ANALYTICITY OF DIMENSIONS FOR HYPERBOLICSURFACE DIFFEOMORPHISMS

M. POLLICOTT

Abstract. In this note we give a simple proof that the HausdorffDimension of the basic set for a real analytic Smale horseshoe mapdepends analytically on the transformation. This method is basedon the use of dynamical zeta functions. We prove analogous state-ments for the value of the pointwise dimension of the measure ofmaximal entropy and then use this to address an interesting ques-tion raised by Damanik and Gorodetski.

1. Introduction

The analytic dependence of the Hausdorff dimension of dynamicallydefined sets on the defining maps has been studied in a number ofsettings. For example, Ruelle’s work on hyperbolic Julia sets [15], whichin turn was inspired by Bowen’s work on limits sets for Quasi-Fuchsiangroups [2]. In this note we describe analogous results for Axiom Adiffeomorphisms on surfaces.

Let ⇤ be a basic set for a real analytic Axiom A diffeomorphismf : M ! M on a compact surface M . We can associate to the set itsHausdorff dimension dimH(⇤f ). Let us consider a one parameter familyof real analytic Axiom A diffeomorphisms f� 2 C!(M), � 2 (�✏, ✏),and let ⇤f� denote the basic sets associated by structural stability.

Theorem 1. The Hausdorff dimension dimH(⇤f�) of the basic set ⇤f�for a C! surface diffeomorphism depends analytically on � 2 (�✏, ✏).

The dependence was shown to be continuous by Manning and Mc-Cluskey [10] and C1 by a result of Mañé [9] (cf. also [7]). The approachin [9] was to first prove the smooth dependence of the structural sta-bility map and the associated potential, and then deduce from thisthe results on dimension. A similar approach was used in [6] to showboth the C1 and C! dependence of the topological entropy for Anosovflows. We do not know a suitable reference in the analytic case, where

M. Pollicott, Mathematics Institute, University of Warwick, Coventry, CV4 7AL,UK.e-mail: mpollic@maths.warwick.ac.uk.

1

2 M. POLLICOTT

an extra complication is that one would have to deal with infinite di-mensional Banach manifolds and their complexifications. There arerelated results stated for complex Hénon attractors in [16] and [17],which both rely on an extension of Mañé’s Banach space approach tothe analytic function (but without providing details cf. p.860 of [16]).However, we bypass this difficulty entirely by adopting a different “finitedimensional approach” using individual closed orbits, which works par-ticularly well in establishing analyticity and is similar in spirit to bothRuelle’s original approach for real repeller’s in [15] and the approachin [6].

We can also consider a related problem. We can associate to themeasure of maximal entropy µf its Hausdorff measure dimH(µf ), i.e.,the infimum of the Hausdorff dimensions of sets of full measure.

Theorem 2. The Hausdorff dimension dimH(µ�) of the maximal mea-sure µf� depends analytically on � 2 (�✏, ✏).

Even in the case that ⇤ is the entire surface, Theorem 2 has non-empty content.

In particular, we have the following corollary, which was conjecturedby Damanik-Gorodetski ([3], §1, Remark (i)).

Corollary 3. The density of states for the weakly coupled FibonacciHamiltonian depends analytically on the potential.

In section 2 we give some basic definitions and preliminary resultson Hausdorff dimension. In section 3 we introduce a complex function,called the differential zeta function, and some properties of its mero-morphic extension. Then in section 4 we show how to use these resultsto prove Theorem 1.

In section 5 we give some basic definitions and preliminary resultsfor pointwise dimension. In section 6 we introduce two new complexfunctions, a zeta function and an eta function, and some properties oftheir meromorphic extensions. Then, in section 7 we use these resultsto prove Theorem 2.

In section 8 we recall how Corollary 3 follows from Theorem 2.In Appendix 9 we sketch the proof of Lemma 16 from section 6.

2. Basic sets and their Hausdorff dimension

We begin by recalling some classical results on Axiom A diffeomor-phisms [1]. Let f : M ! M be a real analytic diffeomorphism of acompact manifold M .

Definition 4. We call a f -invariant closed set ⇤ ⇢ M a basic set if:

ANALYTICITY OF DIMENSION 3

(1) f : ⇤ ! ⇤ is hyperbolic, i.e., there exists a Df -invariant split-ting TM = E+ � E� and constants C > 0, 0 < � < 1 suchthat(a) kDfn|E+k C�n for n � 0; and(b) kDf�n|E�k C�n for n � 0;

(2) there exists an open neighbourhood U � ⇤ such that ⇤ = \n2Zf�nU ;(3) f : ⇤ ! ⇤ is transitive; and(4) the periodic orbits are dense in f : ⇤ ! ⇤.

The hyperbolicity is reflected in stable and unstable manifolds ofAxiom A basic sets.

We next recall the definition of the Hausdorff dimension.

Definition 5. Given a set X, d > 0 and ✏ > 0 we define

Hd✏ (X) = infU

(X

i

diam(Ui)d)

where the infimum is over all open covers U = {Ui} whose elementshave diameter at most ✏ > 0. The Hausdorff dimension is given by

dimH(X) = inf{d � 0 : lim✏!0

Hd✏ (X) = 0}.

Let �f = dimH(⇤f ) denote the Hausdorff Dimension of the limit set⇤.

Definition 6. Given ✏ > 0 we can associate to each point x 2 ⇤the (local) stable and unstable manifolds W+✏ (x) = {y : d(fnx, fny) ✏,8n � 0} and W�✏ (x) = {y : d(f�nx, f�ny) ✏,8n � 0}

In particular, f(W+✏ (x)) ⇢ W+✏ (f(x)) and f�1(W�✏ (x)) ⇢ W�✏ (f�1(x)).Let �+f = dimH(⇤f \W

+f )(x) and �

�f = dimH(⇤f \W

�f (x)) be the

Hausdorff dimension of the intersection of ⇤f with the local stable andunstable manifolds, respectively.

Lemma 7 (Manning-McCluskey). �+f and ��f are independent of x 2

⇤f and we can write �f = �+f + ��f .

This can be deduce from Theorem 1 and Theorem 2 in [10].

3. Zeta functions

There is a useful characterization of the dimensions �+, �� in termsof the differential zeta functions [11]. Let Fix(fn) = {x 2 ⇤ : fnx = x}be the set of fixed points for fn.

4 M. POLLICOTT

Definition 8. For any periodic point fn(x) = x 2 ⇤, we can associatethe two weights

��f (x) = � log |Dxfn|E�x | and �+f (x) = � log |Dxf

n|E+x |.

We can then define the stable differential zeta functions and unstabledifferential zeta functions, respectively, by

⇣+f (s) = exp

0

@1X

n=1

1

n

X

x2Fix(fn)e�s�

+f (x)

1

A

and

⇣�f (s) = exp

0

@1X

n=1

1

n

X

x2Fix(fn)e�s�

�f (x)

1

A .

These converge for Re(s) sufficiently large and |z| sufficiently small.

We can define Lipschitz functions �+ : ⇤ ! R and �� : ⇤ ! R by

�+(x) = � log |Dxf |E�x | and ��(x) = � log |Dxf |E+x |,

respectively. The Lipschitz regularity of the functions is a consequenceof the Lipschitz regularity of the stable and unstable laminations.

We denote for each t 2 R,

P (�t�+(x)) := lim supn!+1

1

nlog

X

x2Fix(�n)exp

�t

n�1X

k=0

�+(fkx)

!

= lim sup

n!+1

1

nlog

X

x2Fix(�n)e�t�

+(x)

and

P (�t��(x)) := lim supn!+1

1

nlog

X

x2Fix(�n)exp

�t

n�1X

k=0

��(fkx)

!

= lim sup

n!+1

1

nlog

X

x2Fix(�n)e�t�

�(x)

A lot is known about the domain of these zeta functions, but for ourpurposes we only require the following:

Lemma 9. The differential zeta functions satisfy the following proper-ties:

ANALYTICITY OF DIMENSION 5

(1) ⇣+f (s) converges to an analytic function on the half-plane Re(s) >�+f and has a meromorphic extension to a neighbourhood of asimple pole at s = �+f ; and

(2) ⇣�f (s) converges to an analytic function on the half-plane Re(s) >��f and has a meromorphic extension to a neighbourhood of asimple pole at s = ��f .

Proof. It is immediate from the definitions that; ⇣+f (s) converges toan analytic function provided P (�Re(s)�+(x)) < 0; and ⇣�f (s) con-verges to an analytic function provided P (�Re(s)��(x)) < 0; anddiverges provided P (�Re(s)��(x)) > 0. Furthermore, by [10], t 7!P (�t�+f ), P (�t�

�f ) are strictly decreasing and P (����+) = 0 and

P (�+�+) = 0, giving the half-planes of convergence.The meromorphic extensions to ⇣�(s) and ⇣+(s) to neighbourhoods

of �� and �s, respectively, follows from a result of Ruelle (Theorem 1of [14], and §5.20 of [13]) (cf. also [12], §5).

⇤

4. Proof of theorem 1

Consider the family of analytic functions f�, �✏ < � < ✏. A key tothe proof of the theorems our approach if the following simple lemma.Lemma 10. There exists neighbourhoods (�✏, ✏) ⇢ U ⇢ C and �� 2V ⇢ C such that U ⇥ V 3 (�, s) 7! ⇣�f�(s)

�1 is bianalytic. There is acorresponding result for ⇣+f�(s)

�1.

Proof. By the basic structural stability theorem for Axiom A diffeo-morphisms there is a natural bijection between the periodic points off = f0 and f = f�. We write fn� (x�) = x� for the associated periodicpoint and then it is easy to see that (�✏, ✏) 3 �! x� 2 M is analyticand, moreover, there exists a common neighbourhood U � (�✏, ✏) forthe complexification U 3 � ! x�, where U ⇢ C works for any pe-riodic point (cf. [6], Proposition 1.1). Thus in the region where thezeta function ⇣�f�(s) converges to non-zero analytic function we see that(�, s) 7! ⇣�f�(s)

�1 is bi-analytic. Thus using Hartog’s theorem [8] wehave that (�, s) 7! ⇣�f�(s)

�1 is bi-analytic ⇤Let us assume without loss of generality that � ⇢ V is a simple

closed curve inside which ⇣�f�(s)�1 contains the single zero ��f� . We can

then use Cauchy’s theorem to write

��f� =1

2⇡i

Z

�

⇣�f�0(⇠)

⇣�f�(⇠)⇠d⇠,

6 M. POLLICOTT

which, when combined with Lemma 10, allows us to deduce that U 3� 7! ��f� is analytic. A similar argument involving ⇣

+f�

(s) applies toshow that � 7! �+f� .

The proof of Theorem 1 is now a simple consequence of Lemma 7since

U 3 � 7! dimH(⇤f�) =: �f = �+f� + ��f�

is analytic, as required.

5. Pointwise dimension

We recall that for a given f -invariant measure µ that the pointwisedimension at x is defined by

dµ(x) = lim✏!0

log µ(B(x, ✏)))

log ✏,

when the limit converges.The stable and unstable manifolds provide a local product structure,

i.e., for sufficiently small ✏ > 0 there exists � > 0 such that for eachx 2 ⇤ the map

[·, ·]x : (W+✏ (x) \ ⇤)⇥ (W+✏ (x) \ ⇤) ! ⇤defined by

[y, y0]x = W�✏ (y) \W+✏ (y0)

is a local homeomorphism.

Lemma 11. In the particular case of the measure of maximal entropy,we can locally write the measure as a product µ+ ⇥ µ�.

We can then write

d�µ (x) = lim✏!0log µ�(B�(x, ✏))

log ✏,

where B�(x, ✏) ⇢ W�(x, ✏) is an ✏-ball in the unstable manifold, and

d+µ (x) = lim✏!0log µ+(B�(x, ✏))

log ✏,

where B+(x, ✏) ⇢ W+(x, ✏) is an ✏-ball in the unstable manifold.A useful characterization of d+µ (x) and d�µ (x) is given by the following

lemma [18].

Lemma 12 (L.-S. Young). For any Axiom A diffeomorphism:(1) The functions d+µ (x) and d�µ (x) are constant a.e. (µ).(2) We have that dµ(x) = d+µ (x) + d�µ (x)

ANALYTICITY OF DIMENSION 7

(3) The pointwise dimensions given by

d�µ (x) =h(f)R

��(x)dµ(x)and d+µ (x) =

h(f)R�+(x)dµ(x)

.

Here the topological entropy h(f) is constant under perturbationand so the remaining problem is to understand the dependence of theLyapunov exponent.

6. Generalized Zeta functions

We can define the following generalizations of the differential zetafunctions.

Definition 13. The generalized stable differential zeta functions andunstable differential zeta functions and defined, respectively, by

⇣+f (z, s) = exp

1X

n=1

zn

n

X

fnx=x

e�s�+f (x)

!

and

⇣�f (z, s) = exp

1X

n=1

zn

n

X

fnx=x

e�s��f (x)

!

which converge to a non-zero analytic function for Re(s) sufficientlylarge and |z| sufficiently small.

The generalization of Proposition 9 is the following.

Lemma 14 (after Ruelle). The zeta functions have the following prop-erties:

(1) The zeta function ⇣�f (z, s) converges to a non-zero analytic func-tion on the domain |z| < exp (P (�Re(s)��)) has a meromor-phic extension to a neighbourhood of (e�h(f), 0). More precisely,

⇣�f (z, s)(1� zeP (�s��)

)

is analytic in a neighbourhood of (e�h(f), 0); and(2) The zeta function ⇣+f (z, s) converges to a non-zero analytic func-

tion on the domain |z| < exp (P (�Re(s)�+)) has a meromor-phic extension to a neighbourhood of (e�h(f), 0). More precisely,

⇣+f (z, s)(1� zeP (�s�+)

)

is analytic in a neighbourhood of (e�h(f), 0)

8 M. POLLICOTT

This is again a consequence of a more general result of Ruelle ([13],§5.29), cf. also [12], Theorem 5.6.

We need to modify the complex functions to take a form which allowsthe Lyapunov exponents to be read off more easily. Let us denote thefollowing routine lemma.

Definition 15. We define:

⌘�f (z) = �@

@slog ⇣�f (z, s)|s=0 =

1X

n=1

zn

n

X

fnx=x

��f (x)

and

⌘+f (z) = �@

@slog ⇣+f (z, s)|s=0 =

1X

n=1

zn

n

X

fnx=x

�+f (x)

which both converges for |z| < e�h(f).

This brings us to the following fairly routine result.

Lemma 16. Let µf is the measure of maximal entropy for f (i.e.,h(µf ) = h(f)).

(1) The function ⌘�f (z) converges to a non-zero and analytic for|z| < e�h(f) has a meromorphic extension to a neighbourhood ofa simple pole at z = e�h(f) with residue e�h(f)

R��dµf .

(2) The function ⌘+f (z) onverges to a non-zero and analytic for |z| <e�h(f) has a meromorphic extension to a neighbourhood of asimple pole at z = e�h(f) with residue e�h(f)

R�+dµf .

This follows easily from the results in [13] and [12], but we sketchthe details in the appendix for the reader’s convenience.

7. Proof of Theorem 2

Consider the family of analytic functions f�, � 2 (�✏, ✏). A key tothe proof of the theorems our approach if the following simple lemma.

Lemma 17. There exists neighbourhoods (�✏, ✏) ⇢ U ⇢ C and e�h(f) 2V ⇢ C such that U ⇥ V 3 (�, s) 7! ⌘�f�(z)

�1 is bianalytic. There is acorresponding result for ⌘+f�(z)

�1.

Proof. The proof if analogues to that of Lemma 10. By the basicstructural stability theorem for Axiom A diffeomorphisms there is anatural bijection between the periodic points of f = f0 and f = f�. Wewrite fn� (x�) = x� for the associated periodic point and then it is easy tosee that (�✏, ✏) 3 �! x� 2 M is analytic and, moreover, there exists acommon neighbourhood U � (�✏, ✏) for the complexification U 3 �!

ANALYTICITY OF DIMENSION 9

x�, where U ⇢ C works for any periodic point. Thus in the region wherethe eta function ⌘�f�(z) converges to non-zero analytic function we seethat (�, z) 7! ⌘�f�(z)

�1 is bianalytic. Thus using Hartog’s theorem [8]we have that (�, z) 7! ⌘�f�(z)

�1 is bianalytic ⇤

Let us assume without loss of generality that � ⇢ V is a simpleclosed curve inside which ⌘�f�(z)

�1 contains the single zero e�h(f). Wecan then use Cauchy’s theorem to write the reside:

e�h(f)Z��dµf� =

1

2⇡i

Z

�

⌘�f�(⇠)d⇠,

which, when combined with Lemma 17, allows us to deduce that U 3� 7!

R��dµf� is analytic. A similar argument involving ⌘

+f�

(z) appliesto show that U 3 � 7!

R�+dµf� is analytic.

The proof of Theorem 2 is now a simple consequence of Lemma 12since

U 3 � 7! dµf = d+µf + d�µf

=

h(f)R��dµf�

+

h(f)R�+dµf�

is analytic, as required.

8. Application to the density of states: Proof ofcorollary 3

We can consider the Fibonacci Hamiltonian:

(HV,! )(n) = (n + 1) + (n� 1) + V �[1�↵,1)({n↵+ !}) (n)

for 2 l2(Z), where 0 ! < 1, ↵ is the Golden Mean and V > 0,sufficiently small, is a parameterization of the family. The spectrum isindependent of the choice of !, by unique ergodicity of irrational rota-tions on the circle and we denote it by Spect(HV ). Since the operatoris self-adjoint the spectrum Spect(HV ) is contained in the real line and,finally, the spectrum is a Cantor set of zero Lebesgue measure.

By the spectral theorem, there is a spectral measure µV,! supportedon Spect(HV ) ⇢ R such that for any continuous compactly supportedcontinuous function g : R ! R we have h�0, g(Hv)�0i =

Rg(t)dµV,!(t).

The density of states measure µV on Spect(HV,!) ⇢ R is then givenby integrating with respect to !, i.e., for any continuous compactlysupported continuous function g : R ! R we have

Z 1

0

h�0, g(HV )�0id! =Z

g(t)dµV (t).

10 M. POLLICOTT

We can associate to µV its pointwise dimension, i.e., for a.e. (µV )x 2 Spect(HV,!) we have

dµV (x) := lim✏!0log µV (B(x, ✏))

log ✏

exists, and is constant.In [3] interesting relationship between the measure µV and the point-

wise dimension of the measure of maximal entropy for a horseshoe. Todefine the corresponding hyperbolic map, we can consider the tracemap T : R3 ! R3 defined by

T (x, y, z) = (2xy � z, x, y).For each V , this map preserves the surface

SV =

⇢(x, y, z) 2 R3 : x2 + y2 + z2 � 2xyz = 1 + V

2

4

�

and we will consider the restrictions T |SV , and let ⌦V ⇢ SV be thenon-wandering set of this map. For V > 0 sufficiently small the set ⌦Vis a basic set of an Axiom A diffeomorphism which depends analyti-cially on V . The Hausdorff dimension of the spectrum coincides withthe Hausdorff dimension if the intersection of ⌦V with local unstablemanifolds and depends analytically (see [4], Theorem 6.5 and its proof;cf. [5]).

As before, we can consider the measure of maximal entropy µ forT : SV ! SV and the pointwise dimension dµ� of its projection alongthe stable direction.

The following nice result relates the value of the pointwise dimen-sion of the density of states measure µV to the value of the pointwisedimension of the maximal measure d�µ on unstable manifolds [3].

Proposition 18 (Damanik-Gorodetski). dµV = d�µDamanik and Gorodetski observed that the value of dµV depends

smoothly on V . We have as a corollary of our theorem that the valueof dµV depends analytically on V .Remark 19 (Multifractal Analysis). Given 0 < V0 < V < V1, for each↵ we can associate

F(↵, V ) = dimH{x : dµV (x) = ↵}.Multifractal analysis gives that we can choose 0 < a < b such that

(a, b)⇥ (V0, V1) ! R(↵, V ) 7! F(↵, V )

is analytic.

ANALYTICITY OF DIMENSION 11

9. Appendix : Proof of Lemma 16

This is a simple exercise using symbolic dynamics (and we refer thereader to [12] for more details). We can model the diffeormorphism bya subshift of finite type � : XA ! XA and consider a Hölder functionr : XA ! R in place of either �� and then we can consider

e⇣�(z, s) = exp 1X

n=1

zn

n

X

�nx=x

e�srn(x)

!.

In particular, we can write e⇣�(z, s) = (z, s)⇣�(z, s) where (z, s) isanalytic in a neighbourhood of (e�h(f), 0). We can then write

e⌘�f (z) := �@

@slog

e⇣�(z, s)|s=0

= ⌘�f (z)�@

@s (z, s)|s=0/ (z, 0)

where e⌘�f (z) and ⌘�f (z) have the same residue at the simple pole z =

e�h(f) (and where we observe h(f) = h(�)). Observe that e⌘�f (z) has apole at z = e�h(f) since

e⇣�(z, s) = ⇢(z, s)1� zeP (�sr) ,

where ⇢(z, s) is analytic and non-zero in a neighbourhood of (e�h(f), 0)cf. [12]. Thus taking the logarithmic derivative in s gives:

⇣@P (�sr)

@s

⌘zeP (�sr)

1� zeP (�sr) +@⇢@s (z, s)

⇢(z, s)

and then setting s = 0 gives that the residue for e⌘�f (z) (and thus ⌘�f (z))

at z = e�h(f) is

lim

z!e�h(f)

�z � e�h(f)

�⇣

@P (�sr)@s |s=0

⌘

1� zeh(f)

= e�h(f) limz!e�h(f)

✓z � e�h(f)

e�h(f) � z

◆✓@P (�sr)

@s

◆

= e�h(f)Z��dµf

since(1) eP (�sr) ! eh(f) as s ! 0;(2) @P (�sr)@s |s=0 = �

R��dµf

12 M. POLLICOTT

(cf. [12], Proposition 4.10). The identity for the residue for e⌘+f (z) (andthus ⌘+f (z)) is proved analogously.

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