AN IMPROVEMENT OF J. RIVERA-LETELIER RESULT ON WEAK HYPERBOLICITY ON PERIODIC ORBITS FOR POLYNOMIALS

We prove that for f : Ī C → Ī C a rational mapping of the Riemann sphere of degree at least 2 and Ω a simply connected immediate basin of attraction to an attracting fixed point, if |(fn)0(p)| ≥ Cn for constants ξ > 0, C > 0 all positive integers n and all repelling periodic points p of period n in Julia set for f , then a Riemann mapping R : ID → Ω extends continuously to ĪD and FrΩ is locally connected. This improves a result proved by J. Rivera-Letelier for Ω the basin of infinity for polynomials, and 5 + ξ rather than 3 + ξ. ∗Supported by Polish KBN grant 2P03A 03425 278 Feliks Przytycki We prove the following Theorem 1. Let f be a polynomial of 1 complex variable of degree at least 2, with connected Julia set. Suppose there are constants C > 0 and ξ > 0 such that for every repelling periodic point p in the complex plane I C of period n,

We prove the following Theorem 1.Let f be a polynomial of 1 complex variable of degree at least 2, with connected Julia set.Suppose there are constants C > 0 and ξ > 0 such that for every repelling periodic point p in the complex plane I C of period n, Then a Riemann map R : Ī C \ Ī D → Ī C \ K(f ) from the complement of the closure of the unit disc ID to the complement of the filled-in Julia set in the Riemann sphere, extends continuously to Ī C \ ID.In particular Julia set is locally connected and there are no Cremer periodic orbits.
In [R] Juan Rivera-Letelier proved this under the assumption The same strategy proves in fact a stronger theorem below, in the setting of [P2], including the case of an arbitrary simply connected immediate basin of attraction to a periodic sink for a rational map of Ī C.
Theorem 2. Let f be a rational mapping on the Riemann sphere Ī C of degree at least 2 and let Ω be a simply connected immediate basin of attraction to an attracting fixed point.Suppose that (*) holds for all repelling periodic points p in Julia set for f .Then any Riemann map R : ID → Ω extends continuously to Ī D and FrΩ is locally connected.
Most part of our proof of Theorems 1 and 2 follows [R].The proof of Theorem 1 uses an invariant measure of maximal entropy.However the right measure to use in more general situations, like in Theorem 2, is an f -invariant measure ω equivalent to a harmonic measure on FrΩ viewed from Ω; it coincides with the measure of maximal entropy in the case of the basin of ∞ for polynomials.
In the situation of Theorem 2 there is however a technical difficulty, namely proving the existence of an expanding repeller X in FrΩ, such that in particular the topological entropy of f | X is arbitrarily close to the measure theoretical entropy h ω (f ), in consequence such that Hausdorff dimension HD(X) is arbitrarily close to HD(ω) = 1, see Lemma 3.This fact is a strengthening of the theorem on the density of periodic points in FrΩ, see [PZ].The proof can be obtained as in [PZ] with the use of Pesin-Katok theory and is omitted here.We devote a separate short paper [P4] to it.In the situation of Theorem 1 the existence of X is also needed in the proof, but this case is easier (see the references in [R]).
Proof of Theorem 1 (and analogously Theorem 2) reduces to checking the summability assumption in the following standard Lemma 1, see [R].Let w 0 ∈ I C \ K(f ) and ω n , n = 1, 2, ... be an increasing sequence of positive real numbers such that Definitions.We call a closed set X ⊂ J(f ) an expanding repeller if f (X) ⊂ X the map f restricted to X is open, topologically mixing and expanding.
Here expanding means that there exist C > 0 and λ > 1, called an expanding constant, such that for every x ∈ X we have C, called a repelling neighbourhood, such that every forward f -trajectory x, f (x), ...f n (x), ... staying in U must be contained in X, see for example [PU1, Ch.5].This easily implies that if {x, f (x), ...
maybe for a constant C bigger than before and U a smaller neighbourhood of X.
Let λ n , n = 1, 2, ... be an increasing sequence of positive real numbers such that for every n, every repelling periodic point p of period n has the multiplier (f n ) 0 (p) of absolute value at least λ n .
In the sequel C will denote various positive constants which can change even in one consideration.
Lemma 2, see [R].Let f be a polynomial of 1 complex variable of degree at least 2. Let X ⊂ J(f ) be an expanding repeller of positive Hausdorff dimension, HD(X) > 0, and λ be its expanding constant.Then there is U , a repelling neighbourhood of X, a "base point" w 0 ∈ I C \ K(f ) and a constant C > 0 such that the following holds.

Sketch of Proof. This Lemma in a slightly different formulation was proved in [R] and in a more rough version in
for an arbitrarily fixed a > 1/HD(X).The latter inequality assures the existence of x.Here Crit(f ) denotes the set of all f -critical points in I C. Fix an arbitrary point ŵ ∈ X and r 0 > 0 such that B 0 := B( ŵ, r 0 ) is well inside U and choose an arbitrary w 0 ∈ B 0 \ K as a base point.
Let be a minimal time such that a component V of f − (B 0 ) intersecting X is in B 00 := B(x, δn −a )), where 0 < δ << 1 is a constant.By construction f is univalent on V and has bounded distortion.Denote the branch of f − leading B 0 to V by F 1 .
(More precisely, F 1 can be constructed in two steps.First, let k be the smallest integer such that f k maps B 00 to a boundedly distorted large disc B 000 .Denote the branch of f −k leading B 000 to B 00 by F 0 1 .Next using the topological transitivity of f on X we find a branch F 00 1 of f −M on B 0 mapping it in B 000 , where M is bounded independently of n.We define , can be composed with F 3 being the composition of at most N branches of f −1 for N bounded independently of n, so that

QED
Proof of Theorem 1.Let X and other constants be as in Lemma 2. Let β 2 ≥ β 1 > 1 be constants such that for all k large enough and all y such that y, ..., f k (y) ∈ U we have Consider an arbitrary w n ∈ f −n (w 0 ).Join x = x(n) to w 0 by a hyperbolic geodesic γ = γ n in I C \ K(f ).Let x n be the end of the component of f −n (γ n ) having one end at w n , different from w n .Then we write By Lemma 2 we have We have where f −n is the branch leading x 0 to x n and w 0 to w n .Note that |x| − 1 ≥ Cd − , where C depends only on | w0 |.We estimate the fraction I by Koebe Distortion Lemma.Namely there is a constant C K depending only on w0 such that We have also, denoting g(z) = z d , using Rg = f R,

In conclusion
Invoking the estimate of we get By Pesin-Katok theory, applied to the measure of maximal entropy equal to ln d, there exists X and its repelling neighbourhood U , such that with ε, hence ε 0 , arbitrarily close to 0. So, if λ n ≥ Cn 3+ξ the assumptions of Lemma 1 are satisfied and Theorem 1 follows.QED

Remark 1 (corresponding to an observation in [R]
).The measure of maximal entropy is optimal in this construction.If µ is any f -invariant ergodic measure on J(f , where h µ (f ) is the measure-theoretic entropy.≈ means that the ratio is arbitrarily close to 1 for appropriate X. Therefore |(f n ) 0 (w n )| ≥ λ n n −2 ln d/h µ (f )−ε 0 , which attains maximum at h µ (f ) = h top (f ) = ln d, the topological entropy, giving (1).
Remark 2. The property (*) excludes an existence of parabolic periodic points in FrΩ.Otherwise we would find periodic orbits spending almost all the time close to such a parabolic point q, so its multiplier would about Cn a , where a = t/(t − 1) ≤ 2 for f m (z) = z + b(z − q) t + ... for some integer m and b 6 = 0, for z close to q.
The absence of Cremer periodic orbits follows from the local connectedness, see [R] and the references there.We do not know whether Siegel discs can exist.The proof given in [PRS] under the assumption of the uniform exponential growth of the multipliers of repelling periodic orbits ω n does not seem to work here.We do not know whether (*) implies a summability condition which would already imply the absence of Siegel discs and Cremer points due to so-called backward asymptotic stability, cf.Lemma 3. Let ν be an ergodic g-invariant probability measure on ∂ID, such that for ν-a.e.ζ ∈ ∂ID there exists a radial limit R(ζ) := lim r%1 R(rζ).Assume that the measure µ := R * (ν) has positive Lyapunov exponent χ µ (f ).Let ϕ : ∂ID → IR be a continuous real-valued function.Then for every ε > 0 there exist Y ⊂ ∂ID a g-invariant expanding repeller in the domain of R and C > 0 such that for every δ > 0 small enough there exists r(δ) < 1, such that for all r : r(δ) ≤ r < 1 and ζ ∈ Y and all positive integers n (i) (ii) X = R(Y ) is an expanding repeller for f and for every r : Then the final estimate in Proof of Theorem 1 replaces by Now we apply HD(µ) = h µ (f )/χ µ (f ) see [PU1, Ch.9] and h ν (g) = h µ (f ), see [P1] and [P2, §4].We get the latter equality for ν equivalent to length (harmonic) measure, where χ ν (g)/h ν (g) = HD(ν) = 1.
Though in this construction w 0 depends on n, this does not influence the result.We can replace at the end w 0 by a base point independent of n which changes the final estimate only by a distortion constant, which can be absorbed by ε 0 for n large enough.QED Remark 3. As in Remark 1 note that the measure ν equivalent to the length is optimal in the sense that for any other g-invariant probability measure of positive Lyapunov exponent (which implies that µ = R * (ν) also has positive Lyapunov exponent, see [P2]), as HD(ν) ≤ HD(∂ID) = 1, we obtain , the estimate which is not better.Let Ω be a simply connected domain in Ī C and f be a holomorphic map defined on a neighbourhood W of FrΩ to Ī C. Assume f (W ∩ Ω) ⊂ Ω, f (FrΩ) ⊂ FrΩ and FrΩ repells to the side of Ω, that is FrΩ. Suppose that (*) holds for all repelling periodic points p in FrΩ.Then any Riemann map R : ID → Ω extends continuously to Ī D and FrΩ is locally connected.
We do not know how to overcome troubles with finding N consecutive branches of f −1 whose composition maps F 2 (B(x, n −a )) deep in B 0 (in the notation in Proof of Lemma 2).Even if we succeed we do not know whether the periodic point p belongs to FrΩ.The problem is that we want to control every backward branch of f −n leading x into Ω, rather than (measure) typical, as in [PZ], or in accordance to some invariant hyperbolic subset of FrΩ.
Note that at least Lemma 3 holds in this setting, see [P4].
[GS] or [P3, Th.B and Remark 3.2] and [PU2, Appendix B].Now we pass to the setting of Theorem 2, where R : ID → Ω is a Riemann mapping.Let g be a holomorphic extension of R −1 • f • R to a neighbourhood of the unit circle ∂ID.It exists and it is expanding on ∂ID, see [P2, §7].Now we formulate a lemma about the existence of appropriate expanding repellers.As we mentioned in Introduction it follows from Pesin-Katok theory.For the detailed proof see [P4], developing [PZ].

Remark 4 .
It would be natural to prove a local version of Theorem 2, in the setting of [P2], assuming (*) only for periodic orbits in FrΩ.More precisely the question is whether the following holds: