Complex Analytic Methods in RealOne-dimensional Dynamics

Weixiao ShenShanghai Center for Mathematical Sciences, Fudan University

7 November 2018, Beijing

Table Of Contents

1 Backgrounds in low dimensional dynamics

2 Results in 1D dynamics

3 Sullivan-McMullen-Lyubich’s theory on Feigenbaumrenormalization

4 Revisits on Milnor-Thurston kneading theory

Time-discrete dynamics

Let M be a compact manifold (phase space) and consider a(smooth) map f : M → M. For each n ≥ 0, let

f n = f f · · · f︸ ︷︷ ︸n

.

For each x ∈ M, the orbit is

orb(x) = f n(x)∞n=0.

The theory of time-discrete dynamical systems studies theorbit structure of maps.

A dynamical system is stable if nearby dynamical systemshave roughly the same orbit structure.

A point x is called periodic if f k(x) = x for some k ≥ 1 andthe minimal positive integer k with this property is called itsperiod.

It happens that orb(x) is dense in the phase space.

Problem.

Describe the dynamical property of ‘most’ dynamical systems.

Are these properties ‘stable’ in suitable sense?

Difficulty. Sensitive dependence on initial values:

d(x , y) 1 6⇒ ∞supn=0

d(f n(x), f n(y)) 1.

Structural stability

Definition

Two maps f : M → M and g : N → N are called topologicallyconjugate if there is a homeomorphism h : M → N such thatg h = h f .

Definition

A C r map f : M → M is called C r -structurally stable if there is aneighborhood U of f in the space C r (M,M) such that all themaps in U are topologically conjugate to each other.

Problems.

Characterize structurally stable maps.

Are “most” maps structurally stable?

In 1960s, Smale posed the notion “Axiom A” and conjecturedthat it is essentially equivalent to structural stability. Ingeneral dimension, this was so far verified in the C 1 topologyfor diffeomorphsim.(Mane, Hayashi, Wen)

Smale also hoped that Axiom A maps are dense, but it wassoon realized that this is not true when dim(M) ≥ 2.

We shall see that in the case dim(M) = 1, Axiom A isessentially equivalent to structural stability and these mapsare dense in the C k topology for any k ≥ 1.

In 1960s, Smale posed the notion “Axiom A” and conjecturedthat it is essentially equivalent to structural stability. Ingeneral dimension, this was so far verified in the C 1 topologyfor diffeomorphsim.(Mane, Hayashi, Wen)

Smale also hoped that Axiom A maps are dense, but it wassoon realized that this is not true when dim(M) ≥ 2.

We shall see that in the case dim(M) = 1, Axiom A isessentially equivalent to structural stability and these mapsare dense in the C k topology for any k ≥ 1.

In 1960s, Smale posed the notion “Axiom A” and conjecturedthat it is essentially equivalent to structural stability. Ingeneral dimension, this was so far verified in the C 1 topologyfor diffeomorphsim.(Mane, Hayashi, Wen)

Smale also hoped that Axiom A maps are dense, but it wassoon realized that this is not true when dim(M) ≥ 2.

We shall see that in the case dim(M) = 1, Axiom A isessentially equivalent to structural stability and these mapsare dense in the C k topology for any k ≥ 1.

Even in the one-dimensional case, Jakobson (1981) showedthat there is a large set of dynamical systems in themeasure-theoretical sense which are not Axiom A, but are stillwell understood from stochastic point of view.

Physical measures

A Borel probability measure µ is called f -invariant if for any Borelset A ⊂ M, we have

µ(f −1(A)) = µ(A).

The invariant measure µ is called ergodic if for each Borel setA ⊂ M,

A = f −1(A)⇒ µ(A) = 0 or 1.

The basin of µ is defined as

B(µ) =

x ∈ M :

1n

∑n−1i=0 δf i (x) → µ as n→∞

in the weak star topology

.

If Leb(B(µ)) > 0 then we say that µ is a physical measure.

Examples.

A periodic point x is called hyperbolic attracting if alleigenvalues of Df k(x) lie inside the unit circle, where k is theperiod. In this case, µ := 1

k

∑ki=1 δf i (x) is a physical measure

since B(µ) contains a neighborhood of x .

If µ is an ergodic invariant probability measure and µ Leb(an acip), then µ is a physical measure. Indeed, by Birkhroff’sergodic theorem, µ(B(µ)) = 1, hence Leb(B(µ)) > 0.

Palis Conjecture (Simplified version)

Given a compact manifold M, in measure-theoretical sense, ‘most’maps f : M → M are stochastic, that is, there exists finitely manyphysical measures µ1, . . . , µm such that

the topological support of µi ’s are pairwise disjoint;

Leb (M \ (⋃m

i=1 B(µi ))) = 0.

Moreover, the statistical property of f is stable under randomperturbation.

Example.

f : T→ T, x 7→ 2x mod 1. For this map the Haar measureon T is the unique physical measure and its basin has fullmeasure in T.

1D Dynamics

Real 1D dynamics: iteration of interval/circle maps

Complex 1D dynamics: iteration of holomorphic maps onRiemann surfaces

Interplay:

A map defined by a real polynomial can be viewed as both areal and a complex 1D dynamical system.A real analytic interval map, via generalized renormalization,can be reduced to a real polynomial (Sullivan, Levin-vanStrien, Graczyk-Swiatek, Lyubich-Yampolsky, Levin, Hu, Shen,Smania, Clark-Treje-van Strien)

The logistic family

Jakobson’s Theorem (1981)

Consider the logistic family Qa(x) = x2 + a, −2 ≤ a ≤ 1/4. Thenthere is a subset J of parameters of positive Lebesgue measuresuch that for a ∈ J , Qa has a unique physical measure which isergodic and absolutely continuous.

Purely real analytic method.

Lyubich’s Dichotomy Theorem (2002)

For almost every a ∈ [−2, 1/4], either Qa satisfies Axiom A or ithas a unique physical measure which is ergodic and absolutelycontinuous.

Mainly complex analytic method. Solved Palis’ conjecture for thelogistic family. Continued by Avila-Lyubich-de Melo, Avila-Moreira,Avila-Shen-Lyubich, Avila-Lyubich

Kozlovski-Shen-van Strien’s Theorem (2007)

For any d = 2, 3 . . ., Axiom A maps form an open and densesubset of the space of real polynomials of degree d.

Here, a map f : R→ R is called Axiom A if for each of its criticalpoints c , f n(c) converges to an attracting periodic orbit or toinfinity. For d = 2, the theorem is due to Lyubich (1997),Graczyk-Swiatek (1997). Mainly complex analytic method.

Corollary (Structural stability conjecture holds in 1D)

Let M = [0, 1] or S1.

For any k ≥ 1, if a C k map f : M → M is C k structurallystable then f satisfies Axiom A.

C k -structurally stable maps are open and dense in the C k

topology.

This solves the second part of Smale’s 11-th problem for the 21stcentury.

Complex analytic methods played important roles in the followingproblems:

Milnor-Thurston’s monotonicity problem (1980s)

Conceptual proof of Feigenbaum renormalization (1988-)

Milnor’s attractor problem (1991)

Density of hyperbolicity (1997-)

Palis conjecture for unimodal maps

Complex analytic methods have NOT played essential roles in thefollowing problems:

Non-existence of wandering intervals

Statistical property of interval maps: existence of acip,stochastic stability

Existence of wild attractor

Problems on interval maps with non-integral critical order

qc map

Definition

A homeomorphic ϕ : U → V between two open sets in C is calledK -qc if

ϕ has locally integrable partial derivatives in the sense ofdistribution; ∣∣∣∣∂ϕ∂ϕ

∣∣∣∣ ≤ K − 1

K + 1, a.e..

A qc map is differentiable a.e., and the classical partialderivatives coincide with the distributional ways.

The space of all K -qc map of C normalized at two distinctpoints is compact.

A unimodal map is a continuous map from an interval I into itselffor which there is a unique point c ∈ I o such that f is strictlyincreasing on the left of c and strictly decreasing on the right of c(or vice versa). E.g.

fc(x) = x2 + c .

fc(x) = |x |` + c , ` > 1.

fc(x) = be−1/|x |`

+ c , b > 0, ` > 1.

Feigenbaum renormalization

Let f : I → I be a unimodal map with a maximum at the criticalpoint 0. f is called Feigenbaum-renormalizable, if there exists aclosed interval J which contains 0 in its interior such that

f 2(J) ⊂ J, f 2(∂J) ⊂ ∂J and f 2 : J → J is unimodal;

the intervals J and f (J), have pairwise disjoint closure.

In fact,

f is Feigenbaum renormalizable ⇔ f 2(0) < 0 < f 4(0) ≤ f 3(0) < f (0).

Let RF f : I → I denote the unimodal map which is affineconjugate to f 2 : J → J and which has a maximum at 0.

It could happen that RF f is again Feigenbaum renormalizableand then we obtain a second renormalization R2

F . If theprocedure can be continued indefinitely then we obtain asequence of unimodal maps Rn

F f and say that f is infinitelyrenormalizable in the sense of Feigenbaum.

By numerical experiments, Feigenbaum found that thesequence Rn

F f converges and moreover, the limit isindependent of the map f where we start with.

It could happen that RF f is again Feigenbaum renormalizableand then we obtain a second renormalization R2

F . If theprocedure can be continued indefinitely then we obtain asequence of unimodal maps Rn

F f and say that f is infinitelyrenormalizable in the sense of Feigenbaum.

By numerical experiments, Feigenbaum found that thesequence Rn

F f converges and moreover, the limit isindependent of the map f where we start with.

More generally, given a permutation σ : 0, 1, . . . , s − 1 oforder s ≥ 2, we say that f is σ-renormalizable if there exists aclosed interval J containing 0 in its interior such that

f (f j(J)) ⊂ f σ(j)(J), ∀0 ≤ j < s

and the intervals J, f (J), . . . , f s−1(J) have pairwise disjointinterior. Let Rσf denote the unimodal map affine conjugateto f s : J → J with maximal at 0.

The Feigenbaum renormalization corresponds to the cases = 2 and σ = (10).

We say that a unimodal map f is infinitely renormalizablewith combinatorics (σn)∞n=0 if fn = Rσn−1fn−1 are well definedfor all n ≥ 1, where f0 = f .

It is called infinitely renormalizable of bounded type ifσn : n ≥ 0 is finite.

Feigenbaum’s Conjecture

Consider a suitable space B of unimodal maps f : [−1, 1] and letB0 be the subspace of all renormalizable maps. Then therenormalization map RF : B0 → B has a unique fixed point f∗.The fixed point is hyperbolic and has codimensional one stablemanifold consisting of all infinitely renormalizable maps in thesense of Feigenbaum.

Lanford gave computer-assisted proof for suitable B.

Sullivan’s conceptual proof

Theorem (Sullivan’s Beau Bounds Theorem)

Let f be an infinitely renormalizable C 3 unimodal map, of boundedtype, with maximum at 0 and such that f ′′(0) 6= 0. Then thesequence Rnf is precompact in C 1 and any possible limit map isreal analytic which has a quadratic-like extension with a universalbounds.

A quadratic-like map is a proper holomorphic map F : U → Vof degree 2, where U,V are topological disks.

Sullivan’s universal bounds refers to a lower bound on µ =mod (V \ U).

By Douday-Hubbard’s straightening theorem, F is K (µ)-qcconjugate to a quadratic polynomial.

Theorem (Levin-van Strien, Lyubich-Yampolsky, Graczyk-Swaitek,Shen, Clark-Treje-van Strien...)

Let f be an arbitrary real analytic interval map and let c be anon-periodic recurrent critical point. Then there is an arbitrarilysmall neighborhood J of c such that the first return map to J has(generalized) polynomial-like extension with universal bounds.

Theorem (Sullivan’s Rigidity Theorem)

Let f , g be quadratic-like infinitely renormalizable unimodal mapsof the same bounded type. Then f and g are qc conjugate.

Sullivan also proved that Rnf converges to a unique fixed point f∗of R and Rnf → f∗ for any infinitely renormalizable f . This proofof this part was simplified and strengthened by McMullen:

Theorem (McMullen 1994)

Let f , g be quadratic-like infinitely renormalizable unimodal mapsof the same bounded type. Then ‖Rnf −Rng‖C0 converges tozero exponentially fast.

Avila-Lyubich generalized the result to unbounded type.

Theorem (Kozlovski-Shen-van Strien 2007)

Let f , g be real polynomials of the same degree d with all criticalpoints real and non-degenerate and without neutral periodicpoints. If f , g are topologically conjugate in R, then they arequasiconformally conjugate in C.

Remark:

The case d = 2 is due to Lyubich (1997), Graczyk-Swiatek(1997), which uses special geometric property of quadraticpolynomials.

Rigidity Problem.

1 Any two combinatorially equivalent rational maps f , g are qcconjugate.

2 A qc conjugacy is conformal a.e. on the Julia set unless f is aLattes example.

Remark:

If f , g are C 1 conjugate, then all the corresponding periodicpoints have the same multiplier which would imply that f , gare conformally conjugate.

A Lattes example is a rational map which is doubly converedby an integral torus endomorphism.

Lyubich’s contribution

Theorem (Lyubich 2002)

The Feigenbaum conjecture holds if we take B bequadratic-like unimodal maps.

Extension of Feigenbaum conjecture to all (real)combinatorics.

In particular, Lyubich gave a simple proof of McMullen’s result:

Let V be a small neighborhood of [−1, 1] and let B denote theBanach space of all bounded holomorphic maps from V to C.

The subspace of B consisting of all infinitely renormalizablemaps is a complex submanifold, denoted by Bs .

By Sullivan’s Beau bounds Theorem, Theorem, there exists Nsuch that RN maps Bs(f∗, 2ε) into Bs(f∗, ε). ThusRN |Bs(f∗, ε) is contracting due to the Schwarz Lemma.

Another important contribution of Lyubich is existence ofunstable direction of f∗.

He deduced it from a Rigidity Theorem via a small orbitsargument: If there were no unstable direction, then therewould be f such that Rnf converges to f∗ slowly. But f ishybrid equivalent to f∗ and this is impossible.

Using his result on renormalization, Lyubich proved that inthe quadratic family, almost every map is NOT infinitelyrenormalizable.

Smania’s approach in the multimodal case

Theorem (Smania, to appear)

Extension of the Feigenbaum conjecture to multimodal case withbounded combinatorics.

Smania’s approach is based on solving the following cohomologousequation:

α(f (z))− Df (z)α(z) = v(z).

Corollary (Smania)

In the real cubic family fa,b(x) = x3 − 3a2x + b, the set ofparameters corresponding to infinitely renormalizable maps ofbounded type has Lebesgue measure zero.

Martens’ result

The Feigenbaum conjecture is expected to hold for unimodalmaps with critical order ` > 1: maps with

f (x)− f (c) = φ(|x − c |`).

But the methods discussed above only applies to the caseswhere ` is an even integer.

For general `, Martens (1996) proved existence of arenormalization fixed point.

He also claimed that the fixed point has a stable manifold ofcodimension ≤ 1.

More on Lyubich’s Dichotomy Theorem

To obtain the Dichotomy Theorem, Lyubich also proved thatamong all non-infinitely renormalizable, non-hyperbolic maps,almost every map has enough expansion, so that acip exists.

The one-dimensional feature of the parameter space of thequadratic family (x 7→ x2 + c) is essentially used in the proof:

the fine structure of the one-dimensional parameter space(para-puzzle), developed by Douady-Hubbardquasiconformality of a transition map from phase space toparameter space.

It is hard to extend the argument to the multimodal case, e.g.fa,b(x) = x3 − 3a2x + b.

Maybe some generalization of Smania’s work applies here.

Kneading sequence

Associate to a unimodal map f there is a sequencei = in∞n=1 ∈ L, c ,RZ

+with

in =

L if f n(c) < c ;c if f n(c) = c ;R if f n(c) > c .

This sequence determines the relative order the post-critical orbitf n(c)∞n=0 and thus the folding pattern of iterates of f .

Topological entropy

Let `n(f ) denote the number of maximal intervals ofmonotonicity of f n. Define the entropy of f to be

htop(f ) = limn→∞

1

nlog `n(f ),

which measures the topological complexity of the dynamicalsystem f .

htop(f ) is determined by the kneading sequence i(f ).

Monotonicity of entropy

Let Qc(x) = x2 + c . Milnor-Thurston (1977) asked whetherΦ(c) := htop(Qc) is a monotone decreasing function.

Theorem (Milnor-Thurston, Sullivan, . . . )

The function Φ is monotone decreasing.

Φ is a continuous function from R onto [0, log 2].

Φ(c) = log 2 when c ≤ −2 and Φ(c) = 0 when c ≥ 0.

We say that Qc is hyperbolic if Qc has an attracting cycle.Near a hyperbolic parameter c0, Q is constant.

Guckenheimer (1980): Φ is Holder continuous.

Dobbs-Mihalache(2016 preprint) Φ′(c) = 0 holds for a.e. c .⇒ Φ is not Lipschitz.

Several proofs of the theorem exist, all using complextechniques. Important: Qc is a holomorphic map from C to C.

quasiconformal mappings, Teichmuller theory

Approach 1: rigidity

Theorem (Sullivan)

Suppose Qc and Qc are two quadratic maps with the samekneading sequence and with Qq

c (0) = Qqc (0) = 0. Then c = c .

⇒ monotonicity of entropy (Milnor-Thurston’s kneading theory)

Outline of proof of Sullivan’s theorem.

By the classical Bottcher’s theorem, Qa1 and Qa2 areconformally conjugate near ∞ and also near their criticalorbits;

Via lifting and taking convergent subsequence, we obtain a qcconjugacy which is conformal in the basin of ∞ and the orbitof 0, i.e. outside the Julia set.

Prove that the Julia set has Lebesgue measure zero, so thatthe qc conjugacy is indeed conformal (Weil’s Lemma).

Approach 2: Teichmuller Theory

Given a finite set P ⊂ C, we say that two qc maps

ϕi : (C,P)→ (C, ϕi (P))

are Teichmuller equivalent if ϕ2 ϕ−11 is homotopic rel ϕ1(P)to the affine map sending ϕ1(P) to ϕ2(P). The Teichmullerspace TP is the set of all qc map mod the Teichmullerequivalence relation.

The space TP is a complex-analytic manifold of dimension#P − 2.

It is also endowed with a Teichmuller metric such that anyholomorphic endomorphism does not increase the metric.

The cotangent space of TP at [ϕ] is naturally identified withthe space of all integrable meromorphic quadratic differentialin C with (simple) poles in ϕ(P) ∪ ∞, endowed with L1

norm.

Thurston’s algorithm

Suppose Q(x) = Qc(x) = x2 + c satisfies Qqc (0) = 0 and let

P = Q jc(0) : j ≥ 0.

Given any qc map ϕ : (C,P)→ (C, ϕ(P)) there exists aquadratic map Q and qc map ψ : (C,P)→ (C, ψ(P)) suchthat

Q ψ = ϕ Q.

The map ϕ 7→ ψ descends to a holomorphic map

σ : TP → TP

which fixes the [id ]. It is non-decreasing with respect to theTeichmuller metric.

Sullivan’s rigidity theorem follows once we prove that σ hasno non-trivial fixed point. In particular, once we prove that σis strictly contracting.

Extremal mapping

Outline of Milnor-Thurston’s proof.

Assume that σf has a non-trivial fixed point [ϕ] and assume ϕis an extremal mapping in the equivalence class, i.e. a K -qcmap in the homotopy class with minimal K .

By a theorem of Teichmuller asserts that there is an integrablemeromorphic quadratic v(z)dz2 with poles in P ∪ ∞ suchthat

∂ϕ

∂ϕ= k

v(z)

|v(z)|.

However, [σf (ϕ)] = [ϕ] ⇒ Q∗(v(z)dz2) = v(z)dz2 ⇒ v(z)has infinitely many poles, a contradiction!

A problem on extremal mappings

Let α : 0 = a1 < a2 < · · · < aq and β : 0 < b1 < b2 < · · · < bq.For θ ∈ (0, 2π), let

Sθ = re it : 0 < t < θ.

Let K (θ) = Kα,β(θ) denote the minimal number K for which thereexists a K -qc map h : Sθ → Sθ such that

h(aj) = bj , h(ajeiθ) = bje

iθ.

Problem. Is K (θ) monotone decreasing?

An affirmative answer to this question will impliesmonotonicity of entropy for the family c 7→ |x |` + c for each` ≥ 2.

K (θ/n) ≤ K (θ) for each integer n ≥ 1.

Yes if q = 2. (Cui)

Approach 3: Transversality

Theorem (Tsujii 1998)

Suppose 0 is of period q for Qa. Then

dQqt (0)dt

∣∣∣t=a

DQq−1a (a)

=

q−1∑n=0

1

DQna (a)

> 0.

⇒ a 7→ `n(Qa)⇒ Φ(a).

Transversality appears in other problems of complex dynamics.Epstein and Levin, among others, worked in this problem.

Tsujii’s proof still uses global analytic structure of the maps

Outline of proof of Tsujii’s theorem.

Write f = Qa and P = f j(0). Let Ω denote the space ofintegrable meromorphic quadratic differentials with poles inP ∪ ∞.The co-derivative of σ : TP → TP at [id ] is given by thepushforward operator

f∗(v(z)dz2) =

∑w∈Q−1(z)

v(w)

f ′(w)2

dz2.

∫V|f ∗(v(z)dz2)| ≤

∫f −1V

|v(z)dz2|.

det(Id − f∗) =

q−1∑n=0

1

Df n(f (0)).

Let V0 = B(0,R) with R >> 1 and let Vn = f −n(V0). Then

area(Vn \ Vn+1)→ 0.

For each v ∈ Ω,∫V0\V1

|f n∗ (v)| ≤

∫Vn\Vn+1

|v | → 0.

⇒ the spectrum radius of f∗ is strictly less than 1.

Approach 4: algebraic

Theorem (Gleason Lemma)

Assume that 0 is a periodic point of Qc(x) = x2 + c of periodic q.Then

dQqt (0)

dt

∣∣∣∣t=c

6= 0.

The proof works also for Qc(x) = |x |` + c with ` > 1 an oddinteger and c ∈ R. But, it does not give the sign.

Outline of proof. Let A be the set of algebraic integers. Thenc ∈ A. Note that

A ∩Q = Z.

Let

∆k =dQk

t (0)

dt

∣∣∣∣t=c

.

Then ∆1 = 1 and ∆k+1 = 2Qkc (0)∆k + 1. If Qq

c (0) = 0 for someq ≥ 1, then c ∈ A. By induction,

∆k ≡ 1 mod 2A,∀k .

Thus ∆k 6= 0.

New results (joint with Levin and van Strien)

We develop method/language to deal with maps which allowspartial complex analytic extension. Roughly speaking, our methodshows

a lifting property ⇒ positive transversality.

Theorem (Levin-Shen-van Strien)

Let g : I → I be a unimodal map with critical point 0 withg(0) = 0. Assume that g |I\0 has extension to a holomorphicmap g : U → V where

U,V are bounded open sets in C;

g : U \ 0 → V \ 0 is an unbranched covering.

If the separation property

V ⊃ B(0; diam(U)) ⊃ U (1)

holds, then the entropy of gt(x) = g(x) + t is monotone in t.

Application of the theorem

A new proof for the family gt(x) = t + x2d , d ∈ Z+: Take Vto be large disk centered at zero.

Familiesgt(x) = be−1/|x |

`+ t, t ∈ [0, β],

where ` ≥ 1, b > 2(e`)1/` and 2β = e1/β`. of unimodal maps.

(with flat critical points)

Main steps of proof

Consider f := gt0 satisfying f q(0) = 0, f j(0) ∈ U \ 0,1 ≤ j < q. It suffices to show

ddt gq

t (0)∣∣t=t0

Df q−1(f (0))> 0.

Let P = f j(0) : 0 ≤ j < q. A holomorphic motion of P overDr is a family of injections hλ : P → C, λ ∈ Dr , such thathλ(0) ≡ 0, h0 = idP , and λ 7→ hλ(p) is holomorphic for eachp ∈ P.

A holomorphic motion of P is called admissible if hλ(p) ∈ Ufor all p ∈ P \ 0. Note that any holomorphic motion, whenrestricted to a small disk, is admissible.

(The lifting property) If hλ is an admissible holomorphicmotion of P over Dr , then it has a lift hλ, which is again aholomorphic motion of P over Dr ,

g(hλ(p)) = hλ(f (p))− hλ(f (0)) ∈ V \ 0

holds for all p ∈ P \ 0 and λ ∈ Dr .

An admissible holomorphic motion is called asymptoticinvariant of order m, if

hλ(p)− hλ(p) = O(λm+1) as λ→ 0.

∃ a non-trivial holomorphic motion asymptotically invariant ofoder m ⇔ gq

t (0) = O(|t − t0|m+1) as t → t0.

(Averaging process improving asymptotical invariance order) Ifthere is an admissible holomorphic motion hλ asymptoticallyinvariant of order m, then there is another one which isasymptotically invariant of order m + 1.Indeed, letting hn

λ be the successive lifts of h0λ = hλ, it suffices

to choose a convergent subsequence

Hλ := limnk→∞

1

nk

nk∑j=1

hjλ.

As gqt (0) is a holomorphic function not-identically zero, we

obtain a contradiction if transversality fails.

A deformation argument shows the sign in the inequality.

The lift process is a variation of Thurston’s pull backalgorithm, modified for maps which are locally defined. Thelift property means that the σn is well-defined in aneighborhood of [id ] in the Teichmuller space TP .

Adam Epstein seems to have a different approach, using somevariation of the Thurston’s pull-back algorithm, to obtaintransversality. It is not clear how to apply his method tolocally defined maps.

The hyperbolic case

Recently, we are able to extend the argument to the hyperbolic andparabolic case (with infinite a post-critical orbit), extending aclassical result of Douady-Hubbard-Sullivan.

Theorem (Levin-Shen-van Strien)

Let g be as in the theorem above. Assume that for some t0,f = gt0 has an attracting cycle attracting the critical point 0. Letλ(t) denote the multiplier of the attracting periodic orbits of gt fort close to t0. Then λ′(t0) 6= 0.

The main challenge in the post-critically infinite case is theAveraging Process. While Hλ exists, it is not clear whether itis a holomorphic motion (for |λ| small).

The solution is as follows: If transversalilty fails, then thecoholomogous equation

α(f (z))− f ′(z)α(z) = v(z)

has a solution holomorphic near the attracting periodic orbit.

So we can construct a holomorphic motion hλ(z), which isalso holomorphic in z near the attracting periodic orbit. Thecorresponding Hλ(z) will be a holomorphic motion by theKoebe distortion principle.

Unfortunately, for the interesting family x 7→ t + |x |`, we only have

Theorem

For each positive integer q, there exists `0(q) > 1 such that if` > `0(q) and if 0 is of period q under the map fa,`(x) = a− |x |`,then

q−1∑n=0

1

Df na (a)

> 0.

Corollary

There exists ε(`) > 0 such that ε(`)→ 0 as `→∞ such that

supa>b

htop(fb,`)− htop(fa,`) > −ε(`).

Thank you for your attention!