Poincaré Series, Pressure, and Periodic Points

Jacek Graczyk

Department of Mathematics University of Paris XI 91405 Orsay Cedex, France

Stas Smirnov

Department of Mathematics Royal Institute of Technology 10044 Stockholm, Sweden

11 February, 2003

Abstract

We study Poincaré series and pressure as functions of multipliers of repelling periodic orbits for rational functions satisfying the summability condition.

1 Introduction

Non-uniform hyperbolicity. Global attractors or repellers of unstable systems often display a very complicated fractal structure which in turn is crucial in understanding underlying dynamics. Julia set of rational function with hyperbolic periodic points is a prototype example of global holomorphic repeller. Even in the simplest quadratic case, Julia sets display a striking complexity. One of the main objectives of the paper is to explore fractal parameters of rational Julia sets through thermodynamical formalism. In the theory of iterations, equilibrium measures for conformal potentials can be used to characterize metric properties of invariant sets. This method has its origins in [2, 9] and led further to a systematic study of the socalled conformal (Sullivan-Patterson) measures, [10, 3]. Our technical arguments are based on non-uniformly hyperbolicity of [4].

Suppose that F is a rational function without parabolic periodic points. Denote by Crit the set of all critical points {c ∈ ℂ̂ : F′(c) = 0} of F. We say that F satisfies the summability condition with an exponent α if for every critical point c ∈ Crit ∩ J, there exists a positive integer n(c) such that,

∑_{n=n(c)}^{∞} |(F^{n})′(c)|^{−α} < ∞

and F does not have parabolic cycles.

For simplicity, we assume that there is not any critical point which belongs to another critical orbit. Otherwise all theorems remain valid with the following amendment: a “block” of critical points

F : c_{1} ↦ … ↦ c_{2} ↦ . . . . . . ↦ c_{k} ,  (1)

of multiplicities μ_{1}, μ_{2}, …, μ_{k} enters the statements as if it is a single critical point of multiplicity ∏ μ_{j}.

It is believed that the class of rational functions which satisfy the summability condition with any exponent α > 0 is generic in the measure theoretical sense in the space of all rational maps of a given degree, see [8]. Generic systems are usually distinguished by some form of hyperbolicity. A system can become hyperbolic if one considers only small pieces of the phase space and a high iterate of the map on each piece. If it is possible to find such pieces almost everywhere, we say that the system induces hyperbolicity or is non-uniformly hyperbolic with respect to a given measure. This approach originates from the work of Jakobson on the abundance of absolutely continuous invariant measures in unimodal maps as well as Benedicks and Carleson’s work on the Hénon attractor. In holomorphic dynamics a similar strategy was followed in [4] to study non-uniform hyperbolicity with respect to the so-called conformal measures. Conformal or Sullivan-Patterson measures are dynamical analogues of Hausdorff measures and capture important (hyperbolic) features of the underlying dynamics. If is F a rational map with the Julia set J then a Borel measure ν supported on J is called conformal with an exponent p (or p-conformal) if for every Borel set A on which F is injective one has

ν(F (B)) = ∫_{B} |F′(z)|^{p} dν .

As observed in [10], the set of pairs (p, ν) with p-conformal measure ν is compact (in the weak-∗ topology). Hence, there exists a conformal measure with the minimal exponent

δ_{conf} := inf{p : ∃ a p-conformal measure on J.}

The minimal exponent δ_{conf} is also called a conformal dimension of J.

In [4] it is proven that if

sup_{c∈Crit∩J} ∑_{n=n_{c}}^{∞} n · |(F^{n}(c))′|^{−α} < ∞ ,

holds then a strong version of non-uniform hyperbolicity is true, namely, there exists a unique absolutely continuous invariant measure σ with respect to a unique δ_{conf}-conformal measure with δ_{conf} = HDim(J). Additionally, σ is mixing and has a positive Lyapunov exponent.

Poincaré series. Patterson-Sullivan’s construction of conformal measures is based on Poincaré series, [10]. This construction was further studied in [4]. In the current paper we will focus on the relations between Poincaré series and fractal geometry of Julia sets.

We call a point z admissible if it does not belong to ⋃_{i=0}^{∞} F^{n}(Crit). Take an admissible point z and assume that F has no elliptic Fatou components and J ≠ ℂ̂. We define the Poincaré series by

Σ_{δ}(z) := ∑_{n=1}^{∞} ∑_{y∈F^{−n}z} |(F^{n})′ (y)|^{−δ} .

The series converges for every δ > δ_{P oin}(z) and the minimal such δ_{P oin}(z) is called the Poincaré exponent (of F at the point z). By the standard distortion considerations, if F is a component of the Fatou set, then for all admissible z ∈ F Poincaré exponents coincide, so we set δ_{P oin}(F) := δ_{P oin}(z). We define the Poincaré exponent by

δ_{P oin}(J) := max {δ_{P oin}(F) } ,

Should J = ℂ̂ then by the definition δ_{P oin}(J) = 2. By Theorem 1 of [4]. if a rational function F satisfies the summability condition with an exponent α < \frac{δ_{P oin}(J)}{μ_{max}+δ_{P oin}(J)} then

δ_{P oin}(z) = δ_{P oin}(c) = δ_{P oin}(J) = inf {δ_{P oin}(x) : x ∈ ℂ̂} .  (2)

for every z which is at a positive distance to the critical orbits and every c ∈ Crit ∩ J of the maximal multiplicity.^{1} Moreover, the Poincaré series with the critical exponent δ_{P oin}(J) diverges for every point z ∈ ℂ̂.

In the current paper we will prove a statistical version of Theorem 1 of [4].

Theorem 1 If a rational function F satisfies the summability condition with an exponent α < \frac{δ_{P oin}(J)}{μ_{max}+δ_{P oin}(J)} then for almost all z ∈ ℂ̂ in the sense of Lebesgue measure,

δ_{P oin}(z) = inf {δ_{P oin}(x) : x ∈ ℂ̂} .

Let HDim(J) and MDim(J) stand for Hausdorff and Minkowski dimensions of J, respectively, and the hyperbolic dimension of a Julia set J_{c} is defined as

HypDim(J) = sup_{K⊂J} HDim(K) ,

where the supremum is taken over all hyperbolic subsets of J. In [3] it was proven that δ_{conf} = HypDim(J) for every rational map F. If a rational function F satisfies the summability condition with an exponent α < \frac{p}{μ_{max}+p} where p is any (e.g., maximal) of the quantities in the formula below, then according to [4],

δ_{conf}(J) = δ_{P oin}(J) = δ_{W hit}(J) = MDim(J) = HDim(J).  (3)

Let P_{n} be a set of all periodic points of period n in the Julia set J. We define the Poincaré series on periodic points as

Σ_{δ}^{Per} = ∑_{n=1}^{∞} \frac{1}{n} ∑_{y∈P_{n}} |(F^{n})′ (y)|^{−δ}

and the corresponding Poincaré exponent as

δ_{P er} := inf{δ : Σ_{δ}^{Per} < ∞} .

1The equality δ_{P oin}(J) = inf{δ_{P oin}(x) : x ∈ ℂ^{}ˆ } can be regarded as an alternative definition of the Poincaré exponent when J = ℂ^{}ˆ .

Theorem 2 Suppose that F is a rational function which satisfies the summability condition with an exponent α < \frac{δ_{P oin}(J)}{μ_{max}+δ_{P oin}(J)}. Then

inf_{c∈Crit∩J} δ_{P oin}(c) = δ_{P er}(J) .

Pressure function. Let ν be an ergodic F -invariant measure. Denote by h_{μ}(F) the ν-entropy of F. The standard pressure function P (t) is defined as

P (t) = sup_{ν}{h_{ν}(F) − t ∫ log |F′|dν} ,

where the supremum is taken over all F -invariant and ergodic measures ν. F. Przytycki proved in [6] (Theorem A2.9.) that P (t) = inf{P (z, t); z ∈ ℂ̂}, where

P (z, t) = lim sup_{n→∞} \frac{1}{n} log ∑_{y∈F^{−n}(z)} |(F^{n})′(y)|^{−t} .

The function P (z, t) is equal to P (t) for z outside of an exceptional set of Hausdorff dimension 0 and the smallest zero of P (t) is equal to HypDim(J). We recall that P (·) is strictly decreasing and convex function for every t ∈ [0, HypDim(J)], see [6].

A pressure function H(t) on periodic points is given by

H(t) = lim sup_{n→∞} \frac{1}{n} log ∑_{z∈P_{n}} |(F^{n})′(z)|^{−t} .

Let L_{n}(η) be a set of all periodic points z ∈ P_{n} such that a branch F^{−n} which fixes z is well defined on the ball B_{η}(z). The periodic points from L_{η} are ‘visible’ from the scale η > 0 and the corresponding pressure function H_{η}(t) is defined by

H_{η}(t) = lim sup_{n→∞} \frac{1}{n} log ∑_{z∈L_{n}(η)} |(F^{n})′(z)|^{−t} .

Theorem 3 Suppose that F is a rational function which satisfies the summability condition with an exponent α < \frac{δ_{P oin}(J)}{μ_{max}+δ_{P oin}(J)}. Then there exists η > 0 so that

P (t) = H(t) = H_{η}(t)

for every t ∈ [0, HypDim(J)].

Observe that the smallest zero of P (t) is a number at which the Poincaré series starts to diverge exponentially and is by definition smaller and equal to δ_{P oin}(J). Therefore, the equality of these two numbers is a reflection of some underlying hyperbolicity of the system. The estimate P (t) ≥ H(t) for all polynomials was obtained recently in [1]. Further development related to Thereom 3 can be found in [7].

Conventions. Many properties will take into account μ_{max} = max_{c∈Crit∩J} μ(c) – the maximal multiplicity of critical points in the Julia set (calculated as in 1, if there are any critical orbit relations).

If the Julia set is not the whole sphere, we use the usual Euclidean metric on the plane, changing the coordinates by a Möbius transformation so that ∞ belongs to a periodic Fatou component, and doing all the reasoning on a large compact containing the Julia set. Alternatively (and also when J = ℂ̂) one can use the spherical metric. For simplicity and convenience of the reader we will write all the distortion estimates for the planar metric, when Köbe distortion theorem has a more familiar formulation. The estimates remain valid in the case of spherical metric, with an appropriate version of Köbe distortion theorem (which differs only by a multiplicative constant, since we work with the scales smaller than some very small R).

Another general convention is following: we call F^{−n}(z), …, z a sequence of preimages of z by F if for every 1 ≤ j ≤ n,

F (F^{−j})(z) = F^{−j+1}(z) .

We will write A≲B whenever A ≤ CB with some absolute (but depending on the equation) constant C. If A ≤ CB and B ≤ CA then we write A ≍ B. We adopt the convention that ∑_{n}(ω_{n})^{−∞} < ∞ means that the sequence ω_{n} tends to zero as n → ∞.

2 Almost everywhere convergence of the Poincaré series

Shrinking neighborhoods. To control the distortion, we will use the method of shrinking neighborhoods. Suppose that ∑_{n=1}^{∞} δ_{n} < 1/2 and δ_{n} > 0 for every positive integer n. Set Δ_{n} := ∏_{k≤n} (1 − δ_{k}). Let B_{r} be a ball of radius r around a point z and {F^{−n}z} be a sequence of preimages of z. We define U_{n} and U′_{n} as the connected components of F^{−n} B_{rΔ_{n}} and F^{−n} B_{rΔ_{n+1}}, respectively, which contain F^{−n}z. Clearly,

F U_{n+1} = U′_{n} ⊂ U_{n} .

If U_{k}, for 1 ≤ k ≤ n, do not contain critical points then distortion of F^{n} : U′_{n} → B_{rΔ_{n+1}} is bounded (the Köbe distortion lemma) by a power of \frac{1}{δ_{n+1}} , multiplied by an absolute constant.

Since ∑_{n} δ_{n} < \frac{1}{2}, one also has ∏_{n} (1 − δ_{n}) > \frac{1}{2}, and hence always B_{r/2} ⊂ B_{rΔ_{n}}.

Specification of orbits. We call F^{−n}(z), …, z a sequence of preimages of z by F if for every 1 ≤ j ≤ n,

F (F^{−j})(z) = F^{−j+1}(z) .

We will estimate expansion along the backward orbits by decomposing them into blocks of different types. We will introduce three types of orbits.

Definition 2.1 Let R′ < R < 1.

1. A sequence F^{−n}(z), ⋯, F^{−1}(z), z of preimages of z is of the first type with respect to critical points c_{1} and c_{2} if (i) Shrinking neighborhoods U_{k} for B_{r}(z), 1 ≤ k < n, avoid critical points and r ≤ 2R′. (ii) The critical point c_{2} ∈ ∂U_{n}, (iii) The critical point c_{1} ∈ F^{−1}B_{R}(F z).

2. Let dist (z, J_{F}) ≤ R′/2. A sequence F^{−n}(z), ⋯, F^{−1}(z), z of preimages of z is of the second type if the ball B_{R′}(z) can be pulled back univalently along it.

3. A sequence F^{−n}(z), ⋯, F^{−1}(z), z of preimages of z is of the third type with respect to the critical point c_{2} if

(i) Shrinking neighborhoods U_{k} for B_{r}(z), 1 ≤ k < n, avoid critical points and r ≤ 2R′ , (ii) The critical point c_{2} ∈ ∂U_{n} .

We say that a backward orbit y = F^{−n}(z), …, z is decomposed into a sequence of blocks if there exists an increasing sequence of integers 0 = n_{0} < … < n_{k} = n so that for every i = 0, …, k − 1 the orbit F^{−n_{i+1}}(z), …, F^{−n_{i}}(z) is of type 1, 2, or 3. Given a pair of integers 0 ≤ r < l ≤ n, we say that a subsequence F^{−n_{l}}(y), …, F^{−n_{r}}(y) yields expansion M if

|(F^{n_{l}−n_{r−1}})′(y)| ≥ M .

Lemma 2.1 Assume that a rational function F satisfies the summability condition with an exponent α ≤ 1 and set β = μ_{max}α/(1 − α). There exist ϵ > 0, parameters R′ < R < 1, and a sequence γ_{n}, ∑_{n=1}^{∞} γ_{n}^{−β} < 1/(16 deg F)^{2} , with the following properties: if z belongs to the ϵ-neighborhood of the Julia set J and a ball B_{Δ}(z) can be pulled back univalently by a branch of F^{−N} then there exist positive constants L′ > L, K independent of z, Δ, and ϵ such that the sequence F^{−N}(z), …, z can be decomposed into blocks of types type 1, 2, and 3, with the parameters R′ < R and

• every type 2 block, except possibly the leftmost one, has the length contained in [L, L′) and yields expansion 6,

• the leftmost type 2 block has the length contained in [0, L] and yields expansion K > 0,

• all subsequences of the form 1 … 13, except possibly the rightmost one, yield expansion

γ_{k_{j}} … γ_{k_{1}}γ_{k_{0}} ,

k_{i} being the lengths of the corresponding blocks,

• the rightmost sequence of the form 1 … 13 yields expansion

γ_{k_{j}} … γ_{k_{1}}γ_{k_{0}} Δ^{(1−μ(c)/μ′_{max})} if a critical point c ∈ B_{Δ}(z) ,
γ_{k_{j}} … γ_{k_{1}}γ_{k_{0}} Δ^{(1−1/μ′_{max})}  if otherwise ,

where μ′_{max} is the largest multiplicity of critical points of F met by shrinking neighborhoods involved in the decomposition procedure of the sequence F^{−N}(z), …, z into blocks of types 1, 2, and 3, with the parameters R′ < R.

Proof: If we replace in the last estimate μ′_{max} by μ_{max} then Lemma 2.1 is verbatim the Main Lemma of [4]. On the other hand it is clear that the critical points of F which are not involved in the construction of blocks of types 1, 2, and 3 do not contribute to the estimates. Therefore, after this change in the formulation of the Main Lemma of [4], the estimates of expansion along the rightmost blocks are sill valid.

□

Conditional estimates. Fix a point z and a positive number Δ. Let H(z, Δ) stand for a set of all preimages of z such that a ball B_{Δ}(z) can be pulled back univalently along the corresponding branch. By Lemma 2.1, every backward orbit of z which terminates at y ∈ H(z, Δ) can be decomposed into blocks of type 1, 2, or 3.

In the decomposition of Lemma 2.1, let x ∈ H(z, Δ) be a point which starts a type 3 block. Denote by I(x|z) = I^{Δ}(x|z) a set of all y ∈ H(z, Δ) which are the endpoints of type 1 blocks preceded by exactly one type 3 block. For example, preimages of x which are endpoints of blocks 13, 113, … belong to I(x|z). Note that the definition depends on the choice of Δ.

We will drop z from the notation of I(x|z) whenever no confusion can arise.

Lemma 2.2 Let β = μ_{max}α/(1 − α). If a rational function F satisfies the summability condition with an exponent α ≤ 1 then there exist ϵ > 0 and a sequence γ_{n} summable with the power −β = μ_{max} \frac{α}{1−α} so that for every point z from ϵ-neighborhood of the Julia set J and every set I(x|z) = I^{Δ}(x|z),

∑_{y∈I(x|z)} |(F^{n(y)})′ (y)|^{−β} < \frac{1}{3} ∑_{k=1}^{∞} γ_{k}^{−β}Δ_{k}(x)^{−β(1−1/μ_{max})}

where

Δ_{k}(x) = \{ \begin{matrix}1 & if x ≠ z , \\ dist (x, F^{k}(Crit ∩ J)) & if x = z . \end{matrix}

Proof: Let γ_{n} be the sequence from the assertion of Lemma 2.1.

Observe that any point y ∈ F^{−k}(z) has at most 4 deg F preimages of a given length which are of the first or the third type. In fact, since pull-backs to the critical values are univalent, there is only one way to hit a specific critical value after particular number of steps, and thus only μ(c) ways to hit a critical point c, but

∑_{c} μ(c) = #{c} + ∑_{c}(μ(c) − 1) ≤ 2(deg F − 1) + 2(deg F − 1) < 4 deg F .  (4)

Therefore, for every sequence k_{0}, k_{1}, …, k_{m} of positive integers there are at most (2 deg F)^{m+1} sequences 1 … 13 with the corresponding lengths of the pieces of type 1 and 3. By Lemma 2.1,

|(F^{n(y)})′ (y)| ≥ γ_{k_{m}} … γ_{k_{1}}γ_{k_{0}}Δ_{k_{0}}(x)^{(1−1/μ_{max})} .

We obtain that

∑_{y∈I(x)} |(F^{n(y)})′ (y)|^{−β} < ∑_{m,k_{0},k_{1},...,k_{m}} (4 deg F)^{m+1} (γ_{k_{m}} … γ_{k_{1}}γ_{k_{0}})^{−β} Δ_{k_{0}}(x)^{−β(1−1/μ_{max})}

< ∑_{m=1}^{∞} (4 deg F ∑_{k_{m}} γ_{k_{m}}^{−β}) · . . . · (4 deg F ∑_{k_{1}} γ_{k_{1}}^{−β})

· (4 deg F ∑_{k_{0}} γ_{k_{0}}^{−β} Δ_{k_{0}}(x)^{−β(1−1/μ_{max})})

< ∑_{m=1}^{∞} (\frac{1}{16 deg F} )^{m} (4 deg F ∑_{k_{0}} γ_{k_{0}}^{−β} Δ_{k_{0}}(x)^{−β(1−1/μ_{max})})
< \frac{1}{3} ∑_{k_{0}} γ_{k_{0}}^{−β} Δ_{k_{0}}(x)^{−β(1−1/μ_{max})} .

This completes the proof of Lemma 2.2.

□

Let L′ > L be the constants supplied by Lemma 2.1. In the decomposition of Lemma 2.1, let x ∈ H(z, Δ) be a point which starts a type 2 block. Denote by II_{l}(x|z) and II_{s}(x|z) correspondingly the sets of all “long” (of order L′ > n(y) ≥ L) and “short” (of order n(y) < L) type 2 preimages y of x obtained in the decomposition of Lemma 2.1. This definition also depends on the choice of Δ, but as Lemma 2.3 shows we will use only estimates independent of Δ, so we simplify the notation by omitting Δ. Lemma 2.3 is Lemma 3.5 of [4].

Lemma 2.3 Assume that the Poincaré series with exponent q is summable for some point v ∈ ℂ̂. Then there exists ϵ > 0 so that for every point z from ϵ-neighborhood of the Julia set J and every set II_{l}(x|z) and II_{s}(x|z),

∑_{y∈II_{l}(x|z)} |(F^{n(y)})′ (y)|^{−q} < \frac{1}{36}  ,
∑_{y∈II_{s}(x|z)} |(F^{n(y)})′ (y)|^{−p} < C(p) for any p .

Conditional Poincaré series. Recall that H(z, Δ) stands for a set of all preimages of z such that a ball B_{Δ}(z) can be pulled back univalently along the corresponding branch.

Proposition 1 Suppose that a rational function F satisfies the summability condition with an exponent

α < \frac{δ_{P oin}(J)}{μ_{max} + δ_{P oin}(J)} .

Then there exists ϵ > 0 so that for every p > δ_{P oin}(J) and every point z in the ϵ-neighborhood of the Julia set, there exists C(ϵ, p) so that

∑_{y∈H(z,Δ)} |(F^{n})′ (y)|^{−p} < C ∑_{k=1}^{∞} γ^{−p} Δ_{k}(z)^{p( \frac{1}{μmax} −1)} .

Proof: The proof follows closely the proof of the self-improving property of the Poincaré series from [4]. Let p ∈ (δ_{P oin}(J), 2] and c be a point of maximal multiplicity. Then, by (2), the Poincaré series Σ_{p}(c) is summable.

We take v := c in the hypotheses of Lemma 2.3. By Lemma 2.3, there exists ϵ > 0 so that for every z from the ϵ-neighborhood of the Julia set of F,

∑_{y∈II_{l}(x)} |(F^{n(y)})′ (y)|^{−p} < \frac{1}{36} .

Also by Lemma 2.3

∑_{y∈II_{s}(x)} |(F^{n(y)})′ (y)|^{−p} < C = C(p) .

We expand ∑_{y∈H(z,Δ)} |(F^{n})′ (y)|^{−p} by grouping preimages of the same kind into clusters. We begin with z obtaining preimages of three kinds: I(z) = I^{Δ}(z), II_{l}(z) and II_{s}(z). Points in II_{s}(z) are terminal while preimages y of the points in I(z) and II_{l}(z) are divided further. We proceed in this fashion down the tree of preimages of z. Using Lemmas 2.2 and 2.3, we obtain that

∑_{y∈H(z,Δ)} |(F^{n})′ (y)|^{−p} = ∑_{z′∈II_{s}(z)} |(F^{n(z′)})′ (z′)|^{−p} + ∑_{z′∈I,II_{l}(z)} |(F^{n(z′)})′ (z′)|^{−p}

· ( ∑_{z′′∈II_{s}(z′)} |(F^{n(z′′)})′ (z′′)|^{−p} + ∑_{z′′∈I,II_{l}(z′)} |(F^{n(z′′)})′ (z′′)|^{−p}

· ( ∑_{z′′′∈II_{s}(z′′)} |(F^{n(z′′′)})′ (z′′′)|^{−p} + …))

≤ C + (\frac{1}{3} + \frac{1}{3} ∑_{k=1}^{∞} γ_{k}^{−p}Δ_{k}(z)^{−p(1−1/μ_{max})}) (C + \frac{2}{3} (C + …))

= C + \frac{1}{3} (1 + ∑_{k=1}^{∞} γ_{k}^{−p} Δ_{k}(z)^{−p(1−1/μ_{max})}) C (1 + \frac{2}{3} + (\frac{2}{3})^{2} + …)

< 3C ∑_{k=1}^{∞} γ_{k}^{−p} Δ_{k}(z)^{−p(1−1/μ_{max})} .

This proves Proposition 1.

□

Proof of Theorem 1 Let σ be the normalized Lebesgue measure on ℂ̂. Observe that for every integer k ≥ 0 and every p ≤ 2,

∫ Δ_{k}(z)^{−p(1−1/μ_{max})} dσ ≤ ∑_{c∈Crit∩J} ∫ |z − F^{k}(c)|^{−p(1−1/μ_{max})} dσ < ∞ .

Hence for every p ∈ (δ_{P oin}(J), 2),

∫ ∑_{k=1}^{∞} γ_{k}^{−p} Δ_{k}(z)^{p(1/μ_{max}−1)} dσ ≤ C ∑_{k=1}^{∞} γ_{k}^{−p} < C′ < ∞

and consequently, for almost all z in the sense of Lebesgue measure σ,

∑_{y∈H(z,Δ)} |(F^{n})′ (y)|^{−p} < C′

is bounded independently from Δ. Passing with Δ to zero we obtain Theorem 1.

2.1 Proof of Theorem 2.

Construction. Let z ∈ J be a periodic point with period n and a sequence

F^{−n}(z), … F^{−1}(z), z

form a chain of preimages, that is F (F^{−i}(z)) = F^{−i+1}(z) for i = 1, … n. We will decompose the chain into blocks of preimages of the types 2 and 1 … 1. Consider shrinking neighborhoods {U_{k}} for B_{2R′}(z). If they do not contain the critical points we form one block of type 2 of the length n. Otherwise, we set r = 0 and increase it continuously until certain shrinking neighborhood U_{k} hits a critical point c, c ∈ ∂U_{k}. It must happen for some 0 < r < 2R′. We set n_{1} := k and z_{1} := F^{−n_{1}}(z). Then z_{1} is a third type preimage of z and the ball B_{r}(z) can be pulled back univalently by F^{n_{1}} along the chain.

Inductive procedure. Suppose we have already constructed z_{j} = F^{−n_{j}}(z) which is of type 1 or 3. We enlarge the ball B_{r}(z_{j}) continuously increasing the radius r from 0 until one of the following conditions is met:

1) for some k ≤ n − n_{j} the shrinking neighborhood U_{k} for B_{r}(z_{j}) hits a critical point c ∈ Crit ∩ J, c ∈ ∂U_{k},

2) radius r reaches the value of 2R′.

In the case 1) we put n_{j+1} := n_{j} + k. Clearly, z_{j+1} := F^{−n_{j+1}}(z) is a type 1 preimage of z_{j}. If 2) holds, we set z_{j+1} := F^{−n}(z_{1}) which is a type 2 preimage of z_{j}. This terminates the construction in this case.

Coding. As a result of the inductive procedure, we can decompose the backward orbit of every point z from a given cycle C into pieces of type 1, 2 and 3. We ascribe to the cycle a code 2 if there is a point z ∈ C so that its backward orbit F^{−n}(z), …, z consists of one block of type 2. Otherwise, for every z ∈ C its backward orbit F^{−n}(z), …, zx must contain a least one block of type 3 or 1. Consider now all critical points which end or begin the blocks of type 3 and 1. Denote μ_{max}(C) a maximal multiplicity of these critical points. Let y ∈ C be a point which begins or ends a type 3 or a type 1 block with the corresponding critical point of multiplicity μ_{max}(C). We attach to C a code of the backward orbit F^{−n}(y), …, y.

This gives a coding of cycles by sequences of 1’s, 2’s. By the construction, only the following three types of codings are allowable: 2, 1 … 1, 21 … 1. We recall that according to our convention, during the inductive procedure we put symbols in the coding from the right to the left.

We attach to every chain of preimages of z the sequence k_{l}, … k_{0} of the lengths of the blocks of preimages of a given type in its coding. Again our convention requires that k_{0} always stands for the length of the rightmost block in the code. Clearly, k_{0} + ⋯ + k_{l} = n.

Estimates. We recall that the last estimate of Lemma 2.1 (μ′_{max} =: μ_{max}(C)) implies that every sequence of the form 11 … 1 with the length of the corresponding pieces k_{l}, … k_{0} yields expansion

γ_{k_{l}} · . . . · γ_{k_{0}}

Let ν be a δ_{conf}-conformal measure and δ ≥ δ_{conf}. Next, observe that for singleton sequences {2} of the length k ≤ n we have the following estimate,

∑_{y∈F^{−k}(x)} \frac{1}{|(F^{k})′(y)|^{δ}} ≲ ∑_{F^{−k}} \frac{ν(F^{−k}(B_{R′}(x)))}{ν(B_{R′}(x))} ≤ \frac{1}{ν(B_{R′}(x))} ≤ K_{R′} ,  (5)

which is independent from k.

Let A_{n} be a set of all periodic points of period n in the Julia set with codes of the form 1 … 1 and 21 … 1. If x starts a block of type 2 in the code of F^{−n}(z), … z then we define n_{x} by F^{n_{x}}(x) = z. Hence,

∑_{y∈A_{n}} \frac{1}{|(F^{n})′(y)|^{δ}} ≤ ∑_{n_{x}=1}^{n} ( ∑_{11...13} \frac{1}{|(F^{n_{x}})′(x)|^{δ}} ) ( ∑_{y∈F^{−n+nx}} \frac{1}{|(F^{n})′(y)|^{δ}} )
≤ K_{R′} ∑_{11...13}(γ_{k_{l}} · . . . · γ_{k_{0}})^{−δ}

For every sequence k_{l}, … k_{0} of positive integers, there is at most (2 deg F)^{l+1} different branches involved in the construction of the codings 11 … 1 with the corresponding lengths of the pieces of type 1 (see the estimate (4)). Every periodic point z defines a unique inverse branch of F^{−n} by the condition that F^{−n}(z) = z. Therefore, we obtain that

∑_{n=1}^{∞} \frac{1}{n} ∑_{y∈A_{n}} \frac{1}{|(F^{n})′(y)|^{δ}} ≲ ∑_{n=1}^{∞} ∑_{l,k_{l}+···+k_{0}=n}(4 deg F)^{l+1}(γ_{k_{l}} · . . . · γ_{k_{0}})^{−δ}

< ∑_{l=1}^{∞} (4 deg F ∑_{k_{l}} γ_{k_{l}}^{−δ}) · . . . · (4 deg F ∑_{k_{0}} γ_{k_{0}}^{−δ})

< ∑_{l=1}^{∞} (\frac{1}{4})^{l} < ∞ .

Let B_{n} be a set of all repelling points of period n coded by a sequence 2. By eventually ‘onto’ property, there exists m > 0 so that every disk with center in J and radius R′ contains a preimage F^{−k}(c), k ≤ m, of a critical point c of the maximal multiplicity μ_{max}. Since the derivative of F is bounded on J (if J = ℂ̂ then we work with the spherical metric), we arrive at the following estimate,

∑_{n=1}^{∞} \frac{1}{n} ∑_{y∈B_{n}} |(F^{y})′(z)|^{−δ} ≲ ∑_{n=1}^{∞} ∑_{k=0}^{m} ∑_{y∈F^{−n−k}(c)} |(F^{n+k})′(y)|^{−δ}

≲ m ∑_{n=1}^{∞} ∑_{y∈F^{−n}(c)} |(F^{n})′(y)|^{−δ} < ∞

provided δ ≥ δ_{P oin}(J) = δ_{conf}, see (2) and (3). Since A_{n} ∪ B_{n} is the set of all periodic points of period n, we obtain that δ_{P er}(J) ≤ δ_{P oin}(J).

The reversed inequality follows from the general theory. Indeed, let X be a Cantor repeller contained in J_{c}. Denote by δ_{P er}(X) the Poincaré exponent of the restriction of F to X. For Cantor repellers we have that

δ_{P er}(X) = δ_{conf}(X) = HDim(X) .

Since δ_{conf}(J) = sup_{X} δ_{conf}(X), for any δ < δ_{conf} there exist a Cantor repeller X ⊂ J and a vicinity W of X so that for any point z ∈ W, the Poincaré series Σ_{δ,X}(z) = ∞. Consequently, for any non-exceptional z ∈ ℂ there exists y ∈ W so that

∑_{n=1}^{∞} \frac{1}{n} ∑_{y∈C_{n}} |(F^{y})′(z)|^{−δ} ≳ Σ_{δ,X}(z) = ∞ ,

which completes the proof of Theorem 2.

2.2 Proof of Theorem 3.

We start with a remark that P (t) ≤ H(t) by Katok’s theory, [5, 6]. Indeed, let X be a Cantor repeller contained in J_{c}. Denote by P_{X}(t) the pressure of the restriction of F to X. Then

P (t) = sup_{X⊂J} P_{X}(t) = sup_{X⊂J} H_{X}(t) ≤ H(t) .

Different types of periodic orbits. As in the proof of Thereom 3, we divide all periodic points of F of period n into two parts, A_{n} and B_{n}. Every cycle in A_{n} has a code of the form 21 … 1 or 1 … 1 with the corresponding lengths of the blocks of 1’s and 2 adding up to n. If the code of a cycle C of period n contains a terminating block of type 2 starting at y ∈ C then we will replace it by a block of type 1 preimages starting at y in the decomposition of the chain F^{−n}(y), …, y. A new code of C is of the form 1 … 1 with the corresponding lengths of the blocks of 1’s adding up to a number in [n, 2n). As in the proof of Theorem 2, we observe that for every sequence k_{l}, …, k_{0} of positive integers there is at most (2 deg F)^{l+1} different branches involved in the construction of the codes 11 … 1 with the corresponding lengths of the pieces associated to a given choice of the critical points. Therefore, denoting H_{n}(t) = ∑_{y∈A_{n}} |(F^{n})′(y)|^{−t}, we obtain that

H_{n}(t) ≲ ∑_{l,n≤k_{l}+···+k_{0}<2n}(4 deg F)^{l+1}(γ_{k_{l}} · . . . · γ_{k_{0}})^{−t}

< ∑_{l=1}^{2n} (4 deg F ∑_{k_{l}} γ_{k_{l}}^{−t}) · . . . · (4 deg F ∑_{k_{0}} γ_{k_{0}})^{−t})

< ∑_{l=1}^{2n} (4 deg F ∑_{i=1}^{n} γ_{i}^{−t})^{l}

≤ ∑_{l=1}^{2n}(2n)^{1− \frac{t}{β}} (\frac{1}{4})^{\frac{lt}{β}} ≲ n^{2} ,

where the estimate in the last line follows from the Hölder inequality. By eventually ‘onto’ property, for every non-exceptional w ∈ ℂ̂, there exists m > 0 so that every disk with center in J and radius R′ contains a preimage F^{−k}(w), k ≤ m. Since the derivative of F is bounded on J, say by M, (if J = ℂ̂ then we work with the spherical metric), we have that

∑_{y∈B_{n}} |(F^{y})′(y)|^{−t} ≲ ∑_{y∈F^{−n−k}(w)} |(F^{n+k})′(y)|^{−t}
≲ M^{m} ∑_{y∈F^{−n−m}(w)} |(F^{n+m})′(y)|^{−t} .

For every w ∈ ℂ̂ and 0 ≤ t < δ_{conf} the pressure P (t, w) is positive, hence

H(t) ≤ lim_{n→∞} \frac{1}{n} log (n^{2} + M^{m} ∑_{y∈F^{−n−m}(w)} |(F^{n+m})′(y)|^{−t}) = P (t, w)

and H_{R′}(t) = H(t). Consequently, H(t) ≤ inf_{w∈ℂ^{}}ˆ P (t, w) for t ∈ [0, HypDim(J)] which completes the proof of Theorem 3.

Acknowledgements. The secodn author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.

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