Version of January 27, 2003

Equality of pressures for rational functions

by Feliks Przytycki, Juan Rivera-Letelier and Stanislav Smirnov ^{1}

Abstract. We prove that for all rational functions f on the Riemann sphere and potential −t ln |f′|, t ≥ 0 all the notions of pressure introduced in [P2] coincide. We prove also that they coincide with the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet-Eckmann condition. In Appendix we discuss the case t < 0.

0. Introduction.

The thermodynamical formalism, the study of various pressures and spectra, proved to be an important tool for solving problems in hyperbolic dynamics and fractal geometry. In this paper we are interested in a setup involving iterations of a rational function on the Riemann sphere. The book [Z] is a good introduction, see also [PU]. Consult Ruelle’s monograph [R1] for a general picture. We discuss several definitions of pressure for potential −t ln |f′|, frequently applied in rational holomorphic dynamics. We prove that all of them coincide (and under an additional assumption are equal to the periodic orbits pressure ).

Part of our motivation comes from the holomorphic dynamics itself. In this setup, the thermodynamic formalism was first applied by Bowen in [B], where among other things it was proved that for the quasi-Fuchsian groups the first zero of the pressure function is equal to the Hausdorff dimension of the limit set. This formula, together with a more detailed study of pressure, was spectacularly applied by Ruelle in [R2] to show that the Hausdorff dimension of the Julia set for hyperbolic rational functions f depends real analytically on f.

Additional motivation comes from perspective applications to complex analysis. Study of pressure already produced interesting examples for problems outside dynamics, see e.g. [E,BS]. Pressure for all rational functions and potentials −t ln |f′| describes the multifractal spectra of the measure of maximal entropy. For polynomials this measure coincides with the harmonic measure for the basin of attraction to infinity, so pressure for polynomial

1 All authors are supported by the European Science Foundation program PRODYN. The first author has been also supported by the Foundation for Polish Sciences, Polish KBN grant 2P03A 00917, and Göran Gustafsson Foundation (KTH). The second author is grateful to IMPAN and KTH for hospitality and is also supported by a Polish-French governmental agreement, Fundacion Andes and a “Beca Presidente de la Republica,” Chile. The third author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.

Julia sets describes the multifractal spectra of the harmonic measure, see [MS1,MS2]. It has been recently announced in [BJ], that the extremal values of such spectra for all planar domains coincide with those for polynomial Julia sets, meaning that several problems in complex analysis can be reduced to evaluating pressure for polynomials. One example of a “dynamical” solution to an “analytical” problem along such lines can be found in [BMS]. We expect future applications of thermodynamical formalism in this setup which stretch beyond purely “dynamical” problems, see [M] for an introduction.

In what follows we will deal with a rational function f : \overline{I} C → \overline{I} C of degree d ≥ 2 on the Riemann sphere. We denote by Crit or Crit(f) the set of critical points, that is f′(x) = 0 for x ∈ Crit, J stands for the Julia set of f. Absolute values of derivatives and distances are considered with respect to the standard Riemann sphere metric. We study various definitions of pressure with potentials −t ln |f′|, with t real. In the main body of the paper the proofs are given for t ≥ 0, with Appendix devoted to t ≤ 0, where some definitions and phenomena are different. In our paper we discuss only dynamical definitions (as given in [P2]), but as was mentioned above, in this setup pressure can also be defined geometrically through multifractal spectra for the measure of maximal entropy – harmonic measure. For the discussion of this approach, consult [M,MS1,MS2].

Definition 0.1. Tree pressure. For every z ∈ \overline{I} C define (on the tree of all its preimages)

P_{tree}(z, t) = lim sup_{n→∞} \frac{1}{n} ln ∑_{f^{n}(x)=z} |(f^{n})′(x)|^{−t}

This limit does not depend on z for typical z, i.e. for z outside the zero Hausdorff dimension exceptional set E = E′ ∪ O(Crit), where O(Crit) = ⋃_{j=1}^{∞} f^{j}(Crit) and E′ = ⋂_{N} ⋃_{n>N} B(f^{n}(Crit), exp −\sqrt{n}), see [P2, Theorem 3.3]. The resulting value for typical z we call tree pressure P_{tree}(t).

Definition 0.2. Hyperbolic pressure.

P_{hyp}(t) := sup_{X} P (f|_{X}, −t ln |f′|),

where the supremum is taken over all compact f-invariant (that is f(X) ⊂ X) isolated hyperbolic subsets of J. We call such sets expanding repellers, following Ruelle. Isolated means that there is a neighbourhood U of X such that f^{n}(x) ∈ U for all n ≥ 0 implies x ∈ X. Hyperbolic or expanding means that there is a constant λ_{X} > 1 such that for all n large enough and all x ∈ X we have |(f^{n})′(x)| ≥ λ_{X}^{n}.

P (f|_{X}, −t ln |f′|) denotes the standard topological pressure for the continuous mapping f|_{X} and continuous real valued potential function −t ln |f′| on X, see for example [W].

Definition 0.3. Hyperbolic variational pressure.

P_{hypvar}(t) := sup_{μ}{h_{μ}(f) − t ∫ ln |f′|dμ},

where the supremum is taken over all ergodic f-invariant measures of positive Lyapunov exponent, i.e. χ_{μ} = χ_{μ}(f) = ∫ ln |f′|dμ > 0.

Definition 0.4. Variational pressure.

P_{var}(t) := sup_{μ}{h_{μ}(f) − t ∫ ln |f′|dμ},

where the supremum is taken over all ergodic f-invariant measures supported on J. Note that good properties of P_{var} rely on χ_{μ} ≥ 0 for all such measures, [P1].

Definition 0.5. (M. Denker & M. Urbański, [DU]). Let V be a neighborhood of Crit ∩ J in J. Define K(V) := J \ ⋃_{n≥0} f^{−n}(V) and observe that f(K(V)) = K(V). As K(V) ∩ Crit = ∅ we can consider the topological pressure P (f|_{K(V)}, −t ln |f′|) for the map f|_{K(V)} and the real continuous function −t ln |f′| on the compact set K(V). Define

P_{DU}(t) := sup_{V} P (f|_{K(V)}, −t ln |f′|),

where the supremum is taken over all V as above.

Definition 0.6. Conformal pressure, [P2]. Set P_{Conf}(t) := ln λ(t), where

λ(t) = inf{λ > 0 : ∃μ, a probability measure on J with Jacobian λ|f′|^{t}}.

We say that φ : J → IR, φ ≥ 0 is the Jacobian for f|_{J} with respect to μ if φ is μ-integrable and for every Borel set E ∈ J on which f is injective μ(f(E)) = ∫_{E} |f′|^{t}dμ. We write φ = Jac_{μ}(f|_{J}). We call μ on J with Jacobian of the form λ|f′|^{t} a conformal measure.

All these pressure functions are Lipschitz continuous, monotone decreasing, and convex. These properties follow easily from the definitions (or have been explained in [P2]), except P_{Conf}(t).

Together with previous results, we prove

Theorem A. For all rational functions the pressures P_{tree}(t), P_{hyp}(t), P_{hypvar}(t), P_{var}(t), P_{DU}(t), P_{Conf}(t) coincide for all t ≥ 0.

Lipschitz continuouity, monotone decreasing, and convexity of P_{Conf}(t) follows of course from Theorem A and the properties of other pressures.

For t ≥ 0, we will denote by P (t) the common value of these functions at t.

In [P2, Appendix 2] it was proved that the so called hyperbolic dimension of f, denoted by t_{0}, was the first zero of each of these functions and that these functions coincide for 0 ≤ t ≤ t_{0}. This extends the mentioned result of Bowen to (non-hyperbolic) rational maps.

The missing inequalities for t ≥ t_{0} in the chain

P_{tree}(t) ≥ P_{hyp}(t) ≥ P_{hypvar}(t) ≥ P_{var}(t) ≥ P_{DU}(t) ≥ P_{Conf}(t) ≥ P_{tree}(t)  (∗)

were P_{hypvar}(t) ≥ P_{var}(t) and P_{DU}(t) ≥ P_{Conf}(t). Here we complete the proof of Theorem A by showing these inequalities, see Section 1.

Recently M. Urbański [U] proved these missing inequalities (and hence the conclusion of Theorem A) for critically non-recurrent maps without parabolic periodic orbits. Our proof in the general case is based on his approach and some ideas of [PRS]. Historically the progress has been made consecutively in [DU], [P1], [P2], [PRS], [BMS], [U].

In Section 2 we give a simple direct proof of the equality P_{tree}(t) = P_{hyp}(t), for t ≥ 0.This yields a new direct proof that the Hausdorff dimension of all variants of conical limit sets defined in [P2] coincide with t_{0} (Remark 2.3).

Section 0.1. Pressure at periodic points. We consider the following additional definition of pressure.

Definition 0.7. Periodic orbits pressure. For n ≥ 1 let Per_{n} be the set of points p ∈\overline{I C} satisfying f^{n}(p) = p. Define

P_{Per}(t) = lim sup_{n→∞} \frac{1}{n} ln ∑_{z∈Per_{n}(f)∩J} |(f^{n})′(z)|^{−t}.

This function of t is also Lipschitz continuous, monotone decreasing, and convex, that easily follows from the definition..

It is reasonable to think that this definition of pressure coincides with the previous ones, for t ≥ 0. However we are only able to prove the following inequality.

Proposition B. For t ≥ 0 we have P_{Per}(t) ≥ P (t).

The reverse inequality was established in [BMS] for polynomials without indifferent periodic orbits. In this paper we prove the reverse inequality for rational functions under the following hypothesis.

Hypothesis H. For every δ > 0 and all n large enough, if for a set P ⊂ Per_{n} for all p, q ∈ P and all i : 0 ≤ i < n dist(f^{i}(p), f^{i}(q)) < exp −δn, then #P ≤ exp δn.

Theorem C. Under hypothesis H we have P_{Per}(t) = P (t) for every t ≥ 0.

Remark. It is reasonable to think that Hypothesis H holds for all rational maps. Of course it holds for Topological Collet-Eckmann Maps (TCE), see [PRS] and the definition of UHP below. Its negation would imply the existence of exponentially many periodic orbits of period n of multiplier exponentially close to 1, for a sequence of n’s tending to ∞ (see Remark 3.9).

Section 0.2. Preliminaries. We shall use the following definitions of TCE, which are equivalent by [PRS].

• TCE. Topological Collet-Eckmann condition. There exist M ≥ 0, P ≥ 1 and r > 0 such that for every x ∈ J(f) there exists a strictly increasing sequence of positive integers n_{j}, for j = 1, 2, ... such that n_{j} ≤ P · j and for each j

#{i : 0 ≤ i < n_{j}, Comp_{f^{i}(x)}f^{−(n_{j}−i)}B(f^{n_{j}}(x), r) ∩ Crit ≠ ∅} ≤ M,

where Comp_{y} means the connected component containing y (above y = f^{i}(x)).

• CE2(z_{0}). Backward or second Collet-Eckmann condition at z_{0} ∈ \overline{I} C. There exist λ_{CE2} > 1 and C > 0 such that for every n ≥ 1 and every w ∈ f^{−n}(z_{0}),

|(f^{n})′(w)| ≥ Cλ_{CE2}^{n}.

(In this case z_{0} is necessarily not in the forward orbit of a critical point.) We write CE2(some z_{0}) if there exists z_{0} ∈ \overline{I} C such that CE2(z_{0}) holds.

• UHP. Uniform Hyperbolicity on repelling Periodic orbits. There exists λ_{Per} > 1 such that every repelling periodic point p ∈ J(f) of period k ≥ 1 satisfies,

|(f^{k})′(p)| ≥ λ_{Per}^{k}.

• Lyapunov. Lyapunov exponents of invariant measures are bounded away from zero. There is a constant λ_{Lyap} > 1 such that the Lyapunov exponent of any ergodic invariant probability measure μ supported on Julia set satisfies χ_{μ}(f) := ∫ ln |f′|dμ ≥ ln λ_{Lyap}.

• Negative Pressure. Pressure for large t is negative. For large values of t the pressure function P_{var}(t) is negative.

Note that TCE implies immediately that P_{var}(t) is strictly decreasing with ”slope” bounded away from 0 (use the condition •Lyapunov).

TCE not satisfied implies immediately by negation of •Negative Pressure, that P_{var}(t) ≡ 0 for all t ≥ t_{0}.

We shall consider sometimes these two cases separately.

Section 1. The inequalities P_{hypvar}(t) ≥ P_{var}(t) and P_{DU}(t) ≥ P_{Conf}(t) in the chain (*).

Proposition 1.1. P_{var}(t) = P_{hypvar}(t) for all t ≥ 0.

Proof. In TCE case the equality holds since the measures under supremum in both definitions are the same (by •Lyapunov).

In the general case, for 0 ≤ t ≤ t_{0} the equality has been proved in [P2] (Let us recall the argument. Suppose t < t_{0}. If h_{μ}(f) − tχ_{μ}(f) is close to P_{var}(t) then it is positive hence, due to χ_{μ}(f) ≥ 0, see [P1], we get h_{μ}(f) > 0. Hence by Ruelle inequality χ_{μ}(f) ≥ h_{μ}(f)/2 > 0, hence this measure is being taken in account also in the definition of P_{hypvar}(t).) For t ≥ t_{0} in not TCE case, both pressures are identically equal to 0 by the monotonicity and negation of •UHP. (One can support invariant measures μ on repelling periodic orbits with χ_{μ}(f) → 0.) QED

Proposition 1.2. P_{DU}(t) ≥ P_{Conf}(t) for all t ≥ 0.

Proof. We shall find a right conformal measure μ repeating the construction from [DU].

For each critical point c ∈ Crit ∩ J choose an arbitrary ergodic probability invariant measure μ_{c} supported on the ω-limit set ω(c) and choose a point y_{c} ∈ ω(c) such that

lim sup_{n→∞}(exp −χ_{μ_{c}}n)|(f^{n})′(y_{c})| ≥ 1  (2.1)

(Use Poincaré Recurrence Theorem, or Birkhoff Ergodic Theorem as in [P1, Proof of Corollary A]. For example: Let C < 1, n_{0} > 0 and E of positive measure μ_{c} such that for all x ∈ E and n ≥ n_{0} we have (exp −χ_{μ_{c}}n)|(f^{n})′(y_{c})| ≤ C. We can shrink E so that the smallest time of return to E is at least n_{0}. Then by Birkhoff Ergodic Theorem for a.e. x ∈ E we get lim_{n→∞} \frac{1}{n} ln(exp −χ_{μ_{c}}(f)n) |(f^{n})′(x)| ≤ μ_{c}(E) ln C < 0, so −χ_{μ_{c}}(f) + χ_{μ_{c}}(f) < 0, a contradiction.)

For each c let

V_{c,n} := ⋃_{k=0}^{∞} f^{−k}B(y_{c}, 1/n) and put V_{n} = ∪_{c∈Crit∩J}V_{c,n}.

Then there is a measure μ_{n} on K(V_{n}) whose Jacobian for f is less or equal to λ_{n}|f′|^{t} (equality holds outside ∂V_{n}), where

P_{n} := P (f|_{K(V_{n})}, −t ln |f′|) = ln λ_{n},

see [DU]. Then let μ be a weak* limit of a subsequence {μ_{n_{j}}}_{j≥1} of these measures.

We will prove that μ is conformal with Jacobian λ|f′|^{t}, where λ = lim λ_{n_{j}} ≤ exp P_{DU}(t). To prove this we only need to check that μ is conformal, with Jacobian equal to λ|f′|^{t} at each y_{c} or that μ(y_{c}) = 0.

To this end it is sufficient to check that for every f invariant ergodic probability measure ν on J we have ln λ + tχ_{ν}(f) ≥ 0. Indeed, this inequality for each ν = μ_{c} implies by (2.1) that

lim sup_{n→∞} λ^{n}|(f^{n})′(y_{c})|^{t} ≥ lim sup_{n→∞}(exp −tχ_{μ_{c}}n) |(f^{n})′(x)|^{t} ≥ 1.

If y_{c} is not periodic then the resulting ∑_{j=0}^{∞} |(f^{j})′|^{t} = ∞, so μ(y_{c}) = 0. The case y_{c} periodic of period, say, k, leads to μ(y_{c}) = μ(f^{k}(y_{c})) ≥ |(f^{n})′(y_{c})|^{t}μ(y_{c}), hence |(f^{n})′(y_{c})| = 1. Next note that μ(y_{c}) ≤ μ(f(y_{c})) ≤ ... ≤ μ(f^{k}(y_{c})) = μ(y_{c}) implies equalities in place of the inequalities, hence μ conformal at y_{c}, see [U].

To prove ln λ+tχ_{ν}(f) ≥ 0 assume first that χ_{ν}(f) > 0. Then there are infinitely many periodic orbits O(p) with χ_{μ_{O(p)}} arbitrarily close to χ_{ν}, where χ_{μ_{O(p)}} is the probability measure equidistributed on O(p), (use Katok-Pesin Theory, cf [P2]). All O(p) except at most one do not contain y_{c}, hence are in K(V_{n}) for n large enough. Therefore

ln λ ≥ lim_{n_{j}→∞} P_{n_{j}} ≥ h_{μ_{O(p)}}(f) − tχ_{μ_{O(p)}} ≥ −tχ_{ν}(f).

If χ_{ν}(f) = 0 we are in the non-TCE case where by non-•UHP there are infinitely many repelling periodic orbits O(p) with \frac{1}{n(p)} ln |(f^{n(p)})′| arbitrarily close to 1, where n(p) is a period of p. Therefore, as above, for all O(p) except at most one

ln λ ≥ − inf_{p} tχ_{μ_{O(p)}} ≥ 0 = −tχ_{ν}(f).

QED

Section 2. The equality P_{tree}(t) = P_{hyp}(t), for t ≥ 0.

Two of the most elementary notions of pressure are P_{tree} and P_{hyp}. Theorem A asserts that these functions coincide for t ≥ 0. In this section we give a simple and nice proof of the inequality P_{tree}(t) ≤ P_{hyp}(t), for t ≥ 0. The proof relies on the same idea as the proof of the implication UHP⇒CE2(z_{0}) in [PRS]: given a backward branch from z_{0}, we capture a periodic orbit (this procedure has names: Bowen’s specification or shadowing). Here we shall capture a large hyperbolic set.

Note that the proof of P_{tree}(t) ≤ P_{hyp}(t) provided in Section 1 via the chain of inequalities (*) has been quite complicated and used results of [P2].

Proposition 2.1. For t ≥ 0 we have P_{tree}(t) = P_{hyp}(t).

Proof. We start with a direct proof of the inequality P_{tree}(t) ≥ P_{hyp}(t) (In [P2] the proof was also indirect). It follows immediately from f^{−n}({z_{0}}) ⊃ f^{−n}({z_{0}})∩X for every f-invariant isolated hyperbolic set X ⊂ J z_{0} ∈ X and from the fact that for expanding repellers X and Hölder φ : X → IR all notions of pressure P (f|_{X}, φ) coincide [W], [PU, Ch.3]. Note that we need z_{0} typical (see Definition 0.1); we can achieve it by replacing z_{0} ∈ X by a point z′_{0} close to z_{0} not necessarily in X (for example in case X is just a repelling periodic orbit). Any backward trajectory of z_{0} in X is close to a backward trajectory of z′_{0} being attracted to X, thus contributing to the sum under logarithm in P_{tree} the same summand, up to a bounded factor.

To prove P_{tree}(t) ≤ P_{hyp}(t) we will follow the proof of Lemma 3.1 in [PRS]. First we find a typical z_{0} ∈ J in an expanding repeller X ∈ J such that

W := Comp_{z_{0}}f^{−l}B(f^{l}(z_{0}), 2δ) ⊂ B(z_{0}, εn^{−α}),

f^{l} is univalent on W, l = l(n) := β ln n and

B(z_{0}, 2n^{−α}) ∩ ⋃_{j=1}^{2n} f^{j}(Crit) = ∅

for constants α, β > 0 depending on X, arbitrary ε > 0 and all n large enough.

By Koebe Distortion Theorem for ε small enough, for every 1 ≤ j ≤ 2n and z_{j} ∈ f^{−j}(z_{0}) we have

Comp_{z_{j}}f^{−j}B(z_{0}, εn^{−α}) ⊂ B(z_{j}, δ).

Let m = m(δ) be be such that f^{m}(B(y, δ/2) ⊃ J for every y ∈ J. Then, putting y = f^{l}(z_{0}), for every z_{n} ∈ f^{−n}(z_{0}) we find a component W_{z_{n}} of f^{−m}(Comp_{z_{n}}f^{−n}(B(z_{0}, εn^{−α})) ⊂ B(f^{l}(z_{0}, \frac{3}{2}δ) on which f^{m+n} is univalent (provided m ≤ n).

Therefore f^{m+n+l} is univalent from W′_{z_{n}} := Comp(f^{−(m+n+l)}(B(f^{l}(z_{0}), 2δ)) ⊂ W_{z_{n}} onto B(f^{l}(z_{0}), 2δ). The mapping

F = f^{m+n+l} : ⋃_{z_{n}∈f^{−n}(z_{0})} W′_{z_{n}} → B(f^{l}(z_{0}), 2δ)

has no critical points, hence Z := ⋂_{k=0}^{∞} F^{−k}(B(f^{l}(z_{0}), 2δ)) is an isolated expanding F - invariant (Cantor) subset of J.

For each z_{n} denote the point in f^{−m}(z_{n})∩W′_{z_{n}} by z′_{n}. We obtain for a constant C > 0 resulting from distortion and L = sup |f′|,

P (F |_{Z}, −t ln |F′|) ≥ ln(C ∑_{z_{n}∈f^{−n}(z_{0})} |(f^{m+n+l})′(z′_{n})|^{−t})

≥ ln(C ∑_{z_{n}∈f^{−n}(z_{0})} |(f^{n})′(z_{n})|^{−t}L^{−t(m+l)}).  (1.1)

Hence on the expanding set Z′ := ⋃_{j=0}^{m+n+l−1} f^{j}(Z) we obtain

P (f|_{Z′}, −t ln |f′|) ≥ \frac{1}{m + n + l} P (F, −t ln |F′|)

≥ \frac{1}{m + n + l} (ln C − t(m + l) ln L + ln ∑_{z_{n}∈f^{−n}(z_{0})} |(f^{n})′(z_{n})|^{−t}).

Passing with n to ∞ and using notation in Definition 1.1 we obtain

P (f|_{Z′}, −t ln |f′|) ≥ P (z_{0}, t) = P_{tree}(t).

QED

Remark 2.2. Notice that the proof of the inequality P_{hypvar}(t) ≤ P_{hyp}(t) for all t ≥ 0 provided in [P2] relying on Pesin-Katok theory (cf. [PU]) uses in fact the same idea as Proof of Proposition 2.1: capturing of a large hyperbolic set using good backward branches.

Remark 2.3. In [P2] several notions of conical limit set have been provided. The one (formally not defined in [P2]) smaller than all other is

Λ_{hyp} := ⋃{X : X ⊂ J, f − invariant isolated hyperbolic}.

The largest is

Λ_{ML1} := {x ∈ J : (∃η, M > 0, n_{j} → ∞), such that each f^{n_{j}}
has degree bounded by M on Comp_{x}f^{−n_{j}}(B(f^{n_{j}}(x), η)}.

For the hyperbolic Hausdorff dimension of J defined by

HD_{hyp}(J) := sup{HD(X) : X ⊂ J f − invariant isolated hyperbolic}

the inequality HD_{hyp}(J) ≤ HD(Λ_{hyp}) is obvious. The inequality HD(Λ_{ML1}) ≤ α(f), where α(f) denotes the smallest exponent α of a conformal measure i.e. probability measure on J with Jacobian |f′|^{α}, easily follows from distortion estimates, see [P2, Proposition A3.7].

The equality t_{0}(P_{hyp}) = HD_{hyp}(J), where t_{0}(P_{hyp}) denotes the first zero of P_{hyp}(t), follows immediately from the definitions and the equality of the zero of pressure and Hausdorff dimension on hyperbolic sets. Also α(f) ≤ t_{0}(P_{tree}), the first zero of P_{tree}(t) easily follows from Patterson-Sullivan type of construction, see [P2, the construction following Remark 2.6].

Therefore Proposition 2.1 gives a new proof of the equality of Hausdorff dimensions of all the conical limit sets to HD_{hyp}(J), omitting delicate considerations with the use of P_{DU}(t) and P_{Conf}(t).

Section 3. Pressure on periodic orbits

The proof of the inequality P_{Per}(t) ≥ P_{tree}(t) for t ≥ 0 (Proposition B), is a repetition of [PRS, Lemma 3.1], (shadowing by periodic orbits) and is in fact contained in Proof of Proposition 2.1. It also follows directly from Proposition 2.1 since for f on isolated hyperbolic sets where the mapping is topologically transitive pressures for all Hölder functions coincide by classical theory.

The main goal of this section is to prove Theorem C, by proving the reverse inequality under Hypothesis H.

We start with a sequence of standard lemmas.

Lemma 3.1. ([P1]) Given a rational function f : \overline{I} C → \overline{I} C there exists C_{1} > 0 such that for every positive integer k and z ∈ Per_{k}(f) ∩ J, it holds dist(z, f(Crit)) ≥ exp −C_{1}k.

Lemma 3.2. ([P1]) There exists C_{2} > 0 such that for every positive integer k and z ∈ Per_{k}(f) ∩ J for every δ > 0

#{j : 0 ≤ j < k, B(f^{j}(z), exp −δk) ∩ Crit ≠ ∅} ≤ C_{2}/δ.

Lemma 3.3. (telescope) ([P3], [PRS]) For every λ > 1 there exists a > 0 and C_{3} > 0 such that for every z ∈ J, k ≥ 0 and r > 0, if diamComp_{f^{j}(z)}f^{−(k−j)}B(f^{k}(z), r) ≤ a for every 0 ≤ j < k, then

diamComp_{z}f^{−k}B(f^{k}(z), r)/r ≤ C_{3}λ^{k}|(f^{k})′(z)|^{−1}

Lemma 3.4. (distortion) For every M ≥ 0 there exists C(M) ≥ 1 such that for every η > 0 small enough, k ≥ 0 and z ∈ J if

#{j : 0 ≤ j < k, Comp_{f^{j}(z)}f^{−(k−j)}B(f^{k}(z), 2η) ∩ Crit ≠ ∅} ≤ M

then for every w ∈ Comp_{z}f^{−k}B(f^{k}(z), η)

|(f^{k})′(w)| ≤ C(M)\frac{η}{diamComp_{z}f^{−k}B(f^{k}(z), η)}.

Lemma 3.5. (Koebe) If in Lemma 3.4 we assume M = 0 then there exists K > 1 such that for every w_{1}, w_{2} ∈ Comp_{z}f^{−k}B(f^{k}(z), η)

|(f^{k})′(w_{1})|/|(f^{k})′(w_{2})| ≤ K.

Definition 3.6. Denote P C^{n} := ⋃_{i=1}^{n} f^{i}(Crit). Fix n and arbitrary x_{0} ∈ J \P C^{n} and r > 0. For every backward trajectory of x_{0}, namely a sequence of points (x_{i}, i = 0, 1, ..., n) such that f(x_{i}) = x_{i−1} do the following procedure. Take the smallest k = k_{1} ≥ 0 such that Comp_{x_{k_{1}}} f^{−k_{1}}B(x_{0}, r) contains a critical point. Next let k_{2} be the smallest k > k_{1} such that Comp_{x_{k_{2}}} f^{−(k_{2}−k_{1})}B(x_{1}, r) contains a critical point. Etc. until k = n. Let the largest k_{j} ≤ n for the sequence (x_{i}) be denoted by k((x_{i})) and the set {y : y = x_{k((x_{i}))} for a backward trajectory (x_{i})} by N(x_{0}) = N(x_{0}, n, r).

Lemma 3.7. ([PRS, Appendix A]) For every ε > 0 for all r_{0} > 0 small enough and n sufficiently large, for every x_{0} ∈ J it holds #N(x_{0}, n, r_{0}) ≤ exp εn.

Proof of Theorem C, P_{Per}(t) ≤ P_{tree}(t) for t ≥ 0.

1^{o} Regular and singular periodic orbits. Fix ε > 0 and r > 0 small enough so that the conclution of Lemma 3.7 holds with r_{0} = 2r.

Definition 3.8. We say that a periodic orbit O of period n ≥ 1 is regular if there exists p ∈ O such that f^{n} is injective on Comp_{p}f^{−n}(B(p, r)). If O is not regular, then we say that O is singular.

(Caution. Given p, this definition depends on n. Indeed a parabolic periodic orbit of period n is regular if r is small enough, but for period kn with k large enough this orbit is singular.)

We denote by Per_{n}^{r} and Per_{n}^{s} the set of points p ∈ Per_{n} ∩ J whose orbit is regular and singular, respectively. We shall prove that for t ≥ 0,

P_{Per}^{r}(t) = lim sup_{n→∞} \frac{1}{n} ln ∑_{p∈Per_{n}^{r}} |(f^{n})′(p)|^{−t} ≤ P_{tree}(t),

P_{Per}^{s}(t) = lim sup_{n→∞} \frac{1}{n} ln ∑_{p∈Per_{n}^{s}} |(f^{n})′(p)|^{−t} ≤ (exp ε)P_{tree}(t).

This implies that P_{Per}(t) ≤ (exp ε)P_{tree}(t), and since ε > 0 can be chosen arbitrarily small we obtain P_{Per}(t) ≤ P_{tree}(t).

The first inequality will be proven in Part 3^{o}. In Part 4^{o} we shall bound the number of singular periodic orbits of period n; this will be the only part where Hypotesis H will be used. In Part 5^{o} we shall complete the proof of the second inequality by majorating |(f^{n})′(p)|, for p ∈ Per_{n}^{s}.

2^{o}. Fix an arbitrary constant δ > 0 and an integer n. Set m := n[1/δ], where [·] is the integer part function. Let X(n, δ) ⊂ J be an (exp −δn)-dense set in J (i.e. ⋃_{x∈X(n,δ)} B(x, exp −δn) ⊃ J), consisting of points x satisfying dist(x, P C^{m}) ≥ exp −2nδ (see Definition 3.6). We can assume that there is a constant C > 0 such that #X(n, δ) ≤ C exp 2nδ.

Fix a typical point z_{0} ∈ \overline{I} C as in Definition 0.1 of P_{tree}. By geometric Lemma 3.1 of [P2] there is a constant C′ > 0 such that for every j : n ≤ j ≤ m the quasi-hyperbolic distance in U_{j} := \overline{I} C \P C^{j} between z_{0} and a point x_{0} outside B(P C^{j}, exp −2nδ) is bounded by

C′\sqrt{j}\sqrt{ln 1/ min\{dist(z_{0}, P C^{j}), dist(x_{0}, P C^{j})\}} ≤ C′\sqrt{j}\sqrt{2nδ} = l_{n,j}.

In this case consider a path γ joining z_{0} and x_{0} with quasi-hyperbolic length bounded by l_{j,n}. Given x_{j} ∈ f^{−j}(x_{0}) let z_{j} ∈ f^{−j}(z_{0}) be the point such that x_{j} and z_{j} are the endpoints of a connected component of f^{−j}(γ). Note that this yields a 1-to-1 correspondence between points in f^{−j}(x_{0}) and points in f^{−j}(z_{0}).

Then for n large enough Koebe Distortion Lemma (Lemma 3.5) implies

|(f^{j})′(x_{j})| ≥ K^{−l_{n,j}}|(f^{j})′(z_{j})|.

3^{o}. Pressure on regular periodic orbits. Let O ⊂ Per_{n}^{r} be a regular periodic orbit and let p_{0} ∈ O such that f^{n} is injective on Comp_{p_{0}}f^{−n}(B(p, r)), see Defintion 3.8. Let x_{0} ∈ X(n, δ) be a point in B(p_{0}, r/2) and consider the (unique) backward orbit (x_{i}, i = 0, ..., n) such that x_{j} ∈ Comp_{f^{n−j}(p_{0})}f^{−j}(B(p_{0}, r/2)). Moreover let z_{n} be the n-th preimage of z_{0} associated to x_{n}, as in Part 2.

Applying Koebe Distortion Lemma (Lemma 3.5) to f^{n} at p_{0} and x_{n} we obtain, for n large enough

|(f^{n})′(p_{0})| ≥ K^{−1}|(f^{n})′(x_{n})| ≥ K^{−1}K^{−l_{n,n}}|(f^{n})′(z_{n})| ≥ K^{−1−C′n\sqrt{2δ}}|(f^{n})′(z_{n})|.

On the other hand, for n large enough

Comp_{x_{n}}f^{−n}B(x_{0}, r/2) ⊂ Comp_{p_{0}}f^{−n}B(p_{0}, 2r/3) ⊂ B(p_{0}, r/4) ⊂ B(x_{0}, r/3),

where the middle inclusion holds by a Fatou argument for large n, as f^{n} is injective in Comp_{p_{0}}f^{−n}(B(p_{0}, r)). By Schwarz Lemma it follows that the map f^{−n} : B(x_{0}, r/2) → B(x_{0}, r/3) has only one fixed point. Therefore the backward orbit (x_{i}, i = 0, 1, ..., n) cannot be associated to a regular periodic orbit, other than O.

Thus for a given z_{n} ∈ f^{−n}(z_{0}) there is at most #X(n, δ) ≤ C exp 2nδ regular orbits associated, at most one for each choice of x_{0} ∈ X(n, δ). We conclude that

∑_{p∈Per_{n}^{r}} |(f^{n})′(p)|^{−t} ≤ Cn(exp 2nδ)K^{t(1+C′n\sqrt{2δ})} ∑_{z_{n}∈f^{−n}(z_{0})} |(f^{n})′(z_{n})|^{−t}.

Since δ > 0 can be chosen arbitrarily small, we conclude that P_{Per}^{r}(t) ≤ P_{tree}(t).

4^{o}. Upper bound on the number of singular periodic orbits.

Consider a periodic orbit O ⊂ Per_{n} ∩ J. Since O is non-attracting we can choose p_{0} ∈ O such that for k ≥ 0 we have |(f^{k})′(p_{k})| ≥ 1, where p_{k} ∈ O is determined by f^{k}(p_{k}) = p_{0}. For ρ > 0 and i ≥ 0 we denote W_{i}(p_{0}, ρ) = Comp_{p_{i}}f^{−i}(B(p_{0}, ρ)).

Choose x_{0} = x_{0}(O) ∈ X(n, δ) such that p_{0} ∈ B(x_{0}, exp −nδ). Assign to each singular periodic orbit O a backward trajectory (x_{i}, i = 0, 1, ..., n) of x_{0}(O), such that x_{i} ∈ W_{i}(p_{0}, exp −nδ).

By Lemma 3.3 applied to λ := exp δ/3, if n is large enough we can prove by induction in i that

diamW_{i}(p_{0}, exp −nδ) ≤ exp −nδ/2,

for all 0 ≤ i ≤ n. In particular we have dist(p_{i}, x_{i}) ≤ exp −nδ/2, for i = 0, ..., n. Therefore, Hypothesis H implies that the number of singular periodic orbits O ⊂ Per_{n}^{s} with the same backward trajectory (x_{0}, ..., x_{n}) assigned, is bounded by exp nδ/2. (We have not used yet the assumption the orbits are singular.)

4^{o}1. Fix k = k((x_{i})) given by Lemma 3.7. Denote by X_{k,x_{k}} the set of all backward trajectories (x_{i}, i = 0, 1, ..., n) assigned to singular periodic orbits O ⊂ Per_{n}^{s}, given k and x_{k}. We shall prove that #X_{k,x_{k}} ≤ k(2d − 2), where d is the degree of f.

Write W′_{i} := Comp_{x_{k+i}}f^{−i}B(x_{k}, 2r) for i = 0, 1, ..., n − k. By the definition of k (no hitting of critical points going backward to W′_{n−k} along trajectories with this k) the sets W′_{n−k} are in 1-to-1 correspondence with (x_{0}, ..., x_{n}) ∈ X_{k,x_{k}}, and they are pairwise disjoint as different components of one f^{−(n−k)}(W′). For each i : 0 < i ≤ k at most #Crit ≤ 2d − 2 pairwise disjoint sets f^{−i}(W′_{n−k}) capture critical points, so summing over i we see that the number of such W′_{n−k} is majorated by k(2d − 2). Observe now that dist(p_{k}, x_{k}) ≤ exp −nδ/2 implies B(x_{k}, 2r) ⊃ B(p_{k}, r). Pulling this inclusion back we obtain W′_{i} ⊃ W_{i}(p_{k}, r), as x_{k+i} ∈ W′_{i} and x_{k+i} ∈ f_{p_{k+i}}^{−i}(W_{k}(p_{0}, exp −nδ)) ⊂ W_{i}(p_{k}, r) for all i = 0, ..., n−k. In particular W_{i}(p_{k}, r) does not capture critical points for i = 1, ..., n−k.

Therefore W_{i}(p_{k}, r) captures a critical point for some i : n−k < i ≤ n as O is singular, hence also f^{−i}(W′_{n−k}) contains this critical point.

We conclude that #X_{k,x_{k}} ≤ k(2d − 2).

4^{o}2. Considering that x_{0} ∈ X(n, δ), the estimate in Part 4^{o}1 and the definition of N(x_{0}) = N(x_{0}, n, 2r) (Definition 3.6) we obtain, by Lemma 3.7, that the number of backward trajectories (x_{0}, .., x_{n}) associated to a singular periodic orbit O ⊂ Per_{n}^{s} is majorated by

#X(n, δ) · N(x_{0}) · n(2d − 2) ≤ C(exp n(2δ + ε))n(2d − 2).

Since the number of singular periodic orbits assigned to a given backward trajectory is at most exp nδ/2, we conclude that for n big

#Per_{n}^{s} ≤ n^{2}(2d − 2)C exp n(δ/2 + 2δ + ε) ≤ exp n(3δ + ε).  (3.1)

5^{o}. Bound of the multiplier of a singular periodic orbit. Consider a singular periodic orbit O. Since O is non-attracting we can choose p_{0} ∈ O such that for k ≥ 0 we have |(f^{k})′(p_{k})| ≥ 1, where p_{k} ∈ O is determined by f^{k}(p_{k}) = p_{0}.

Recall that m := n[1/δ]. Consider W_{i} := W_{i}(p_{0}, 2 exp −δn) and

\frac{1}{2}W_{i} := W_{i}(p_{0}, exp −δn) for i = 0, 1, ..., m, as defined in Part 4.

Set λ = exp(nδ/3m) ≤ exp(δ^{2}/2) and consider a as in Lemma 3.3. By increasing n if necessary we may assume that exp −nδ/2 ≤ a. So we can prove by induction in i that diamW_{i} ≤ exp −nδ/2, for i = 0, …, m. So, by Lemma 3.2, the number of integers i : 0 < i ≤ m such that W_{i} contains a critical point, is bounded by M = C_{2}(m/n)(2/δ) ≤ C_{2}3/δ^{2}. Note that M is independent of n.

Choose x_{0} ∈ X(n, δ) in B(p_{0}, exp −nδ) and a backward trajectory (x_{i}, i = 0, 1, ..., m) such that x_{i} ∈ \frac{1}{2}W_{i}. For 0 ≤ i ≤ m let z_{i} ∈ f^{−i}(z_{0}) be chosen as in Part 2, for x_{i}.

Fix an arbitrary κ : 0 < κ < 1/2 and set m_{1} = n[1/δ^{κ}]. Then there are two cases.

5^{o}1. Case 1. diam\frac{1}{2}W_{m_{1}} ≥ λ^{−m} exp −C_{1}n.

By Lemma 3.4 we have

|(f^{m_{1}})′(x_{m_{1}})| ≤ C(M)\frac{exp −δn}{diam\frac{1}{2}W_{m_{1}}} ≤ C(M) exp C_{1}n ≤ exp 2C_{1}δ^{κ}m_{1}.

Thus, letting δ_{1} = 2C_{1}δ^{κ} + C′\sqrt{δ} ln K, we have
|(f^{m_{1}})′(z_{m_{1}})| ≤ |(f^{m_{1}})′(x_{m_{1}})|K^{C′\sqrt{m_{1}}\sqrt{2nδ}} ≤ exp δ_{1}m_{1}

Observe that δ_{1} → 0 as δ → 0.

Since |(f^{n})′(p_{0})| ≥ 1 we have

|(f^{n})′(p_{0})|^{−t/n} ≤ (exp δ_{1}t)|(f^{m_{1}})′(z_{m_{1}})|^{−t/m_{1}}.

5^{o}2. Case 2. diam\frac{1}{2}W_{m_{1}} < λ^{−m} exp −C_{1}n.

Put m_{2} := m_{1}[1/δ^{κ}]. Considering that |(f^{k})′(p_{k})| ≥ 1, for k ≥ 0, Lemma 3.3 implies that for

W′′_{i} := Comp_{p_{i}}f^{−(i−m_{1})}B(p_{m_{1}}, 2λ^{−m} exp −C_{1}n)),

we have

diamW′′_{i} < exp −C_{1}n for all m_{1} ≤ i ≤ m_{2}.

Hence Lemma 3.1 implies W′′_{i} ∩ Crit = ∅ and we can use Koebe Distortion Lemma (Lemma 3.5) to obtain

|(f^{m_{2}−m_{1}})′(x_{m_{2}})| ≤ K|(f^{m_{2}−m_{1}})′(p_{m_{2}})|.
Since |(f^{m_{1}})′(x_{m_{1}})| ≤ L^{m_{1}}, where L := sup |f′|, we obtain, writing

λ_{p_{0}} := |(f^{n})′(p)|^{1/n} ≥ 1,

|(f^{m_{2}})′(x_{m_{2}})| ≤ K(λ_{p_{0}})^{m_{2}−m_{1}}L^{m_{1}} ≤ Kλ_{p_{0}}^{m_{2}} exp m_{2}(2δ^{κ} ln L).

Letting δ_{2} = 2δ^{κ} ln L + C′\sqrt{δ} ln K we have

|(f^{m_{2}})′(z_{m_{2}})| ≤ |(f^{m_{2}})′(x_{m_{2}})|K^{C′\sqrt{m_{2}}\sqrt{2nδ}} ≤ Kλ_{p_{0}}^{m_{2}}(exp δ_{2}m_{2})

Observe that δ_{2} → 0 as δ → 0.

Equivalently we have

|(f^{n})′(p_{0})|^{−t/n} ≤ (K^{t/m_{2}} exp δ_{2}t)|(f^{m_{2}})′(z_{m_{2}})|^{−t/m_{2}}.

5^{o}3. Let p_{0} ∈ Per_{n}^{s} with minimizing |(f^{n})′(p)| and put m_{0} = m_{1} or m_{2} and δ_{0} = δ_{1} or δ_{2} depending on whether we are in Case 1 or 2. Using the estimate (3.1) and the estimates in 4.1 and 4.2, we have

( ∑_{p∈Per_{n}^{s}} |(f^{n})′(p)|^{−t})^{1/n} ≤ exp(3δ + ε)|(f^{n})′(p_{0})|^{−t/n} ≤

exp(3δ + ε)(Const(exp δ_{0}t)|(f^{m_{0}})′(z_{m_{0}})|)^{−t/m_{0}}.

Letting δ → 0 we obtain P_{Per}^{s}(t) ≤ (exp ε)P_{tree}(t), as desired. QED

Remark 3.9. It is easy to see that if Hypothesis H used here does not hold one can find 0 < δ_{1} ≤ δ_{2} < δ_{3}, and a sequence of n’s such that for each n from the sequence there exists P ⊂ Per_{n} with #P ≥ exp δ_{1}n and there exists x ∈ J satisfying the following

1. For each i : 0 ≤ i < n, f^{i}(P) ⊂ B(f^{i}(x), exp −δ_{3}n)

2. f^{n} is injective on the disc B′ = B(x, exp −(δ_{3} − δ_{2})n), in particular there are no critical points for f^{n} in B′, and the distortion of f^{n} on B := B(x, exp −δ_{3}n) (i.e. K from Lemma 3.5 for w_{1}, w_{2} ∈ B) is bounded by 1 + C exp(−δ_{2}n).

3. Neither f^{n}(B′) contains the closure of B′ nor it is contained in B′.

4. For each p ∈ P, |(f^{n})′(p) − 1| ≤ C exp −δ_{2}n,

Indeed. The existence of the constants δ_{1}, δ_{3}, C and P, x to satisfy the property 1 follows immediately from the negation of Hypothesis H. We can assume that all p ∈ P satisfy |(f^{k})′(p_{k})| ≥ 1 for all k = 0, ..., n with p_{k} in the periodic trajectory of p and f^{k}(p_{k}) = p as in Proof of Theorem C, Part 4 (choosing p_{0} the cardinality of P divides at most by n). Next choose a point p_{0} ∈ P so that for U_{k} = Comp_{p_{i}}f^{−k}(B(x, exp(−nδ_{3}))) and 2U_{k} := Comp_{p_{i}}f^{−k}(B(f^{i}(x), 2 exp(−nδ_{3}))), the following holds:

• diam2U_{k} ≤ exp(−\frac{1}{2}δ_{3}n), where (use Lemma 3.3 as in Proof of theorem C);

• The number of critical points for f^{n} in 2U_{n} is bounded by Constδ_{3}^{−1} (use Lemma 3.2);

• #(P ∩U_{n}) ≥ Const exp δ_{1}n. (Only if U_{i} is exp −δ_{3}-close to f(Crit(f)) the component U_{i+1} can loose a portion of points from the periodic orbits of P; this happens only a number of times bounded by Constδ_{3}^{−1}, again by Lemma 3.2. We always choose the component with the largest portion. This influences the choice of p_{0}. )

Using Lemma 3.1 we conclude that P is disjoint with B_{Crit} := B(Crit(f^{n}), exp −C_{1}n). We cover, with bounded multiplicity, U_{n} \ B_{Crit} by a family of exp \frac{1}{2}δ_{1}n discs of diameters proportional to the distances of their origins to Crit(f^{n}) ∩ U_{n}, and find in the family a disc W intersecting U_{n} so that #(P ∩ W) ≥ Const exp(nδ_{1}/2). P ∩ W is our final P. Of course this procedure changes δ_{1} and δ_{3} and yields δ_{2}.

The bound 1 + C exp(−δ_{2}n) for distortion follows from standard Koebe’s distortion estimate.

The property 3 holds since otherwise either B′ contains only one repelling fixed point for f^{n} due to the absence of critical points, by Schwarz Lemma, or B′ contains an attracting fixed point for f^{n}.

The property 4 follows from the property 3 for B in place of B′ and from Koebe’s estimate.

Remark 3.10. A strategy of Proof of Theorem C is similar to [BMS]. Regular and singular periodic orbits have been introduced in [BMS], under the names: good and bad. The estimate of the number of bad orbits by exp εn, in absence of indifferent cycles, has been achieved in [BMS] with the help of Markov puzzle structure. Hypothesis H, and even a stronger property, followed from an estimate for the degree of appropriate polynomial-like maps f^{n} : B → f^{n}(B) on thickened puzzle pieces B.

Appendix. Pressure for t < 0.

Section A.1. Comparison with standard pressure.

First remind the standard definition of pressure for a continuous mapping f : X → X for a compact metric space X and a function ϕ : X → IR ∪ {−∞}, see [W] and [K] where ϕ taking values −∞ has been considered.

Definition A1.

P (f, ϕ) := lim_{ε→0} lim sup_{n→∞} \frac{1}{n} ln sup_{S_{n,ε}} ∑_{x∈S_{n,ε}} exp ∑_{k=0}^{n−1} ϕ ∘ f^{k}(x)

where S_{n,ε} denote (n, ε)-separated sets, that is for any two x ≠ y belonging to S_{n,ε} we have max_{k=0,1,...,n−1}{ρ(f^{k}(x), f^{k}(y))} ≥ ε for the spherical metric ρ.

We use it for ϕ = −t ln |f′| and X = J, Julia set, obtaining

P (f|_{J}, ϕ) := lim_{ε→0} lim sup_{n→∞} \frac{1}{n} ln sup_{S_{n,ε}} ∑_{x∈S_{n,ε}} |(f^{n})′(x)|^{−t}.

Note that for t > 0 in presence of critical points in J where ϕ = ∞, the notion is not interesting because it is always equal to ∞.

Theorem A2. For all t < 0 we have P (f|_{J}, −t ln |f′|) = P_{var}(t).

This theorem follows immediately from the following version of the Variational Principle, see [K, Theorem 4.4.11]

Theorem A3. Variational Principle. Let f : X → X and ϕ : X → IR ∪ {−∞} be as above, with ϕ upper semicontinuous. Then

P (f, ϕ) = sup_{μ}{h_{μ}(f) + ∫ ϕdμ}

supremum taken over all ergodic f-invariant probability measures on X.

For t < 0 a part of the assertion of Theorem A, see Introduction, holds, namely Theorem A4. For all t < 0

P_{tree}(t) = P_{hyp}(t) = P_{hypvar}(t) = P_{var}(t).

Proof. The proof of P_{hypvar}(t) = P_{var}(t) is the same as for 0 ≤ t ≤ t_{0}, see Proof of Proposition 1.1 and [P2]. Namely P_{var}(t) > 0 implies h_{μ}(f) − tχ_{μ}(f) > 0 for μ such that this expression is close to sup_{μ}. This yields either χ_{μ}(f) > 0 or h_{μ}(f) > 0 which also implies χ_{μ}(f) > 0.

(Note that unlike in the t > 0 case, we do not need to rely on χ_{μ}(f) ≥ 0 [P1]. Note also that for t < 0 the possibility h_{μ}(f) = 0 can happen. Consider for example f(z) = z^{2} − 2 and −t large. Then μ supported on the repelling fixed point z = 2 is the only measure where h_{μ}(f) − tχ_{μ}(f) is close (and actually equal) to the supremum.)

Also the proof of P_{hyp}(t) = P_{hypvar}(t) holds for t < 0 as well as for t ≥ 0, see Remark 2.2 and [P2].

The proof of P_{tree}(t) = P_{tree}(z, t) ≥ P_{hyp}(t) for typical z is immediate, see the beginning of Proof of Proposition 2.1. The opposite inequality follows from P_{hyp}(t) = P (f|_{J}, −t ln |f′|), Theorems A.2 and A.4, and from

P_{tree}(z, t) ≤ P (f|_{J}, −t ln |f′|)  (A1)

for every z ∈ J. The latter inequality can be proved similarly to this inequality with −t ln |f′| replaced by ϕ real-valued, for z ∈ J, see [P3, Lemma 4]. QED

Remark. The inequality (A1) holds in fact for all z ∈ \overline{I} C. The proof uses geometric Lemma 3.1 of [P2] and in the case where z is superattracting, the proof uses the fact that there is no contribution to pressure along its periodic orbit, since for negative t we have |(f^{n})′(z)|^{−t} = 0. We do not provide details since we are interested only in z ∈ J.

We denote all the pressures in Theorems A2, A4 for t < 0 by P (t).

It is not interesting to consider P_{DU}(t) since one can consider pressure as in Definition A1 for ϕ = −t ln |f′| directly on J.

Section A.2. Conformal pressure.

To find conformal measures to study P_{Conf}(t), one can use Perron-Frobenius operator L : C(X) → C(X),

L(h)(x) = ∑_{y∈f^{−1}(x)} |f′(y)|^{−t}h(y).

The conjugate operator L^{∗} acts on measures and we find a probability eigenmeasure μ, i.e. such that L^{∗}(μ) = λμ for an eigenvalue λ > 0, see [W]. It follows from the definition that for every Borel set E ⊂ J on which f is injective

λμ(E) = ∫_{f(E)} |f′((f|_{E})^{−1}(x))|^{−t}dμ(x).  (A2)

This property is equivalent to being an eigen-measure for L^{∗}. We call it backward conformal with Jacobian λ^{−1}|f′|^{−t}.

Note that since f′ is equal to 0 at critical points, μ need not be forward quasi-invariant; we cannot rewrite the above property to μ(f(E)) = ∫ λ|f′|^{t}dμ as in Definition 0.6 in case μ has an atom at a critical value, because at critical points |f′|^{t} is infinite. This can happen indeed; consider f(z) = z^{2} − 2 and μ supported at {−2, 2} and λ = 4^{−t}.

We gather a part of this discussion in

Proposition A5. For t < 0 any probability measure μ on J is conformal with Jacobian λ|f′|^{t} iff it is backward conformal with Jacobian λ^{−1}|f′|^{−t} and has no atoms at critical values.

Proof. For μ conformal and any c ∈ Crit(f) μ(c) > 0 would imply μ(f(c) = ∫_{{c}} λ|f′(c)|^{t}δμ = ∞ which is impossible, and μ(c) = 0 implies μ(f(c) = ∫_{{c}} λ|f′(c)|^{t}δμ = 0. The integral conditions for conformality and backward conformality become equivalent. QED

In the case of t < 0 we replace in Definition 0.6 of P_{Conf}, sup λ by inf λ and conformal by backward conformal.

Definition A6. Backward Conformal pressure for t < 0. P̄_{BConf}(t) = ln λ(t), where

λ(t) := sup{λ > 0 : ∃μ backward conformal on J with Jacobian λ|f′|^{−t}}.

Now we can complete Theorem A4.

Theorem A7. For t < 0 we have P_{tree}(t) = P̄_{Conf}(t).

Proof. By definition we obtain for every backward conformal measure μ with Jacobian λ^{−1}|f′|^{−t} and every n

∫ ∑_{f^{n}(x)=z} |(f^{n})′(x)|^{−t}dμ(z) = ∫ L^{n}(11)dμ = λ^{n},  (A3)

hence

λ^{n} ≤ sup_{z∈J} ∑_{f^{n}(x)=z} |(f^{n})′(x)|^{−t} ≤ Const exp n(P (f|_{J}, −t ln |f′|) + δ)  (A4)

for an arbitrary δ > 0, by [P3, Lemma 4] (allowing the value −∞ does not change the proof.)

We conclude with ln λ ≤ P (f|_{J}, −t ln |f′|).

To prove the opposite inequality we find an appropriate backward conformal measure by Patterson-Sullivan method. We just repeat the construction from [P2, Remark 2.6] for z_{0} ∈ J(f) regular non-periodic.

Let us recall the construction. We define μ := lim μ_{λ}, where

μ_{λ} = ∑_{n≥0} ∑_{f^{n}(x)=z_{0}} D_{x} · b_{n}λ^{−n}|(f^{n})′(x)|^{−t}/B_{λ},

where D_{x} is Dirac measure, λ ↗ exp P (f|_{J}, −t ln |f′|) and b_{n}/b_{n+1} → 1 such that the series for λ replaced by exp P (f|_{J}, −t ln |f′|) is divergent, and B_{λ} is the normalizing factor, making the measure of J(f) equal to 1.

Clearly μ is backward conformal with Jacobian (exp −P (f|_{J}, −t ln |f′|))|f′|^{−t}. QED

Remark A8. We cannot replace in Theorem A7 P̄_{BConf}(t) defined with sup λ by \underline{P}_{BConf}(t) defined with inf λ, since it can happen that ln λ < P_{tree}(t).

Consider for example again f(z) = z^{2} −2 and negative t. Then P_{tree}(t) > P (0) = ln 2, whereas for μ supported on {−2, 2} we have ln λ = −t ln 4 close to 0 for t close to 0.

P̄_{Conf}(t) > \underline{P}_{Conf}(t) can happen only if there is a backward conformal measure which is not conformal. This will follow from Proposition 11 below.

Denote by X the set of all points z_{0} ∈ J whose every backward trajectory z_{1}, z_{2}, ... contains a critical point. The existence of a backward conformal measure not being conformal, is equivalent to X ≠ ∅. The proof is easy, use Proposition A5.

In the example f(z) = z^{2} − 2, for −1 < t < 0 we have \overline{P}_{BConf}(t) = (−t + 1) ln 2 > \underline{P}_{BConf}(t) = −t ln 4. For t < −1 we have \overline{P}_{BConf}(t) = −t ln 4 attained on μ above and no conformal measure exists.

The parameter t = −1 is called the ”phase transition” parameter. The phenomena related to it from the point of view of equilibrium measures and the nondifferentiability of P (t) at −1 have been comprehensively described in [MS0] and [MS1].

Remind one more equality for σ(L) being the spectral radius of Perron-Frobenius operator L, following in fact from (A4) and Theorems A2, A4.

Theorem A9. ([MS1], [P3]) log σ(L) = P (t).

We end Section A.3 with the proof that for conformal measures all ∈ λ’s coincide with P (t) (remember however that the set of conformal measures can be empty). We start with

Lemma A10. The inequality P_{tree}(z, t) ≥ P (t) for t < 0 holds for all z ∈ \overline{I} C \ X.

Proof. For every z ∈ \overline{I} C \ X there exists k ≥ 0 and z_{k} ∈ f^{−k}(z) which is typical in the sense of Definition 0.1 and such that f^{j}(z_{k}) ∩ Crit(f) = ∅ for all j = 0, 1, ..., k − 1, see [P2, Proof of Theorem 3.3]. Then

P (t) = P_{tree}(z_{k}, t) = lim sup_{n→∞} \frac{1}{n} ln ∑_{f^{n}(x)=z_{k}} |(f^{n})′(x)|^{−t}

= lim sup_{n→∞} \frac{1}{n} ln ∑_{f^{n}(x)=z_{k}} |(f^{n+k})′(x)|^{−t}|(f^{k})′(z_{k})|^{t} ≤ P_{tree}(z, t),

since |(f^{k})′(z_{k})|^{t} < ∞ as all f^{j}(z_{k}), j = 0, ..., k − 1 are non-critical. QED

Proposition A11. For every conformal measure μ with Jacobian λ|f′|^{t}, t < 0, we have ln λ = P (t).

Proof. ln λ ≤ P_{tree}(t) follows from Theorem A7. To prove the opposite inequality notice that Tchebyshev inequality and (A3) yield for every δ > 0

μ{z ∈ J : ∑_{f^{n}(x)=z} |(f^{n})′(x)|^{−t} ≥ λ^{n} exp δn} ≤ exp −δn.

Hence by Borel-Cantelli Lemma and letting δ → 0 we get P_{tree}(z, t) ≤ ln λ for all z in a full μ-measure set B. By Proposition A5, repeating the proof by induction along critical orbits, we see that μ has no atoms in ⋃_{i=1}^{∞} f^{i}(Crit) ⊃ X. So B \ X ≠ ∅ and for z ∈ B \ X we have by Lemma A10 that P (t) = P_{tree}(z, t) ≤ ln λ. QED

Section A.3. P_{tree}(t) ≤ P_{hyp}(t) revisited.

Below we give another direct proof of P_{tree}(t) ≤ P_{hyp}(t) trying to repeat Proof of Proposition 2.1; that proof must be however slightly modified. We should be especially careful in presence of eventually periodic critical points in J.

Let O_{1}, ..., O_{I} be periodic orbits in J contained in ⋃_{j=0}^{∞} f^{j}(Crit). Denote λ_{j} := |(f^{k_{j}})′(p_{j})|^{1/k_{j}} for p_{j} ∈ O_{j} of period k_{j}, for j = 1, ..., I.

If there exists j such that −t ln λ_{j} ≥ P_{tree}(t), then we choose as a hyperbolic set X = O_{j} which proves P_{tree}(t) ≤ P_{hyp}(t). So we can assume that all −t ln λ_{j} are less than a constant P_{0} < P_{tree}(t).

Let Y = ⋃_{j=1}^{m} f^{j}(Crit), for m as in Proof of Proposition 2.1. For every k = 0, ..., n define for a constant ε > 0

X_{k} := {z_{n} ∈ f^{−n}(z_{0}) : dist(f^{s}(z_{n}), Y) ≥ ε for s = n − k
and dist(f^{s}(z_{n}), Y) < ε for all s = 0, ..., n − k − 1}.

Consider also k = −1, where the condition for s = n − k should be omitted.

Note that if ε is small enough, then n − k > m_{0} := #Crit(f) · m implies for z_{n} ∈ X_{k} there exists j such that

dist({f^{s}(z_{n}), O_{j}) < ε for all s = m_{0} + 1, ..., n − k}.  (A1)

Denote

R_{k} := ∑_{x∈f^{−n}(z_{0})∩X_{k}} |(f^{k})′(f^{n−k}(x))|^{−t}

and

S_{k} = ∑_{x∈f^{−k}(z_{0})} |(f^{k})′(x)|^{−t}.

Observe that due to (A1) for ε small enough there exists a bound A (independent of n) such that for every k and z_{k} ∈ f^{−k}(z_{0}) we have

#{z_{n} ∈ X_{k} : f^{n−k}(z_{n}) = z_{k}} ≤ A.

We have for L = sup |f′|

S_{n} = ∑_{k=−1,...,n} ∑_{f^{n}(x)=z_{0},x∈X_{k}} |(f^{n})′(x)|^{−t}

≤ L^{−tm_{0}}( ∑_{k=n−m_{0},...,n} R_{k} + ∑_{k=−1,...,n−m_{0}−1}(exp(n − k − m_{0})P_{0})R_{k}).

Above we have got exp(n − k − m_{0})P_{0} due to (A1). The constant C is neede in the case only a piece of O_{j} is followed by f^{s}(z_{n}).

By the definition of P_{tree}(t) for an arbitrary constant a > 0 there is k(a) such that for all k ≥ k(a) we have S_{k} ≤ exp k(P_{tree}(t) + a), hence R_{k} ≤ A exp k(P_{tree}(t) + a). Assume also that n satisfies S_{n} ≥ exp nP_{1} for P_{1} = \frac{P_{0}+P_{tree}(t)}{2}. Fix finally a = P_{tree}(t) − P_{0}.

Suppose that for all k : \frac{1}{4}n ≤ k ≤ n we have R_{k} ≤ exp kP_{0}. Then we obtain

exp nP_{1} ≤ CL^{−tm_{0}}(Const + ∑_{k_{0}<k≤n/4} A exp((n − k − m_{0})P_{0}) exp k(P_{tree}(t) + a))+

∑_{n/4<k≤n−m_{0}} exp((n − k − m_{0})P_{0}) exp kP_{0})

≤ CL^{−tm_{0}}(Const + \frac{n}{4}A exp nP_{tree}(t) + \frac{3n}{4} exp nP_{0}).

which on the other hand is obviously false for n large enough.

Therefore there exists k : n/4 < k ≤ n such that R_{k} ≥ exp kP_{0}. For all z_{k} ∈ f^{n−k}(X_{k}) we find z′_{k} as in Proof of Proposition 2.1 and construct invariant sets Z, Z′, with n replaced by k.

Recall that for all x_{k} ∈ f^{n−k}(X_{k}) we have dist(x_{k}, Y) ≥ ε. Hence there exists ε′ > 0 such that f′(f^{s}(z′_{k})) ≥ ε′ for all s : 0 ≤ s ≤ m. We can also assume that f′(x) ≥ ε′ for x ∈ B(X, 2δ).

We can repeat now and modify the calculation (1.1).

P (F |_{Z}, −t ln |F′|) ≥ ln(C ∑_{z_{k}∈f^{n−k}(X_{k})} |(f^{m+k+l})′(z′_{k})|^{−t}) ≥

ln(C ∑_{z_{k}∈f^{n−k}(X_{k})} |(f^{k})′(z_{k})|^{−t}(ε′)^{−t(m+l)} ≥ ln C − t(m + l) ln ε′ + ln A^{−1} + kP_{0}.

Passing to Z′ we finish the proof as in Proposition 2.1. QED

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[Z] M. Zinsmeister Formalisme thermodynamique et systèmes dynamiques holomorphes. Panoramas et Synthèses, 4. Société Mathématique de France, Paris, 1996. English translation available from the AMS.

ADDRESSES:

Feliks Przytycki, Institute of Mathematics Polish Academy of Sciences, ul. Śniadeckich 8, 00950 Warszawa, Poland. e-mail: feliksp@impan.gov.pl

Juan Rivera-Letelier, Institute for Mathematical Sciences SUNY, Stony Brook NY 11794-3660, U.S.A. e-mail: rivera@math.sunysb.edu

Stanislav Smirnov, Department of Mathematics, Royal Institute of Technology, Stockholm 10044, Sweden. e-mail: stas@math.kth.se