yanalogously on the shortest path ∅→ v.

u

v

(u ∧ v)x

(u ∧ v)y

(u ∧ v)zv zu z∅u

RANd(0)

FIGURE 4.4: Shortest path from u to v. Blue edges are on the shortestpath from u, v to the initial graph, green edges are other edges present

in the graph.

Let zu be the vertex where the shortest path u → v intersects the ancestral line ofu∧ v from (u∧ v)x, thus zu is somewhere in between u∧ v and z∅

u (could be equal toone of them). We define zv analogously from v and assume without loss of generalitythat |zu| ≤ |zv| (zu and zv could be the same).

Observe that by part (a) there is an edge zu − zv. Indeed, for some i ∈ Σd we havezu = Ti((u ∧ v)x). For this same i we also have Tizv = zu because zv is an ancestorof (u ∧ v)x. Moreover, with analogous reasoning, it follows that the four verticesu∧ v, zu, zv, and z∅

u form a complete graph (keeping in mind that potentially any twoof these vertices are the same). Thus, we can write the shortest path u→ v as

u→ (u ∧ v)x− zu − zv − (u ∧ v)y→ v,

and also another u→ v path going through u ∧ v

u→ (u ∧ v)x− zu − u ∧ v− zv − (u ∧ v)y→ v,

whose length is precisely Hop(u, u ∧ v) + Hop(v, u ∧ v). This is because there is no(u ∧ v)x− u ∧ v edge unless u ∧ v = zu. The assertion now follows.

4.3.2 Distances in RANs and EANs: the main idea

With the help of the grouping of the edges as above, we can determine the distancebetween two arbitrary vertices u, v with codes u, v as follows:

First, determine the generation of their latest common ancestor u∧ v. Then deter-mine the length of their code below u ∧ v. Finally, determine how fast can we reachthe latest common ancestor along the shortcut edges in these two branches, i.e., whatis the minimal number of hops we need to go up from u and v to u ∧ v?

If we pick u, v u.a.r., then we have to determine the typical length of codes in the treeand the typical number of shortcut edges needed to reach the typical common ancestor.If on the other hand we want to analyse the diameter or the flooding time, we have tofind a ‘long’ branch with ‘many’ shortcut edges. Clearly, one can look at the vertex of


maximal depth in the tree: but then - by an independence argument about the gs in thecode and the length of the code - with high probability the code of the maximal depthvertex in the tree will show typical behaviour for the number of shortcut edges. On theother hand, we can calculate how many slightly shorter branches are there in the tree.Then, since there are many of them, it is more likely that one of them has a code withmore shortcut edges needed than typical. Hence, we study the typical depth and alsohow many vertices are at larger, atypical depths of a branching process that arisesfrom the forward edges of RANs. The effect of the shortcut edges on the distances isdetermined using renewal theory (also done in [AM08]) and large deviation theory.Finally, we optimize parameters such that we achieve the maximal distance by anentropy vs energy argument.

4.3.3 Combinatorial analysis of shortcut edges

Now we investigate the effect of shortcut edges on this tree. Lemma 4.3.2 part (a)says that the shortcut edges of a vertex u in the tree lead exactly to Tiu, the prefixesof u achieved by chopping the code after the last occurrence of symbol i in the code.Recall that Piu = u(i) . . . un denotes the postfix of u that starts with the last occurrenceof the symbol i ∈ Σd in the code of u, while Tiu = u1 . . . u(i)−1.

Moreover, recall the operator that gives the prefix with length mini |Tiu| is Tminand further, denote the length of the maximal cut by

YANd (u) := |u| − |Tminu| = max

i∈Σd|Piu|. (4.3.2)

This is the length of the maximal hop we can achieve from the vertex u towards theroot in the tree via a shortcut edge.

Consecutively using the operator Tmin we can decompose u into independentblocks, where each block, when reversed, ends at the first position when all the sym-bols in Σd have appeared. We call such a block full coupon collector block. E.g. foru = 113213323122221131 this gives 1|132|1332|31222

∣∣21131. Let us denote the totalnumber of blocks needed in this decomposition by

N(u) = maxk + 1 : (Tmin)ku 6= ∅. (4.3.3)

Note that this is not the only way to decompose the code in such a way that we alwayscut only postfixes of the form Piu: e.g. 1|132|1332|31222211

∣∣31 gives an alternative cutwith the same number of blocks.

The following (deterministic) claim establishes that the decomposition along repet-itive use of Tmin (longest possible hops) is optimal.

Claim 4.3.3. Suppose we have an arbitrary code u of length n with symbol from Σd, that wewant to decompose into blocks in such a way that from right to left, each block ends at thefirst appearance of some symbol in that block. Then, the minimal number of blocks needed isgiven by N(u).

Proof. Consider two different decompositions of u into blocks: in the first decompo-sition use the operator Tmin consecutively, while in the second one we suppose that atleast one block is not a full coupon collector block. Without loss of generality we mayassume that this is the first block from the end of the code u. The endpoint of the firsthop in the first decomposition is Tminu := Ti∗u, while in the second decompositionthe endpoint is Tju for some j 6= i∗, with |Tju| > |Tminu|. Hence, there is a w suchthat Tju = (Tminu)w. Conclude from (4.3.1) that Hop(Tminu,∅) ≤ Hop(Tju,∅): thusthe number of blocks in the second decomposition can not be smaller than N(u).


Note that YANd (·) and N(·) are deterministic operators when applied to a fixed

code u. Next we state the distributional properties of YANd (u) and N(u) when u is

the code of a uniformly chosen active clique. The reason for the need of this is thatboth in the evolution of RAN and EAN, once a clique with code u becomes inactiveand is replaced with the vertex with code u, the d + 1 new cliques that become activeare exactly the direct descendants (children) of the vertex u in the d-ary tree. At eachstep in the evolution of the RAN, the clique to become inactive is chosen u.a.r. amongthe active cliques, and in the EAN an independent coin flip with success probabilityqn (not depending on the code itself) determines for each active clique if it becomesinactive or stays active for the next step.

Claim 4.3.4. Let u and v be two active vertices chosen u.a.r. from a RANd(n) or EANd(n).

(a) Then, conditioned on the length of u, the symbols in Σd are distributed uniformly at eachcoordinate of u.

(b) Then, conditioned on the length of u and v, the symbols in Σd are independent and dis-tributed uniformly at each coordinate after the |u ∧ v|+ 1-th in both u and v.

Proof. Assertion (a) follows from the symmetry of the dynamics. Assume there is acoordinate in which the symbols of Σd are not uniformly distributed, i.e. there is asymbol with greater probability than the others. If we permute the labels of Σd thenanother symbol has greater probability in that coordinate. However, the role of eachsymbol is symmetric due to the dynamics, therefore all symbols in that coordinateshould also have the same probability, giving a contradiction.

Due to the construction, two active cliques either share at most a hyperplane orone is contained in the other. In the former case, the evolution of the graph withinthe two cliques is independent, since the restriction of the uniform distribution to anysubset is also uniform. This is the case with the two active cliques u1u2 . . . u|u∧v|+1and v1v2 . . . v|u∧v|+1. Within these two cliques we can simply think that we start twoindependent RANs. Conditioned on the lengths of u and v, it now follows frompart (a) that the symbols in Σd are independent and distributed uniformly at eachcoordinate in the postfixes of u and v after the |u ∧ v|+ 1-th coordinate.

For every k ≥ 1, let us define the random variable

Hk := max` : ∑`

j=1 Y(`)d ≤ k, (4.3.4)

where Y(`)d are i.i.d copies of Yd in (4.1.2).

Lemma 4.3.5. Suppose u is a code of length k with symbols chosen u.a.r. from Σd at eachposition. Then

YANd (u) d

= minYd, k,N(u) d

= Hk + 1.

Proof. The last occurrence of any symbol i ∈ Σd in a uniform code is the first occurrencefrom backwards of the same symbol. Hence, reverse the code of u, and then |Piu| isthe position of the first occurrence of symbol i in a uniform sequence of symbols oflength k, since |u| = k. Clearly |Piu| = k if the symbol i does not occur in u. Asa result, |Piu| has a geometric distribution with parameter 1/(d + 1) truncated at k.Maximizing this over all i ∈ Σd we get the well-known coupon collector problem, thathas distribution Yd, truncated again at k. For the second part, since N(u) cuts down


full coupon collector blocks from the end of the code of u consecutively, the maximalnumber of cuts possible is exactly the number of consecutive full coupon collectorblocks in the reversed code of u, an i.i.d. code of length k. Since the length of eachblock has distribution minYd, k, and they are independent, the statement followsby observing that the last, non-full block of the reversed code corresponds to the +1in the statement.

Recall µd, σ2d from (4.1.3). From basic renewal theory [Fel68] the following central

limit theorem holds as k→ ∞:

Hk − k/µd√kσ2

d /µ3d

d−→ N (0, 1). (4.3.5)

Furthermore, the expected value of Hk satisfies [Fel68, Chapter XII. 12. Problem 22.]

E [Hk] =k

µd+

2σ2d + µd − µ2

d2µ2

d+ o(1) =

kµd

+ O(1). (4.3.6)

4.4 Distances in RANs and EANs

In light of the main idea of the proof in Subsection 4.3.2 we begin with the analysis ofthe tree created by the forward edges of the graph.

4.4.1 A continuous time branching process

There is a natural embedding of the evolution of the RAN into the evolution of acontinuous time branching process (CTBP) [AN04], or a Bellman-Harris process.

Namely, consider a CTBP where the offspring distribution is deterministic: eachindividual (equivalently, a vertex) has d + 1 children and the lifespan of each indi-vidual is i.i.d. exponential with mean one. Thus, after birth a vertex is active for theduration of its lifespan, then splits, becomes inactive and at that instant gives birthto its d + 1 offspring that become active for their i.i.d. Exp(1) lifespan. The processstarts with a single individual that is called the root and who dies immediately at t = 0giving birth to its d + 1 children.

The bijection between the CTBP at the split times and a RANd is the following:the individuals that have already split in the CTBP are the vertices already present(inactive vertices) in the RANd, while the active (alive) individuals in the CTBP cor-respond to the active vertices (active cliques) in the RANd. This holds since at everystep of a RANd, d + 1 new active cliques arise in place of the one which becomes inac-tive. Furthermore, in a RANd an active clique is chosen u.a.r. in each step which is –by the memoryless property of exponential variables – equivalent to the fact that thenext individual to split in the CTBP is an active individual chosen u.a.r.

We write GU(m) for the generation of a uniformly chosen active individual in theCTBP after m individuals have split, i.e., its graph distance from the root. The nexttwo propositions describe the growth of our CTBP in terms of the typical size ofGU(m) as well as the degree of relationship of two active individuals chosen u.a.r. inProposition 4.4.1 and the maximal size of GU(m) together with its tail behaviour inProposition 4.4.3.

4.4. Distances in RANs and EANs 101

Proposition 4.4.1. Let Z denote a standard normal random variable. Then as m→ ∞

GU(m)− d+1d log m√

d+1d log m

d−→ Z. (4.4.1)

Further, let GU , GV denote the generations of two active vertices chosen u.a.r. in the CTBPafter the m-th split, and let us write GU∧V for the generation of the latest common ancestor of

U, V. Then the marginal distribution GUd= GU(m), and

GU − GU∧V − d+1

d log m√d+1

d log m,

GV − GU∧V − d+1d log m√

d+1d log m

d−→ (Z, Z′), (4.4.2)

where Z, Z′ are independent standard normal distributions.

The proposition is an application of [Büh71, Theorems 2.5, 4.2] to the CTBP herewith deterministic offspring distribution (d + 1 children). Before the proof, we need alemma, that originates from Bühler [Büh71, Theorem 3.3], and the first part can alsobe found e.g. in [BHH11]. First some notation: let us write Di, Si for the numberof children of the i-th splitting vertex and the number of active individuals after thei-th split in a CTBP, and for an event A and random variable X let us write shortlyPm(A) := P(A|Di, Si, i = 1, . . . , m), Em[X] := E[X|Di, Si, i = 1, . . . , m].

Claim 4.4.2. The generation GU(m) of an active individual U chosen u.a.r. after the m-thsplit in a CTBP satisfies the following indicator representation

GU(m)d=

m

∑i=1

11i, where Em [11i] =Di

Si, (4.4.3)

and the indicators are independent conditioned on the sequence Di, Si, i = 1, . . . , m.Secondly, let us denote (U, V) a pair of individuals chosen u.a.r. after the m-th split. Let us

further assume that the latest common ancestor U ∧V of U and V reproduced at the τU∧V-thsplit. Then, conditioned on τU∧V the following two variables are independent and their jointdistribution can be written as

(GU−GU∧V, GV−GU∧V)d=

(m

∑i=τU∧V

1i,m

∑i=τU∧V

1′i

), (4.4.4)

wherePm((1i,1′i) = (1, 0)|τU∧V < i

)=

Di

Si

Si − Di

Si − 1

Pm((1i,1′i) = (0, 1)|τU∧V < i

)=

Di

Si

Si − Di

Si − 1

Pm((1i,1′i) = (1, 1), τU∧V = i|τU∧V ≤ i

)=

Di(Di − 1)Si(Si − 1)

(4.4.5)

and conditioned on τU∧V , different indices are independent.

Proof. A proof of the first statement using the ancestral line can be found in [Büh71,Section 3.A] (see also Section 2.A for clearer explanations), but a proof based on in-duction can also be worked out. Here we give the core idea of the proof of Bühler. Theancestral line of an individual in a CTBP is the unique path from the individual to theroot. For the time interval between the i-th and i + 1-th split we can allocate a unique


individual on the ancestral line that was active in this time interval. For the followingobservations, we condition on Di, Si, i = 1, . . . , m. Then Gm = 11 + 12 + · · · + 1m,where the indicators 1i are conditionally independent and 1i = 1 if and only if theancestor that was alive in the time interval between the i-th and i + 1-th split wasnewborn (born at the i-th split). Recall that the individual that splits at the i-th step ischosen u.a.r., as well as U is also chosen u.a.r. among the Sm many active individualsafter the m-th split. Since in the interval between the i-th and (i+ 1)-th split there wereexactly Di many individuals newborn, and Si many alive, and the ancestor of U isequally likely to be any of them, this yields the probability P(1i = 1|Di, Si) = Di/Si.The proof of the second statement follows from [Büh71, Section 3.B] in a similar man-ner: after time τU∧V , we write GU − GU∧V as sums of indicators, where 11i is 1 if andonly if the individual alive between the i-th and (i + 1)-th on the ancestral line of U isnewborn (born at the ith split). We do the same for GV −GU∧V using 11′is. Conditionalon Di, Si, i = 1, . . . , m the pairs (11i, 11′i) become independent and their joint distribu-tion is the one given in (4.4.5), since at each step, each pair of active individuals isequally likely to be the ancestors of U and V, and the ancestral lines merge preciselywhen the ancestors of U and V are two children of the vertex that splits at step i,giving the last line of (4.4.5).

Proof of Proposition 4.4.1. The proposition follows from Claim 4.4.2. More precisely,we note that in our case Di = d + 1 and Si = di + 1 are deterministic, hence

GU(m)d=

m

∑i=1

11i, where P (11i = 1) =d + 1di + 1

, (4.4.6)

From this identity the expectation and variance of GU follows:

E [GU(m)] =d + 1

dlog m + O(1), D2 [GU(m)] = E [GU ] + O

(m−1

). (4.4.7)

The central limit theorem (4.4.1) holds for the standardization of GU(m) since thecollection of Bernoulli random variables 11im

i=1, m = 1, 2, . . . satisfies Lindeberg’scondition.

For the second statement, GUd= GU(m) is obvious by noting that the marginal of

a uniformly chosen pair of vertices is a uniformly chosen vertex. Next, note that theevent (1i,1′i) = (1, 1) means that the ancestral lines of U and V merge at the i-th split.To see that τU∧V has a limiting distribution we can use the following:

P(τU∧V ≤ k) =m

∏i=k+1

(1−P(τU∧V = i|τU∧V ≤ i)) , (4.4.8)

where the factors on the right hand side are the probabilities that the two ancestrallines do not merge at the i-th split. This tends to a proper limiting distribution sinceby (4.4.5)

∞

∑i=1

P(τU∧V = i|τU∧V ≤ i) =∞

∑i=1

d + 1(di + 1)i

< ∞.

Hence, τU∧V has a limiting distribution, i.e. the limit limm→∞ P(τU∧V ≤ k) exists forevery k. Choosing k = mα we have limm→∞ P(τU∧V ≤ mα) = 1 for any α > 0, thuslog τU∧V/ log m → 0 in probability. Note that GU∧V also has a limiting distribution,independent of m, since GU∧V is the generation of the individual that splits at theτU∧V-th split.


From here, one can show the joint convergence of (4.4.4) using Lindeberg CLT forlinear combinations of ∑m

i=τU∧V+1(α1i + β1′i) and get that the two variables in (4.4.2)tend jointly to a two-dimensional standard normal variable.

Recall the definition of the function fd(c) from (4.1.5) and the constant cd thatsatisfies cd > (d + 1)/d, fd(cd) = −1. We will need the next proposition in the proofof Theorem 4.2.3.

Proposition 4.4.3. The exact asymptotic tail behaviour of GU(m) is given by

limm→∞

log (P (GU(m) > c log m))

log m= fd(c). (4.4.9)

Further, after m splits the deepest branch in the CTBP satisfies

maxi≤m GU(i)log m

P−→ cd. (4.4.10)

Proof. Let Λm(θ) := log E[eθGU(m)/ log m

]. Using (4.4.6) elementary calculation yields:

Λm(θ log m) =m

∑i=1

log(

1 +d+1

di + 1(eθ − 1)

).

Hence, from the series expansion of log(1 + x) we can see that

limm→∞

1log m

Λm(θ log m) =d + 1

d(eθ − 1),

which is the cumulant generating function of a ξ = Poi((d + 1)/d) random variable.The rate function of such a random variable is − fd(c). Hence, the conditions of theGärtner-Ellis theorem [DZ10, Subsection 2.3] are satisfied, which implies (4.4.9).

Our CTBP is a special case of so-called random lopsided trees [CG01; KR89]. Themaximal depth of such trees was studied by Broutin and Devroye [BD06] in a moregeneral framework. Thus (4.4.10) is just an application of [BD06, Theorem 5 and Re-mark afterwards] with our notation.

Remark 4.4.4. To see that cd should be the right constant in (4.4.10) we can arguethat from (4.4.9) it follows that the sum ∑m P (GU(m) > c log m) < ∞ for any c > cd.Thus by the Borel-Cantelli lemma, for any such c there are only finitely many m suchthat the event GU(m) > c log m holds, giving the whp upper bound cd log m on thedepth of the CTBP.

4.4.2 Proof of Theorem 4.2.1 and 4.2.5

Proof of Theorem 4.2.1. Pick a pair of active vertices u, v u.a.r. from a RANd(n). Wewrite |u|, |v| for their generation. As before, we write u ∧ v for their latest commonancestor, i.e., the longest common prefix of their codes. Let us define the distinctpostfixes u, v after u ∧ v by

u =: (u ∧ v)u, v =: (u ∧ v)v.


By Lemma 4.3.2 (c) and Claim 4.3.3 the length of the shortest paths between u, v (upto a small additive error, which cancels in the limit) satisfies

dist(u, v) = N(u) + N(v),

and Proposition 4.4.1 describes the typical distance between u and v along the tree(i.e., only using forward edges and no shortcut edges). Since u and v were chosenuniformly at random among the active vertices after n splits. Hence, we can write

|u| d= GU(n), |v| d

= GV(n), where U, V denotes two uniformly chosen alive individualsin the CTBP in Section 4.4.1. By the same reasoning (and dropping the dependence ofn for shorter notation),

|u ∧ v| d= GU∧V , |u| d

= GU − GU∧V , |v| d= GV − GU∧V . (4.4.11)

Further, by Claim 4.3.4, the symbols in the codes u and v are i.i.d. uniform on Σd(after the first symbol in both of the codes, which has to be different by the definitionof u ∧ v). Hence, by Lemma 4.3.5,

dist(u, v) d= HGU−GU∧V + HGV−GU∧V . (4.4.12)

Using (4.3.6) we have

E [HGU−GU∧V ] = E [E [HGU−GU∧V |GU − GU∧V ]] =E [GU − GU∧V ]

µd+ O(1).

Furthermore, (4.4.7) and the fact from the proof of Claim 4.4.2 that GU∧V has a limitingdistribution implies that for any am → +∞ sequence

E [HGU−GU∧V ] =1

µd

d + 1d

log m + O(am).

To obtain a central limit theorem for HGU−GU∧V observe that

HGU−GU∧V − 1µd

d+1d log m

√d+1

d log m σ2d /µ3

d

=HGU−GU∧V − 1

µd(GU − GU∧V)√

(GU − GU∧V)σ2d /µ3

d

·√

GU − GU∧Vd+1

d log m

+

1µd(GU − GU∧V)− 1

µd

d+1d log m

√d+1

d log m σ2d /µ3

d

.

(4.4.13)

The first factor on the right hand side, conditionally on GU−GU∧V with GU−GU∧V →∞, tends to a standard normal random variable independent of GU by the renewalCLT in (4.3.5) and the second factor tends to one in probability by (4.4.2). By (4.4.2)again, the second term tends to a N (0, µd/σ2

d ). Since the length of the codes u, v areindependent of the symbols in these codes, HGU−GU∧V |GU − GU∧V is independent ofGU −GU∧V . As a result, the two limiting normals arising from the two summands onthe right hand side of (4.4.13) are also independent, thus

HGU−GU∧V − 1µd

d+1d log m

√d+1

d log m σ2d /µ3

d

d−→ N (0, 1 + µd/σ2d ). (4.4.14)


By conditioning first on GU∧V , (as in the proof of Proposition 4.4.1) and using that thesymbols in the code of u, v are all i.i.d. uniform in Σd, one can show that (HGU−GU∧V , HGV−GU∧V )tend jointly to two independent copies ofN (0, 1+ µd/σ2

d ) variables. By (4.4.12) it fol-lows that Hop(n) = HGU−GU∧V + HGV−GU∧V , the first statement of the Theorem 4.2.1immediately follows by normalising such that the total variance is 1.

The second statement follows by calculating how many active cliques a vertexwith degree k is contained in: a vertex with degree d + 1 is contained in d + 1 cliques,and when the degree of a vertex v increases by 1, then the number of cliques contain-ing v increases by d− 1, thus a vertex with degree k ≥ d + 1 is contained in exactly

Qk = 2 + (k− d)(d− 1) (4.4.15)

active cliques. This means that the inactive vertex v is connected to exactly Qk manyactive vertices with an edge. It is clear that the total number of active vertices aftern steps is A(n) = dn + d + 1. This implies that choosing two inactive vertices x, yaccording to the size-biased distribution given in (4.2.2) is equivalent to choosing twoactive cliques U, V chosen u.a.r. that are neighbouring these vertices. The distancebetween x, y is then between N(u) + N(v) − 2, N(u) + N(v) since by Lemma 4.3.2x = Tiu for some i ∈ Σd, hence we can gain at most 1 hop by considering x instead ofthe clique U and the same holds for y and V. Hence, the CLT for U, V implies a CLTfor two vertices picked according to the probabilities in (4.2.2).

Remark 4.4.5. Let us denote the generation of the m-th splitting vertex by Gm. Since ateach split in the CTBP exactly one new inactive vertex is created, namely, a uniformly

chosen active vertex becomes inactive, we have Gmd= GU(m− 1). Hence, if we would

like to choose an inactive vertex of RANd(n) uniformly at random, then its distance fromthe root has distribution GU(X) where X is a random variable uniform in the set0, 1, . . . , n − 1, with GU(0) = 1. With a similar argument than the one in Claim4.4.2, one can obtain that the latest common ancestor of two inactive vertices chosenu.a.r. also has a limiting distribution, and if Hop(n) denotes the distance between

them, one can obtain Hop(n)/(2 d+1d log n) P−→ 1. But it is also not hard to see that

the CLT does not hold anymore, (since it does not hold for GU(X) for X uniform in0, 1, . . . , n− 1).

Proof of Theorem 4.2.5. The proof follows analogous lines to the proof of Theorem 4.2.1,hence we give only the sketch. The main idea here is that the tree can be viewed as aCTBP where at step i, each active individual splits with probability qi or stays activefor the next step with probability 1 − qi. Hence, Proposition 4.4.1 can be modifiedas follows: the generation of an active individual picked u.a.r. after the m-th splitsatisfies

GU(m)d=

m

∑i=1

1i,

where 1i = 1 if and only if the individual on the ancestral line of U is newborn at thei-th step. Note that in this case, the indicators are independent even without condi-tioning, and P(1i = 1) = qi, since at each step each individual splits with the sameprobability, independently of each other. Since splitting happens with probability qiat step i, the CLT for GU(m) holds by Lindeberg CLT. Now, for two individuals U, Vpicked u.a.r., with GU∧V , τU∧V as in Proposition 4.4.1, we have

(GU − GU∧V , GV − GU∧V)d=

(m

∑i=τU∧V

1i,m

∑i=τU∧V

1′i

),


where different indices are independent and conditioned on τU∧V , 1i, 1′i are indepen-dent indicators with P(1i = 1) = P(1′i = 1) = qi. Since the variance ∑i qi(1− qi) →∞, the joint CLT follows in a similar manner then for Proposition 4.4.1 if we can showthat τU∧V has a limiting distribution. For this note that similarly as in (4.4.8),

P(τU∧V ≤ k) =m

∏i=k+1

(1−P (τU∧V = i|τU∧V ≤ i)) , (4.4.16)

and the factors on the right hand side express that the two ancestral lines of U, V donot merge yet at step i. Let us write Ai for the number of active vertices at step i. Thenat step i there are Zi := Bin(Ai, qi) many vertices that split, each of them producingd + 1 new active vertices, and hence the probability that the two ancestral lines mergeat step i, conditioned on Ai, Ai+1 equals

P (τU∧V = i|τU∧V ≤ i, Ai, Ai+1) =Zi · (d + 1)d

Ai+1(Ai+1 − 1), (4.4.17)

where Ai+1 = Ai + d Zi, the new number of active vertices after the ith split. Weobtained the rhs of (4.4.17) by observing that if U, V was chosen uniformly at random,each pair of individuals at step i, Ai+1(Ai+1 − 1)/2 in total, is equally likely to be theancestors of them, and there are Zi(d+ 1)d/2 many pairs that make the ancestral linesmerge. If the sum in i ∈ N on the right hand side of (4.4.17) is a.s. finite then (4.4.16)ensures that τU∧V has a proper limiting distribution. Hence we aim to show that thisis the case whenever the total number of inactive vertices N(n)→ ∞, i.e.,

∞

∑i=1

Zi(d + 1)dAi+1(Ai+1 − 1)

< ∞ a.s. on N(n)→ ∞.

Since Ai+1 = N(i + 1)d + d + 1, and Zi = N(i + 1)− N(i), we can approximate theabove sum by

(d + 1)∞

∑i=1

d(N(i + 1)− N(i))(dN(i + 1))2 .

Now we can interpolate N(i) with a continuous function and then this sum is almostsurely finite if and only if

limT→∞

∫ T

1

N′(x)N(x)2 dx = lim

T→∞

1N(T)

− 1N(1)

is almost surely finite. In particular, this holds when N(n) → ∞. Further, as longas ∑n∈N qn = ∞, N(n) → ∞ holds a.s. by the second Borel-Cantelli lemma: in eachstep we add at least a new vertex with probability qn. The CLT then for the distancesfollows in the exact same manner as in the proof of Theorem 4.2.1.


We need some preliminary statements before the proof. Recall from (4.1.4) the def-inition of the large deviation rate function Id(x) of Yd and also Hk as the number ofconsecutive occurrences of full coupon collector blocks in a code of length k from(4.3.4).


Lemma 4.4.6. For 1 ≤ β ≤ µd/(d + 1), Hk satisfies the large deviation

limk→∞

1k

log(

P(

Hk >β

µdk))

= − β

µdId

(µd

β

). (4.4.18)

Proof. Let Y(i)d be i.i.d. distributed according to Yd. Since

P(

Hk >β

µdk)= P

( kβ/µd

∑i=1

Y(i)d <

( β

µdk)· µd

β

),

we can apply Cramér’s theorem [DZ10, Subsection 2.2] to obtain (4.4.18).

We have seen at the end of the proof of Theorem 4.2.1 that switching from inactivevertices to neighbouring active vertices/cliques only changes the distances by at most2, hence we rather investigate the diameter of the graph by active vertices. Let usdenote the set of active vertices at step n by An. We index An by vertices u anddenote one picked u.a.r by U. We have seen that |An| = dn + d + 1. Our aim is toestimate the expected number of u ∈ An with distance at least xcd log n/µd from theroot, for some x ≥ 1.

Recall the definition of the function g(α, β) from (4.1.6), and let us define for anx ≥ 1

(α(x), β(x)) := arg supα,βg(α, β) : αβ = x. (4.4.19)

Claim 4.4.7. For any x ≥ 1, define the indicator variables for each vertex u ∈ An

Ju(x) := 11[N(u) > xcd log n/µd

].

Then, with (α(x), β(x)) as in (4.4.19),

limn→∞

1log n

log E[

∑u∈An

Ju(x)]= g(α(x), β(x)). (4.4.20)

Proof. Note that

|An|E[ 1|An| ∑

u∈An

Ju(x)]= (dn + d + 1)P

(HGU(n) ≥ xcd log n/µd

), (4.4.21)

where we used Lemma 4.3.5 for the distributional identity N(u) d= HGU(n) for a uni-

formly chosen u ∈ An (see also the argument above (4.4.11)).

P(HGU(n)≥x

µdcd log n) ≥ P(GU(n) > α(x)cd log n)P

(Hα(x)cd log n >

β(x)µd

α(x)cd log n),

where we used that x = α(x)β(x) and that the symbols in a uniformly chosen u arei.i.d. uniform in Σd, and Hk is increasing in k. Finally, multiplying both sides bydn + d + 1, taking the logarithm, and dividing by log n, applying (4.4.9) and (4.4.18)(with k = α(x)cd log n and only dividing by log n instead of α(x)cd log n) we arrive at

limn→∞

log E[

∑u∈AnJu(x)

]

log n≥ 1+ fd(α(x)cd)− α(x)cd

β(x)µd

Id(µd/β(x)) = g(α(x), β(x)).


For the upper bound, let us fix a small ε > 0, and set

i?(ε) := maxi : (α(x)− (i + 1)ε)cd ≥ (d + 1)/d,i?(ε) := mini : (α(x)− iε)cd ≥ xcd

Then we can decompose the event HGU(n) ≥ xcd log n/µd according to which ε-length interval GU(n)/(cd log n) falls into, and use the monotonicity of Hk in k to get

P(HGU(n) ≥x

µdcd log n) ≤ P

(H(d+1) log n/d ≥ xcd log n/µd

)

+ P(GU(n) > xcd log n)

+i?(ε)

∑i=i?(ε)

P(

H(α(x)−iε)cd log n ≥ xcd log n/µd)

·P((α(x)− (i + 1)ε)cd log n < GU(n) < (α(x)− iε)cd log n)(4.4.22)

For the first term, we use (4.4.18) to see that

limn→∞

log(P(

H(d+1) log n/d ≥ xcd log n/µd))

log n= − xcd

µdId

((d + 1)µd

xcdd

). (4.4.23)

For the i-th summand in the third term, we use an upper bound by dropping theupper restriction on GU(n), use (4.4.18) again and also (4.4.9) to get

log(

P(

H(α(x)−iε)cd log n ≥ xcd log n/µd)·P((α(x)− (i + 1)ε)cd log n < GU(n))

)

≤ log n ·(− xcd

µdId(

µd(α− iε)x

) + fd((α(x)− (i + 1)ε)cd

))(1 + o(1)),

(4.4.24)where the (1 + o(1)) disappears when dividing by log m and taking the limit as n →∞. The second term can be treated similarly, except that there there is no part comingfrom the LDP of the H·.-s. This is not surprising since this is the point where thelength of the code becomes so large that a typical number of shortcut edges alreadyexceeds xcd log n/µd.

To finish the upper bound, note that setting i = i?(ε) + 1, the rhs of (4.4.24) exactlygives the rhs of (4.4.23), while setting i = i?(ε) yields the second term, since in thiscase the rate function Id(·) vanishes. Further, note that the terms in (4.4.22) are addi-tive. This implies that when taking logarithm and dividing by log n, the largest termwill dominate and determine the leading exponent. As a result,

limn→∞

P(HGU (n) ≥ xcd log n/µd)

log n

≤ maxi∈[i?(ε),i?(ε)+1]

− xcd

µdId(

µd(α(x)− iε)x

) + fd((α(x)− (i + 1)ε)cd

).

To finish, let ε → 0, and note that the ith term on the rhs is g(z, x/z) − 1 for somez ∈ R. Since the maximum of this expression is taken at z = α(x), this finishes theproof.

Claim 4.4.8 (Monotonicity of g(α(x), β(x)) in x). The function g(α(x), β(x)) is continu-ous and strictly monoton decreasing for x > (d + 1)/(dcd).


Proof. Recall that g(α, β) := 1 + fd(αcd)− αβ cdµd

Id

(µdβ

). The continuity follows from

the fact that g(α, β) is differentiable. For the monotonicity, consider x1 > x2 > 1. Wehave to show that the maximum of the function g(α, β) on the hyperbola β = x1/α issmaller than that and on β = x2/α. Let g1 := g(α(x1), β(x1)), g2 := g(α(x2), β(x2)).Note that fd(αcd) < 0 and monoton decresing in α as long as α > (d+ 1)/(dcd), whilethe second term −αβId(µd/β) < 0 and monoton decreasing in β as long as β > 1.Since x1 > (d + 1)/(dcd), at least one of the inequalities α(x1) > (d + 1)/(dcd) andβ(x1) > 1 must hold.

Suppose first that α(x1) > (d+ 1)/(dcd) holds. Then, if x2/β(x1) > (d+ 1)/(dcd),then clearly g2 = g(α(x1), x1/α(x1)) < g(x2/β(x1), β(x1)) ≤ g1 and we are done. Ifon the other hand x2/β(x1) < (d + 1)/(dcd), then look at the point on the hyperbola(d + 1)/(dcd), x2/(d + 1)/(dcd). Since we decreased both coordinates, g1 < g((d +1)/(dcd), x2/(d + 1)/(dcd)) holds as long as x2/(d + 1)/(dcd) > 1. This must holdsince otherwise the whole hyperbola β = x2/α would be in the region α < d +1/(dcd) ∪ β < 1, which would mean that x2 < (d + 1)/(dcd) which contradictsour original assumption.

If β(x1) > 1, then the argument is similar by first decreasing β to x2/α(x1) or to 1(whichever is larger), and in case we had to decrease it to 1 than we further decreaseα(x1) to x2 and again using that x2 > 1 implies that x2 > (d + 1)/dcd.

Proof of Theorem 4.2.3. First, we wish to choose the largest possible x in Ju(x) so thatlimn→∞ P

(∑u∈An

Ju(x) > 0)> 0, that is, there is at least one active clique that has

distance xcd log n/µd from the root. Note that if x is such that g(α(x), β(x)) < 0, thenby Claim 4.4.7 and Markov’s inequality we have

P(

∑u∈Am

Ju(x) > 0)≤ E

[∑

u∈An

Ju(x)]= ng(α(x),β(x))(1+o(1)) → 0. (4.4.25)

Thus necessarily x has to have g(α(x), β(x)) ≥ 0. Next we work out the lower bound.For this, we shall need a upper bound on the second moment

E[(

∑u∈An

Ju(x))2]

= E[ ∑u∈An

Ju(x)]+ ∑

u,v∈An,u 6=vE[Ju(x)Jv(x)]

Note that the second term equals

(dn + d + 1)2P(HGU > xcd log n/µd, HGV > xcd log n/µd)

where HGU and HGV is the minimal number of hops needed to reach the root fromtwo active vertices chosen independently and u.a.r.. As before, let us write U ∧V forthe latest common ancestor of U and V. Then we can write

P(HGU > xcd log n/µd, HGV > xcd log n/µd)

=P(HGU∧V + HGU−GU∧V > xcd log n/µd, HGU∧V + HGV−GU∧V > xcd log n/µd).

Pick any function ω(n)→ ∞ that also satisfies ω(n) = o(log n) (for instance, ω(n) =log log n will do), then We can bound the right hand side from above as follows:

P(HGU∧V > ω(n))+P(HGU−GU∧V >x

µdcd log n−ω(n), HGV−GU∧V >

xµd

cd log n−ω(n))

(4.4.26)Using the proof of Claim 4.4.2, we know that the joint distribution of two active in-dividuals U, V picked u.a.r. satisfies that their common ancestor GU∧V has a limiting


distribution. Hence, for any ω(n)→ ∞,

P(HGU∧V > ω(n))→ 0. (4.4.27)

Further, conditioned on the splitting time τu∧v of U ∧V, the joint distribution of GU −GU∧V , GV − GU∧V can be described as the sum of indicators, see (4.4.4). Further, thetwo sums are asymptotically independent, and also the symbols in the code u, v ofU and V are independent and uniform in Σd after u ∧ v, the code of U ∧ V. Hence,choosing n large enough, the ω(n) term becomes negligible and we get that

P(HGU−GU∧V > xcd log n/µd −ω(n), HGV−GU∧V > xcd log n/µd −ω(n))= P(HGU−GU∧V > xcd log n/µd) ·P(HGV−GU∧V > xcd log n/µd)(1 + o(1))

= P(HGU > xcd log n/µd)2(1 + o(1)),

(4.4.28)

where the o(1) → 0 as n → ∞, and in the last equation we used again the fact thatGU∧V has a limiting distribution. Combining (4.4.27) with (4.4.28) to bound (4.4.26),we arrive at

E[(

∑u∈An

Ju(x))2]

= E[

∑u∈An

Ju(x)]+ (nd + d + 1)2 (P(HGU > xc log n/µd)

2(1 + o(1)) + o(1))

= E[

∑u∈An

Ju(x)]+E

[∑

u∈An

Ju(x)]2(1 + o(1)).

(4.4.29)From a Cauchy-Schwarz inequality followed by (4.4.29), we get

P(

∑u∈An

Ju(x) > 0)≥ E[∑u∈An

Ju(x)]2

E[(

∑u∈AnJu(x)

)2]

≥ E[∑u∈AnJu(x)]2

E[∑u∈AnJu(x)

]+ E[∑u∈An

Ju(x)]2(1 + o(1)),

(4.4.30)

and the right hand side is strictly positive in the limit as n→ ∞ if and only if g(α, β) ≥0 (using Claim 4.4.7 again for each term on the rhs). From this and the monotonicityof g(α(x), β(x)) in x (see Claim 4.4.8 it is immediate that the largest diameter canbe achieved when picking x := x so that g(α(x), β(x)) = 0. Apply (4.4.25) withx = x(1 + ε) and (4.4.30) with x := x(1− ε) to finally conclude that as n→ ∞

maxu∈An HGU(n)

log nP−→ cd

µdx. (4.4.31)

The statement of Theorem 4.2.3 for the flooding time now follows from the fact that ifU is an active clique picked u.a.r. after the n-th step of the evolution of the RAN, then

Flood(n) d= HGU−GU∧V + max

v∈A(n)HGV−GU∧V .

Now, the proof of Theorem 4.2.1 (or Proposition 4.4.1) implies that the CLT holds forgeneration GU − GU∧V , and since the symbols are uniform in the code of U, similarly


as in (4.4.13), the CLT holds for HGU(n) as well. Further, since in Flood(u, v) we max-imise the distance over the choice of the other vertex V, clearly whp we can pick Vsuch that the latest common ancestor U ∧V is the root itself. This combined with thefact that the distance changes only by at most 2 if we consider active cliques instead ofvertices in the graph implies and the statement of the theorem follows from the distri-

butional convergence of HGU(n)/ log n d−→ (d + 1)d/µd and (4.4.31). For the diameterwe have

Diam(n)log n

d= 2

maxu∈A(n) HGu

log n,

since for any ε > 0, whp there are at least two vertices that are not closely related toeach other and both satisfy HGu / log n > cd

µdx(1− ε), but whp there are no vertices

that satisfy HGv / log n > cdµd

x(1 + ε).

We are left to analyse the maximization problem. First of all, it is elementary to see(e.g. using Claim 4.4.8 or elementary two-dimensional calculus) that solving (4.4.19)and then choosing x so that g(α(x), β(x)) = 0 is equivalent to the maximization prob-lem in (4.2.4). However, two dimensional techniques give a better understanding ofthe solution x = α(x)·β(x). For short we write α(x) := α, β(x) := β.

Lemma 4.4.9. The maximization problem (4.2.4) has a unique solution (α, β) ∈ (0, 1] ×[1, µd

d+1 ], and further this solution satisfies

α =1cd

d + 1d

exp− I′d(µd/β)

,

β

µdId

(µd

β

)=

1 + fd(αcd)

αcd.

Proof. Define the Lagrange multiplier function L(α, β, λ) := αβ− λg(α, β). Necessar-ily the optimal (α, β) satisfies ∇L(α, β, λ) = 0. The partial derivative L(α, β, λ)′λ = 0simply gives the condition g(α, β) = 0. Further, the optimising λ can be expressedfrom L(α, β, λ)′α = 0 and L(α, β, λ)′β = 0 and satisfies

λ =β

∂∂α g(α, β)

=α

∂∂β g(α, β)

.

After differentiation of g(α, β) = 1 + fd(αcd)− αcdβ

µdId

(µdβ

), rearranging terms and

using that f ′d(x) = − log( d

d+1 x)

we obtain the first condition. To check the sufficiencywe look at the bordered Hessian

0 ∂g∂α

∂g∂β

∂g∂α

∂2αβ∂α2

∂2αβ∂α∂β

∂g∂β

∂2αβ∂α∂β

∂2αβ∂β2

=

0 ∂g∂α

∂g∂β

∂g∂α 0 1∂g∂β 1 0

.

Its determinant is(∂2g(α, β)/∂α∂β

)2> 0, thus the condition is also sufficient. We

note that the solution can be approximated by numerical methods.

Remark 4.4.10. We mention here the difficulties in the analysis of the diameter andflooding time of EANs: the main difficulty here is to understand the proper corre-lation structure of the codes (and shortcut edges) on the vertices of the BP: (a) Thecorresponding BP tree is fatter than the BP for RAN as soon as n−1 = o(qn). (b) Ineach step each vertex splits independently of the past with probability qn. (a) and (b)together imply that even though we do understand the marginal distribution of the


symbols of a clique U picked u.a.r. is uniform in Σd, still it is more likely that the’neighbouring codes’ are also present in the graph and hence codes for which N(u)is large are more likely to appear. Hence we expect that the diameter will have alarger constant in front of ∑ qi than the constant in front of log n for RAN. (Com-pare it to the diameter of the deterministic AN: with qn ≡ 1 it is not hard to see thatDiam(ANd(n)) = 2n/(d + 1)).

113

Appendix A

Basic dimension theoreticdefinitions

Here we collect the various types of fractal dimensions used in the thesis. For furtherreference, see any of the excellent books [Fal85; Fal90; Fal97; Mat95].

Let F be a subset of Rd and s ≥ 0. For any δ > 0 we define

Hsδ(F) = inf

∑

i|Ui|s : F ⊆

⋃

i

Ui, |Ui| ≤ δ

, (A.0.1)

where |U| denotes the diameter of U. Any such collection Ui in the definition iscalled a δ-cover of F. The s-dimensional Hausdorff measure of F isHs(F) = limδ→0Hs

δ(F).The limit exists (can be 0 or infinity), because the infimum increases. The Hausdorffdimension of F is

dimH F = inf s ≥ 0 : Hs(F) = 0 = sup s ≥ 0 : Hs(F) = ∞ . (A.0.2)

Hausdorff dimension has many favorable properties, though usually it is hard to cal-culate.

On the other hand box-counting dimension, also commonly called Minokwski-dimension,is easier to calculate or estimate, but it has serious drawbacks. For example countablesets can have dimension strictly larger than zero. The lower and upper box dimen-sions of a set F are defined by

dimBF = lim infδ→0


and dimBF = lim supδ→0


, (A.0.3)

respectively, where Nδ(F) is the smallest number of sets required for a δ-cover of F.Nδ can be replaced with various other definitions based on covering or packing F atscale δ, see [Fal90, Section 3.1]. We shall always use what is most convenient for us.If the limit exists, i.e. dimBF = dimBF, then this common value is called the boxdimension of F, denoted dimB F.

We do not define packing dimension using packing measures, because we will notuse it directly. Instead we mention that there is an equivalent definition using theupper box dimension. Namely

dimP F = inf

sup

idimBFi : F ⊆

∞⋃

i=1

Fi

,

where the infimum is taken over all countable partitions Fi of F. This has the fol-lowing consequence [Fal90, Corollary 3.9], which we will use later.

114 Appendix A. Basic dimension theoretic definitions

Proposition A.0.1. Let F ⊂ Rd be a compact set such that every open set intersecting Fcontains a bi-Lipshitz image of F, we get that dimP F = dimBF.

The last dimension we define is the affinity dimension, introduced by Falconer [Fal88a].The singular values 0 < αd ≤ . . . ≤ α1 of a non-singular d× d matrix A are the pos-itive square roots of the eigenvalues of AT A, where AT is the transpose of A. Thegeometric interpretation of the singular values is that the linear map x 7→ Ax mapsthe unit disk to an ellipse with principal semi-axes of length equal to the singularvalues. Hence, roughly speaking, the singular values indicate how much the mapcontracts (or expands) in different directions. For s ∈ [0, d] define the singular valuefunction (introduced in [Fal88a])

φs(A) = α1α2 . . . αdse−1αs−dse+1dse . (A.0.4)

Given a finite collection A of contracting matrices the affinity dimension is

sA = sA(A) = inf

s :

∞

∑k=1

∑i1,...,ik

φs(Ai1 · . . . · Aik) < ∞

. (A.0.5)

When the matrices define a self-affine set Λ, we use the notation dimA Λ.Now we define the local dimension and Hausdorff dimension of a measure.

Definition A.0.2. Let µ be a Borel probability measure on Rd and x ∈ spt(µ). We define thelocal dimension of the measure µ at x by

dµ(x) := limr→0

log µ(B(x, r))log r

,

if the limit exists. Otherwise we take lim inf and lim sup instead of lim and we obtain thelower local dimension dµ(x) and the upper local dimension dµ(x), respectively.

The measure µ is exact dimensional if for µ-almost every x the limit exists and isequal to a constant.

Definition A.0.3. Let µ be a mass distribution. The upper and lower Hausdorff dimen-sion of µ are defined

dimHµ := infdimH E : µ(Ec) = 0,dimHµ := infdimH E : E is a Borel set with µ(E) > 0.

When µ is exact dimensional, then the quantities are equal, simply called theHausdorff dimension of µ. This is the case throughout the thesis.

115

Appendix B

No Dimension Drop is equivalentto Weak Almost Unique Coding

In this appendix, we prove that for self-similar IFSs on the line and Bernoulli mea-sures the separation conditions No Dimension Drop (NDD) and Weak Almost UniqueCoding (WAUC) are equivalent. We recall notation and definitions.

Notation

Let H = hı(x) := rıx + uıMı=1 be a contractive self-similar IFS on the real line with

attractor ΛH. The symbolic space is ΣH = 1, 2, . . . , MN and the natural projectionis ΠH(ı) := limn→∞ hı|n(0) for ı ∈ ΣH. Define a partition of ΣH by

ξ(ı) := Π−1H ΠH(ı).

As we noted earlier in this paper, ξ is a measurable partition of Σ. We write ξ for theσ-algebra generated by the measurable partition ξ. Then the elements of ξ are unionsof the elements of ξ. For a probability vector q = (q1, . . . , qM) we denote the Bernoullimeasure on ΣH by µq. Then there exists a ΣH ⊂ ΣH, with µq(ΣH) = 1 such that forall ı ∈ ΣH there exists a probability measure µξ(ı) defined on ξ(ı) such that

• For all A ⊂ Σ Borel set the mapping ı 7→ µξ(ı)(A) is ξ-measurable and

• for all Borel sets U ⊂ ΣH we have

µq(U) =∫

µξ(ı)(U)dµq(ı). (B.0.1)

The push forward measure νq = (ΠH)∗µq is the self-similar measure with supportΛH. The entropy and Lyapunov exponent of the system are

hµq = − log〈q〉q and χνq = −M

∑ı=1

qı log rı = − log〈r〉q,

respectively, where 〈c〉q = ∏Mı=1 cqi

i . Now we recall two separation conditions fromDefinition 2.1.9.

116 Appendix B. No Dimension Drop is equivalent to Weak Almost Unique Coding

Definitions

We say that H has No Dimension Drop (NDD) if for all probability vectors q withstrictly positive entries we have

dimH νq =hµq

χνq

.

We say thatH has Weak Almost Unique Coding (WAUC) if for all probability vectorsq with strictly positive entries there exists a set BH ⊂ ΣH (may depend on q) forwhich

µq(BH) = 0 and for every ı ∈ ΣH \ BH : #(ξ(ı) \ BH) = 1.

Proposition B.0.1. For any self-similar IFS on the line the conditions NDD and WUAC areequivalent.

Let δı denote the Dirac-delta measure concentrated on the point ı ∈ ΣH. We showthe assertion in two steps. Namely, we prove that

NDD ⇐⇒ µξ(ı) = δı for µq-a.e. ı ∈ ΣH ⇐⇒ WAUC. (B.0.2)

Proof of first equivalence in (B.0.2) . The result of Bárány–Käenmäki [BK17, Theorem 2.3.]for dimension d = 1 states that for every Bernoulli measure µq its push forward νq isexact dimensional. Moreover,

dimH νq =hµq − H

χνq

, where H = −∫

log µξ(ı)([ı1])dµq(ı) ≥ 0.

From the definition of NDD we get that

NDD ⇐⇒ H = 0 ⇐⇒ µξ(ı)([ı1]) = 1 for µq-a.e. ı ∈ ΣH.

Thus it suffices to show that

µξ(ı)([ı1]) = 1 for µq-a.e. ı ∈ ΣH ⇐⇒ µξ(ı) = δı for µq-a.e. ı ∈ ΣH. (B.0.3)

The⇐= direction in (B.0.3) is obvious. In the other direction we show that for µq-a.e.ı ∈ ΣH

µξ(ı)([ı1]) = 1 =⇒ µξ(ı)([ı1 . . . ın]) = 1 for every n =⇒ µξ(ı) = δı.

To see the first implication fix n. Let H(n) := hı1...ın : (ı1 . . . ın) ∈ 1, . . . , Mnand Σ(n)

H be the symbolic space of infinite sequences of n-tuples (ı1 . . . ın). There is anatural one-to-one bijection between the elements of ΣH and Σ(n)

H . A Bernoulli mea-sure µq on ΣH naturally defines a Bernoulli measure µ

(n)q on Σ(n)

H by µ(n)q ([ı1 . . . ın]) =

∏nj=1 µq([ıj]). Applying [BK17, Theorem 2.3.] to this system yields the first implica-

tion. The second implication follows from the Monotone Convergence Theorem

1 = limn→∞

µξ(ı)([ı1 . . . ın]) = µξ(ı)(ı) =⇒ µξ(ı) = δı for µq-a.e. ı ∈ ΣH.

Proof of second equivalence in (B.0.2).

Appendix B. No Dimension Drop is equivalent to Weak Almost Unique Coding 117

=⇒ direction We claim that the conditions in the definition of WAUC are satisfiedwith

BH := (ΣH \ ΣH)⋃

ı ∈ ΣH : µξ(ı) 6= δı

.

By assumption µq(BH) = 0. Moreover, for any ı ∈ ΣH \ BH we have ı ∈ ΣH,so the probability measure µξ(ı) exists and µξ(ı) = δı. If ∈ ξ(ı) \ BH thenξ( ) = ξ(ı), thus

δı = µξ(ı) = µξ( ) = δ.

That is ı = . We showed that for every ı ∈ ΣH \ BH : ξ(ı) \ BH = ı.

⇐= direction Clearly, WAUC is equivalent to the existence of BH with µq(BH) = 0such that

if ı 6∈ BH then ξ(ı) \ BH = ı . (B.0.4)

Using (B.0.1) for BH we obtain that the set

ΣH :=

ı ∈ ΣH : µξ(ı)(BH) = 0

has full measure:µq(ΣH) = 1, (B.0.5)

where we remind the reader that ΣH is the set of those ı ∈ ΣH for which theconditional probability measure µξ(ı) exists. Assume that ı ∈ ΣH. Then

µξ(ı) (ξ(ı) \ BH) = 1 and by (B.0.4): ξ(ı) \ BH = ı .

That is µξ(ı)(ı) = 1 whenever ı ∈ ΣH. Combining this with (B.0.5) we get thatfor a µq-full measure set of ı we have µξ(ı) = δı.

119

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