MULTIFRACTAL ANALYSIS FOR PARABOLICALLY SEMIHYPERBOLIC GENERALIZED POLYNOMIAL-LIKE MAPS

BERND O. STRATMANN AND MARIUSZ URBAŃSKI

Abstract. In this paper we study parabolically semihyperbolic generalized polynomial-like maps and give a finer fractal analysis of their Julia sets. We discuss various generalizations of the classical notion of topological pressure to situations in which the underlying potentials are not necessarily continuous or bounded. Subsequently, we investigate various types of conformal measures and invariant Gibbs states, which then enables us to deduce analytic properties for the generalized pressure functions. On the basis of these results, we finally derive our multifractal analysis, and then show that for the special case in which the Julia set does not contain critical points, this general multifractal analysis has a more transparent geometric interpretation in terms of the local scaling behaviour of the canonically associated equilibrium state.

1. Introduction and statement of main results

In this paper we give a finer multifractal analysis of Julia sets J(f) for parabolically semihyperbolic generalized polynomial-like maps f (see section 2 for the definition of these maps). First, we give a detailed discussion of various extensions of the classical notion of topological pressure P_{top}(f, ϕ). Different from the classical situation, which requires the potential ϕ to be continuous, these extensions P(t, ϕ) are associated to potentials of the form −t log |f′| + ϕ, which are in general (that is, if the critical points are of dynamical significance) neither continuous nor bounded (section 2.3 and section 4). This discussion is then followed by investigations of various types of conformal measures and invariant Gibbs states (section 3). Subsequently, on the basis of these considerations, we then derive our multifractal analysis for parabolically semihyperbolic generalized polynomial-like maps. We remark that the results in this paper are significant extensions of our results obtained in [15], and furthermore they provide further generalizations of the results in [7] and [14] where totally different methods have been employed. In order to state the main results, we need to introduce the following. For a Hölder continuous function ϕ : J(f) → IR, the lower and upper rate of ϕ at x ∈ J(f) are defined by

\underline{ρ}_{ϕ}(x) := \underline{lim}_{n→∞} \frac{S_{n}(P_{top}(f, ϕ) − ϕ(x))}{log |(f^{n})′(x)|} and \overline{ρ}_{ϕ}(x) := \overline{lim_{n→∞}} \frac{S_{n}(P_{top}(f, ϕ) − ϕ(x))}{log |(f^{n})′(x)|}.

Research partially supported by the NSF Grant DMS 0100078.

If these two rates coincide, then we write ρ_{ϕ}(x) to denote their common value. In order to study the fine-scale geometry of J(f), we then follow the foot steps of the classical multifractal formalism and consider the (ϕ, α)-level sets K_{ϕ}(α), which are defined by, for α ∈ IR,

K_{ϕ}(α) := {x ∈ J(f) : ρ_{ϕ}(x) = α}.

The following theorem gives the first main result of this paper.

Theorem 1. Let f be a parabolically semi-hyperbolic generalized polynomial-like map, and let ϕ : J(f) → IR be a Hölder continuous potential such that sup(ϕ) < P_{top}(f, ϕ) = 0. Then the following holds, where μ_{ϕ} refers to the equilibrium state of ϕ.

(i) For μ_{ϕ}-a.e. x ∈ J(f), we have that ρ_{ϕ}(x) exists and

ρ_{ϕ}(x) = \frac{− ∫ ϕdμ_{ϕ}}{∫ log |f′|dμ_{ϕ}} .

(ii) There exists a function T : (0, 1] → IR, uniquely determined by P(T (q), qϕ) = 0, such

that – T is real-analytic and T′ is strictly negative,

– HD(K_{ϕ}(−T′(q))) = T (q) − qT′(q), for every q ∈ (0, 1).

We then continue by investigating analytic properties of the multifractal ϕ-spectrum k_{ϕ}, which is defined for α ∈ IR by

k_{ϕ}(α) := HD(K_{ϕ}(α)).

The following theorem gives the second main result of this paper. Note, throughout we let h refer to the Hausdorff dimension HD(J(f)) of J(f).

Theorem 2. Let f be a parabolically semi-hyperbolic generalized polynomial-like map, and let ϕ : J(f) → IR be a Hölder continuous potential such that sup(ϕ) < P_{top}(f, ϕ) = 0. If the equilibrium state μ_{ϕ} is not equivalent to the h-conformal measures ν_{h} of f, hence in particular if f has a parabolic point or a non-exceptional critical point, then the domain of the function k_{ϕ} contains a non-degenerate interval on which k_{ϕ} is real-analytic.

Finally, we consider the special class of parabolically semi-hyperbolic generalized polynomiallike maps f for which J(f) does not contain critical points of f. Maps of this type will be referred to as parabolic generalized polynomial-like maps, and we show that for these maps the results of the multifractal analysis in Theorem 1 and Theorem 2 have a more transparent geometric interpretation. More precisely, we obtain the following theorem which states the third main result of this paper. In here, p_{max} refers to the maximal number of petals a parabolic fixed point of f can possibly have.

Theorem 3. Let f be a parabolic generalized polynomial-like map, and ϕ : J(f) → IR a Hölder continuous potential with Hölder exponent exceeding p_{max}/(p_{max} + 1) and such that P_{top}(f, ϕ) > inf_{n∈IN} \{\frac{1}{n} sup S_{n}ϕ\}. Then we have for the equilibrium state μ_{ϕ} associated with ϕ, and for any q ∈ (0, 1),

HD (\{z ∈ J(f) : lim_{r→0} \frac{log μ_{ϕ}(B(x, r))}{log r} = −T′(q)\}) = T (q) − qT′(q).

(Here, T refers to the function which we already considered in Theorem 1).

The paper is organized as follows.

1. Introduction and statement of main results

2. Preliminaries 2.1. Parabolically semihyperbolic generalized polynomial-like maps 2.2. Conformal graph directed Markov systems and GPL-maps 2.3. Topological pressure functions

3. Invariant Gibbs states

4. Real analyticity of the topological pressure

5. Multifractal analysis 5.1. The general case of a parabolically semi-hyperbolic GPL-map 5.2. The parabolic case without critical points in the Julia set

Throughout, we use the following conventions to describe the relationship between two positive numbers a and b. We write a ≍ b if the ratio of a and b is uniformly bounded away from zero and infinity. Similarly, we write a ≪ b, or a ≫ b respectively, if a/b is uniformy bounded away from infinity, or zero respectively.

Acknowledgement: We should like to thank the Mathematical Department at the University of North Texas for its warm hospitality and excellent working conditions while doing part of the work to this paper. Also, we should like to thank the NSF for financial support.

2. Preliminaries

2.1. Parabolically semihyperbolic generalized polynomial-like maps.

In this section we give a brief introduction into parabolically semihyperbolic generalized polynomial-like maps. We begin with recalling the definition of a generalized polynomial-like mapping, which will be abbreviated throughout as a GPL-map. Note that we have adopted the notation of [15]. For U ⊂ C I an open Jordan domain with smooth boundary, let U := ⋃_{i∈I} U_{i} be a finite union of Jordan domains U_{i} which are fully contained in U and which have pairwise disjoint closures. A GPL-map f is a map

f : U → U

which has a holomorphic extension to an open neighbourhood of U such that for each i ∈ I the restriction of this extension to U_{i} is a surjective branched covering map. We let J(f) refer to the Julia set of f. Let Ω denote the set of parabolic periodic points of f given by

Ω := {ω ∈ U : f^{q}(ω) = ω and (f^{q})′(ω) = 1 for some q ∈ IN}.

Without loss of generality, we may assume that the parabolic periodic points of f are in fact fixed points of f, and that f′(ω) = 1 for each ω ∈ Ω (this is achieved as usual, by taking a suitable iterate of f; note that this does not affect our analysis here since P_{top}(f, −t log |f′|) = \frac{1}{n}P_{top}(f, −t log |(f^{n})′|), for each n ∈ ℕ). Also, we define

Crit(f) := {c : f′(c) = 0} and Crit(J(f)) := J(f) ∩ Crit(f).

It will be convenient to split up the index set I in the following way.

I_{o} := \{i ∈ I : \overline{U_{i}} ∩ ⋃_{n≥1} f^{n}(Crit(f)) = ∅\} (‘post-critical free indices’),

I_{p} := \{i ∈ I : Ω ∩ \overline{U_{i}} ≠ ∅\} (‘parabolic indices’),
I_{c} := {i ∈ I : U_{i} ∩ Crit(f) ≠ ∅} (‘critical indices’),
I_{r} := I \ (I_{c} ∪ I_{p}) (‘regular indices’).

With this decomposition of the finite index set I, we define

U_{o} := ⋃_{i∈I_{o}} U_{i}, U_{p} := ⋃_{i∈I_{p}} U_{i}, U_{c} := ⋃_{i∈I_{c}} U_{i}, U_{r} := ⋃_{i∈I_{r}} U_{i}.

Definition 2.1. A GPL-map f is called parabolically semihyperbolic if and only if the following conditions are satisfied.

(a) I_{c} ⊂ I_{o}, (b) \overline{U_{o} ∪ U_{r}} ⊂ U, (c) ⋃_{n∈IN} f^{n}(Crit(f)) ⊂ U_{r}, (d) I_{p} ≠ I.

Note that in this definition we do not rule out the possibility that Ω = ∅. That is, we let the class of semihyperbolic GPL-maps be contained in the class of parabolically semihyperbolic GPL-maps. Also, recall that a GPL-map f is called critically non-recurrent if for each c ∈ Crit(J(f)) we have that U_{i} ∩ {f^{n}(c) : n ∈ IN} = ∅, where i ∈ I is uniquely determined by the fact that c ∈ U_{i}. Hence, by (a) in the definition above, a parabolically semihyperbolic GPL-map is always critically non-recurrent. In fact even more can be said, namely that a GPL-map is in particular always parabolically subhyperbolic and critically tame (see [18] for the definitions). Also, note that for a parabolically semihyperbolic GPL-map the sets I_{o}, I_{p} and I_{r} are always pairwise disjoint. The following lemma is an immediate consequence of the fact that a GPL-map is critical non-recurrent in combination with the topological exactness of its Julia set.

Lemma 2.2. For a critically non-recurrent GPL-map f we have that the closure of the forward orbit of Crit(f) is a nowhere dense in J(f).

Throughout, we shall assume that for i ∈ I_{p} the map f : U_{i} → U is a conformal homeomorphism. By Schwarz’s lemma, we then have that Ω ∩ \overline{U_{i}} is a singleton, denoted by ω_{i}, so that we have in particular that ω_{i} = ∂U_{i} ∩ ∂U. Also, with f_{i}^{−1} : \overline{U} → \overline{U_{i}} referring to the inverse branch of f for which f_{i}^{−1}(ω_{i}) = ω_{i}, the Denjoy-Wolf theorem implies that f_{i}^{−n}(z) converges to ω_{i} uniformly, for each z ∈ U. Since f_{i}^{−1} has an analytic extension to an open neighbourhood of ω_{i} and since (f_{i}^{−1})′(ω_{i}) = 1, the Taylor expansion of this extension for z close to ω_{i} is of the form, for some fixed a_{i} ≠ 0 and p(ω_{i}) ∈ ℕ,

f_{i}^{−1}(z) = z + a_{i}(z − ω_{i})^{p(ω_{i})+1} + ….

Using this, it follows (see e.g [1]) that for each compact set F ⊂ U there exists a constant C_{F} ≥ 1 such that, for every n ∈ IN and for all z ∈ F,

C_{F}^{−1}n^{− \frac{p(ω_{i})+1}{p(ω_{i})}} ≤ |(f_{i}^{−n})′(z)| ≤ C_{F} n^{− \frac{p(ω_{i})+1}{p(ω_{i})}} .  (2.1)

Clearly, the geometric meaning of p(ω_{i}) is that it is the number of petals at ω_{i}. Throughout we let p_{max} := max{p(ω_{i}) : i ∈ I_{p}} denote the maximal number of petals which can possibly occur at parabolic points of f.

2.2. Conformal graph directed Markov systems and GPL-maps.

The analysis in section 4 of analytic properties of the pressure function will make use of the fact that a parabolically semihyperbolic GPL-map is closely related to the concept of a conformal graph directed Markov system (abbreviated as a CGDM-system). In order to explain this relationship in greater detail, we now first recall from [10] the definition a CGDM-system. The combinatorical spine of a graph directed Markov system is represented by a directed multigraph (V, E, i, t, A), consisting of a finite set V of vertices, a countable set E of directed edges, two functions i, t : E → V, and a transition matrix A : E × E → {0, 1}. Here, i(e) refers to the initial vertex and t(e) to the terminal vertex of an edge e ∈ E. In our special context here, the matrix A = (A_{uv}) has the property that A_{uv} = 1 if and only if t(u) = i(v). The associated symbolic space is then defined as follows.

E := {(e_{1}, e_{2}, …) ∈ E^{∞} : A_{e_{i}e_{i+1}} = 1 for all i ∈ IN}.

Furthermore, there is a set {X_{v} : v ∈ V } of non-empty compact connected subsets X_{v} of C I, and a set Φ = {ϕ_{e} : X_{t(e)} → X_{i(e)}}_{e∈E} of univalent contractions, all with some fixed Lipschitz constant 0 < s < 1. Each of these maps ϕ_{e} is assumed to have a conformal extension from some connected open neighbourhood W_{t(e)} of X_{t(e)} to some connected open neighbourhood W_{i(e)} of X_{t(e)}. If additionally Φ satisfies the ‘open set condition’ as well as the ‘cone condition’ (see [10], Section 4.2), then we say that Φ is a CGDM-system. Note, in this situation the limit set J_{Φ} of Φ is given as follows. For τ = (τ_{1}, τ_{2}, …) ∈ E and n ≥ 1, let

ϕ_{τ|_{n}} := ϕ_{τ_{1}} ∘ … ∘ ϕ_{τ_{n}} : X_{t(τ_{n})} → X_{i(τ_{1})}.

Since Φ consists of s-Lipschitz contractions, it follows that π(τ) := ⋂_{n∈IN} ϕ_{τ|_{n}}(X_{t(τ_{n})}) is a singleton. This procedure gives a map π : E → ⋃_{v∈V} X_{v}, and we let

J_{Φ} := π(E).

The following proposition states the main result of this section. The proof introduces some notation which will be relevant also in section 4. Furthermore, recall from [10] that ‘finitely primitive of order 2’ means that for each pair u, v ∈ V_{f} there exists a, b ∈ E_{f} such that A_{a,b} = 1 and i(a) = u, t(b) = v.

Proposition 2.3. Let f be a parabolically semihyperbolic GPL-map. Then there exists a finitely primitive order 2 CGDM-system Φ_{f} with J_{Φ_{f}} ⊂ J(f) such that

J_{Φ_{f}} ∩ U_{o} = J(f) ∩ U_{o} \ ⋃_{n≥0} f^{−n}(Ω ∪ ⋂_{k≥0} f^{−k} (U_{r})) .

Proof. For the proof it suffices to show how to associate to f a CGDM-system. For this we define U_{(i,j)} := f_{j}^{−1}(U_{i}), for each (i, j) ∈ (I_{p} × I_{r}) ∪ (I_{p} × I_{p} \ {diag.}). Here {diag.} denotes the diagonal in I_{p} × I_{p}, and f_{j}^{−1} : U → U_{j} refers the inverse of the map f|_{U_{j}}. Using condition (c) in the definition of a parabolically semihyperbolic GPL-map, it follows that

U_{(i,j)} ∩ ⋃_{n∈IN} f^{n}(Crit(f)) = ∅.  (2.2)

Let V_{f} := I_{o} ∪ (I_{p} × I_{r}) ∪ (I_{p} × I_{p} \ {diag.}) be the set of vertices. The conformal univalent contractions of our system are given as follows. By (2.2) and the definition of the set I_{o}, we have that for each v ∈ V_{f} the holomorphic inverse branches of any iterate of f are well-defined on U_{v}. Hence, for v ∈ V_{f} and n ∈ IN we consider all holomorphic inverse branches f_{∗}^{−n} : U_{v} → U of f^{n} for which f_{∗}^{−n}(U_{v}) ⊂ U_{w} for some w ∈ V_{f}, and for which f^{k}(f_{∗}^{−n}(U_{v})) ∩ (⋃_{s∈V} U_{s}) = ∅ for all 1 ≤ k < n. In this situation we write ϕ_{e} : U_{t(e)} → U_{i(e)} instead of f_{∗}^{−n} : U_{v} → U_{w}, where t(e) = v and i(e) = w. Also, we define N(e) := n. Now, let

Φ_{f} := \{ϕ_{e} : \overline{U_{t(e)}} → \overline{U_{i(e)}}\}_{e∈E_{f}} ,

where E_{f} is some countable auxiliary set parametrizing the family Φ_{f}. Note that the set V_{f} of vertices is finite, whereas in general the set E_{f} of edges is infinite. Let E_{f} refer to the corresponding symbolic space. Since \overline{U_{v}} ∩ \overline{⋃_{n∈IN} f^{n}(Crit(f))} = ∅, it follows that for each v ∈ V_{f} there exists an open connected simply connected set \overline{U_{v}} ⊂ W_{v} ⊂ U such that if e ∈ E_{f} and t(e) = v, then ϕ_{e} has a univalent holomorphic extension to W_{v} and ϕ_{e}(W_{v}) ⊂ U_{i(e)} (for later use, we also introduce accordingly W and W_{o} := ⋃_{i∈I_{o}} W_{i}). Since for each i ∈ I_{p} we have that ⋂_{n≥0} f^{−n}(J(f) ∩ \overline{U_{i}}) = \{ω_{i}\}, we immediately obtain from the construction of Φ_{f} that

J_{Φ_{f}} ∩ U_{o} = J(f) ∩ U_{o} \ ⋃_{n≥0} f^{−n}(Ω ∪ ⋂_{k≥0} f^{−k} (U_{r})) .

We remark that the cone condition is satisfied, since for each v ∈ V the boundaries of the disc \overline{U_{v}} is smooth. Also, the open set condition follows immediately from the construction of Φ_{f}, noting that the elements of Φ_{f} are inverse branches of forward iterates of f. Finally, since for each pair u, v ∈ V there exist a, b ∈ E_{f} such that i(b) ∈ I_{o} and such that i(a) = u, t(b) = v and A_{a,b} = 1, it follows that the system Φ_{f} is finitely primitive of order 2.

2.3. Topological pressure functions.

In this section we give a discussion of various definitions of the concept of a ‘pressure function’ associated with a dynamical system. We shall see that in the context of a parabolically semihyperbolic GPL-map all these different notions of pressure coincide. Let us begin with recalling the classical definition in ergodic theory of the notion pressure. We refer to [2] for further details. Let T : X → X be a continuous automorphism of a compact metric space (X, d). Also, let d_{n} refer to the metric on X which is given, for x, y ∈ X and n ≥ 0, by

d_{n}(x, y) := max{d(T^{i}(x), T^{i}(y)) : 0 ≤ i ≤ n}.

Then a set F ⊂ X is called (n, ϵ)-separated, for n ≥ 0 and ϵ > 0, if it is separated with respect to the metric d_{n} (that is d_{n}(x, y) ≥ ϵ for all distinct x, y ∈ X). With (F_{n}(ϵ))_{n∈ℕ} denoting a sequence of maximal (in the sense of inclusion) (n, ϵ)-separated sets, the topological pressure P_{top} of a continuous potential function ϕ : X → IR is then defined by

P_{top}(T, ϕ) := lim_{ϵ→0} lim sup_{n→∞} \frac{1}{n} log ( ∑_{x∈F_{n}(ϵ)} exp ∑_{j=0}^{n−1} ϕ ∘ T^{j}(x)) .

Note that the concept of topological pressure has its origin in topological dynamics. Closely related to it is the measure theoretical entropy h_{μ}(T), which is central in ergodic theory. It is well-known that the link between these two important notions is given by the following so called variational principle

P_{top}(T, ϕ) = sup{h_{μ}(T) + ∫ ϕdμ}.

In here, the supremum is taken with respect to all T -invariant (ergodic) Borel probability measures μ supported on X.

For more general situations in which the potentials are no longer continuous or bounded, this classical definition of pressure fails. More precisely, for a GPL-map f such that J(f) has nontrivial intersection with Crit(f), we are led to consider potentials of the form −t log |f′|+ϕ, for t ≥ 0 and ϕ : X → IR continuous. One easily verifies that potentials of this type are in general neither continuous nor bounded. A priori it is not clear how to adapt the above definition of pressure to this more general situation. However, in [11] Przytycki suggested, in the context of rational maps, several ways to generalize the concept of topological pressure associated with the potential −t log |f′|. For a GPL-map f and for potentials of the type −t log |f′| + ϕ, we now start our discussion of how to amend the classical definition of topological pressure, by giving the following generalization of one of Przytycki’s suggestions.

(P1) Point pressure. For each z ∈ J(f), t ≥ 0 and ϕ : J(f) → IR a continuous potential, we let

P_{z}(t, ϕ) := lim sup_{n→∞} \frac{1}{n} log ∑_{x∈f^{−n}(z)} |(f^{n})′(x)|^{−t} exp(S_{n}ϕ(x)).

The point pressure P_{P}(t, ϕ) is then defined by

P_{P}(t, ϕ) := inf_{z∈J(f)} P_{z}(t, ϕ).

Before stating further possible generalizations of the notion pressure, we first give a brief discussion of the point pressure just defined. For this the following technical observations will turn out to be useful.

Lemma 2.4. Let θ > 0 be given. Then there exists a constant B_{θ} ≥ 1, depending on θ, such that for each ϵ > 0 sufficiently small, and for every n ∈ IN, z ∈ J(f) and f^{n}(z) ∉ B(Ω, θ), we have

diam(C_{n}(z, B(f^{n}(z), ϵ))) ≤ B_{θ} n^{− \frac{pmax+1}{pmax}} .

Proof. The proof is an immediate adaptation of the proof of Lemma 4.3 in [18]. (Note that by property (c) in the definition of parabolically semi-hyperbolic GPL maps in section 2.1, the factor ξ in formula (4.7) of [18] can be neglected).

The lemma has the following immediate consequence.

Lemma 2.5. Let θ > 0 and ϕ : J(f) → IR be a Hölder continuous function with Hölder exponent exceeding p_{max}/(p_{max}+1). Then there exists a constant C_{θ} ≥ 0, depending on θ, such that for each ϵ > 0 sufficiently small, and for every n ∈ IN, z ∈ J(f) and f^{n}(z) ∉ B(Ω, θ), we have

|S_{n}ϕ(x) − S_{n}ϕ(y)| ≤ C_{θ}.

For a given continuous potential function ϕ : J(f) → IR, let \overline{ϕ} and ϕ̂ be defined by

\overline{ϕ} := inf_{n∈IN} \{\frac{1}{n} sup S_{n}ϕ\} and ϕ̂:= max\{∫ ϕdμ : μ ∘ f^{−1} = μ\}.

Clearly, for every n ∈ IN and every f-invariant Borel probability measure μ, we have

∫ ϕdμ = \frac{1}{n} ∫ S_{n}ϕdμ ≤ \frac{1}{n} sup S_{n}ϕ.

This implies ∫ ϕdμ ≤ \overline{ϕ}, from which we deduce that

ϕ̂ ≤ \overline{ϕ}.  (2.3)

In order to proceed, we require the following simple observation.

Lemma 2.6. For every ϵ > 0 there exists q ∈ IN such that sup\{S_{n}ϕ\} ≤ (\overline{ϕ} + ϵ)n, for all n ≥ q.

Proof. By definition of \overline{ϕ}, for every ϵ > 0 there exists m ∈ IN such that \frac{1}{m} sup\{S_{m}ϕ\} < \overline{ϕ}+ \frac{ϵ}{2}. Now, if n ≥ m such that n = sm + r, for 0 ≤ r ≤ m − 1 and s ∈ IN, then it follows

sup\{S_{n}ϕ\} ≤ sup\{S_{r}ϕ\}+sup\{S_{sm}ϕ\} ≤ (m−1)||ϕ||_{∞}+s sup\{S_{m}ϕ\} ≤ (m−1)||ϕ||_{∞}+sm (\overline{ϕ} + \frac{ϵ}{2}) .

This implies, for n sufficiently large,

\frac{1}{n} sup\{S_{n}ϕ\} ≤ \frac{(m − 1)||ϕ||_{∞}}{n} + \frac{sm}{n} (\overline{ϕ} + \frac{ϵ}{2}) ≤ \frac{(m − 1)||ϕ||_{∞}}{n} + \overline{ϕ} + \frac{ϵ}{2} ≤ \overline{ϕ} + ϵ.

By the previous lemma, we can now define the following critical exponents, for s ∈ IR and ϕ : J(f) → IR a continuous potential,

δ(ϕ, s, z) := inf\{t ≥ 0 : P_{z}(t, ϕ) ≤ s\}, δ(ϕ, z) := δ(ϕ, \overline{ϕ}, z) and δ(ϕ) := inf_{z∈J(f)} δ(ϕ, z).

For the rest of this paper we shall assume from now on, if not stated otherwise, that the potential ϕ : J(f) → IR is a Hölder continuous function with Hölder exponent α which exceeds p_{max}/(p_{max} + 1). The following lemma gives a generalization of a result of Przytycki (cf. Lemma 3.3 in [11]).

Lemma 2.7. There exists a set E ⊂ J(f) of Hausdorff dimension equal to zero such that, for all z ∈ J(f) \ E and t ≥ 0,

P_{z}(t, ϕ) = P_{P}(t, ϕ) and δ(ϕ, z) = δ(ϕ).

Proof. For n ∈ IN, we define

P(z, t, ϕ, n) := ∑_{x∈f^{−n}(z)} |(f^{n})′(x)|^{−t} exp(S_{n}ϕ(x)).

The proof is an immediate adaptation of the first part of the proof of Theorem 3.3 in [11]. The reader is referred to this proof in [11], where one should insert the following changes. Everywhere in this proof replace Crit(f) by Crit(f) ∪ Ω. Also, in the notation of [11], choose the parameter θ of Lemma 2.4 such that θ = \frac{1}{2} min\{diam(B_{j}) : j = 1, 2, …, k\}. It then follows that dist(Ω, B_{1} ∪ B_{2} ∪ … ∪ B_{k}) > 0, and hence, by Koebe’s Distortion Theorem and Lemma 2.5, we have

\frac{P(z_{2}, t, ϕ, n)}{P(z_{1}, t, ϕ, n)} ≤ Δ̂^{αk}

for some suitable constant Δ̂ ≥ 1. With these modifications one can now follow word by word the proof of Theorem 3.3 in [11].

For the following lemma, recall that for t ≥ 0 and s ∈ IR, a Borel probability measure m_{t,ϕ} supported on J(f) is by definition a (e^{s}, t, ϕ)-conformal Gibbs state if and only if f is non-singular with respect to m_{t,ϕ} and

\frac{d(m_{t,ϕ} ∘ f)}{dm_{t,ϕ}} = e^{s}|f′|^{t} exp(−ϕ).

Note that in section 3 we will discuss this type of measure in greater detail.

Lemma 2.8. If there exists a (e^{s}, t, ϕ)-conformal Gibbs state m, then P_{z}(t, ϕ) ≤ s and δ(ϕ, s, z) ≤ t, for m-almost every z ∈ J(f).

Proof. In section 3 (Lemma 3.1) we will show that m (⋃_{k∈IN} f^{k}(Crit(f))) = 0. This implies that for every n ∈ IN the Julia set J(f) is contained in the union of the closures of open disks U_{1}, U_{2}, …, U_{q} (for some finite q ∈ IN) such that the U_{i} are pairwise disjoint and such that the boundary of every U_{i} is contained in ⋃_{k=1}^{n} f^{k}(Crit(f)). Let f_{i,j}^{−n} : U_{i} → C I refer to the holomorphic inverse branches of f^{n} defined on U_{i}, for i = 1, 2, …, q and j = 1, 2, …, deg^{n}(f). We then have

1 ≥ ∑_{i=1}^{q} ∑_{j=1}^{deg^{n}(f)} m(f_{i,j}^{−n}(U_{i})) = ∑_{i=1}^{q} ∑_{j=1}^{deg^{n}(f)} ∫_{U_{i}} |(f_{i,j}^{−n})′|^{t} exp(S_{n}ϕ ∘ f_{i,j}^{−n} − s)dm

= e^{−sn} ∑_{i=1}^{q} ∫_{U_{i}} P(z, t, ϕ, n)dm = e^{−sn} ∫_{J(f)} P(z, t, ϕ, n)dm.

Therefore ∫_{J(f)} P(z, t, ϕ, n)dm ≤ e^{sn}, and for arbitrary ϵ > 0, we have that

m({z : P(z, t, ϕ, n) ≥ e^{s+ϵn}}) ≤ e^{−ϵn}.

Applying the Borel-Cantelli Lemma, it now follows that for m-a.e. z ∈ J(f) we have P_{z}(t, ϕ) ≤ s + ϵ. Since ϵ was arbitrary, this implies that for m-a.e. z ∈ J(f) we have P_{z}(t, ϕ) ≤ s as well as δ(ϕ, s, z) ≤ t.

The following gives a list of other possible generalizations of suggestions of Przytycki in [11] of how to ammend the notion of topological pressure in situations in which Crit(f) plays a crucial role. (P2) Variational pressure.

P_{V}(t, ϕ) := sup{h_{μ}(f) + ∫ (−t log |f′| + ϕ)dμ},

where the supremum is taken with respect to all ergodic f-invariant measures supported on J(f).

(P3) Hyperbolic variational pressure.

P_{HV}(t, ϕ) := sup{h_{μ}(f) + ∫ (−t log |f′| + ϕ)dμ},

where the supremum is taken with respect to all ergodic f-invariant measures supported on J(f) such that the Lyapunov exponent is positive, i.e. such that ∫ log |f′|dμ > 0.

(P4) Hyperbolic pressure.

P_{H}(t, ϕ) := sup{P_{top}(f|_{X}, −t ∫ log |f′| + ϕ)},

where the supremum is taken with respect to all f-invariant hyperbolic subsets X of J(f) such that some iterate of f|_{X} is topologically conjugate to a subshift of finite type. (Recall that a forward invariant compact set X ⊂ J(f) is called hyperbolic if there exists n ∈ IN such that |(f^{n})′(x)| > 1, for each x ∈ X).

(P5) DU-pressure.

P_{DU}(t, ϕ) := sup{P_{top}(f|_{K(V)}, −t ∫ log |f′| + ϕ)},

where the supremum is taken with respect to all open subsets V of J(f) for which J(f) ∩ Crit(f) ⊂ V, and where we have set K(V) := J(f) \ ⋃_{n≥0} f^{−n}(V). (Note that K(V) is compact, f-invariant and disjoint from Crit(f)).

(P6) Conformal pressure.

P_{C}(t, ϕ) := log λ(t, ϕ),

where λ(t, ϕ) is defined as the infimum of the set of all positive λ for which there exists a Borel probability measure m such that d(m ∘ f)/dm = λ|f′|^{t}e^{−ϕ}.

The following theorem gives the main result of this section. We show that for a parabolically semihyperbolic GPL-map f all notions of pressure introduced in (P1) up to (P6) coincide. For the remainder of this paper we shall then refer to the common value established in this theorem as to the topological pressure P(t, ϕ) of the potential −t log |f′| + ϕ.

Theorem 2.9. Let f be a parabolically semihyperbolic GPL-map, and ϕ : X → IR a Hölder continuous potential with Hölder exponent α exceeding p_{max}/(p_{max} + 1). We then have, for every t ∈ [0, δ(ϕ)),

P_{P}(t, ϕ) = P_{V}(t, ϕ) = P_{HV}(t, ϕ) = P_{H}(t, ϕ) = P_{DU}(t, ϕ) = P_{C}(t, ϕ).

Proof. Without loss of generality, we can assume that δ(ϕ) > 0. Clearly, we have that P_{P}(t, ϕ) ≥ P_{H}(t, ϕ). Also, we have P_{H}(t, ϕ) ≥ P_{HV}(t, ϕ) (c.f. [13]), as well as P_{HV}(t, ϕ) ≥ P_{H}(t, ϕ). The latter inequality is an immediate consequence of the variational principle. Summarizing, we now have

P_{P}(t, ϕ) ≥ P_{H}(t, ϕ) = P_{HV}(t, ϕ).  (2.4)

Next we show that P_{DU}(t, ϕ) > \overline{ϕ} implies

P_{HV}(t, ϕ) ≥ P_{DU}(t, ϕ).  (2.5)

For this let ϵ > 0 be chosen sufficiently small such that P_{DU}(t, ϕ) − ϵ > \overline{ϕ}. The variational principle gives the existence of a f-invariant Borel probability measure μ supported on some set K(V) such that h_{μ}(f)−tχ_{μ} + ∫ ϕdμ ≥ P_{DU}(t, ϕ)−ϵ > \overline{ϕ} (where χ_{μ} refers to the Lyapunov exponent). Hence, we are left to show that χ_{μ} > 0. In order to see this, we use (2.3) which gives

h_{μ}(f) − tχ_{μ} > \overline{ϕ} − ∫ ϕdμ ≥ ϕ̂ − ∫ ϕdμ ≥ 0.

Hence, we have h_{μ}(f) > tχ_{μ} ≥ 0, and therefore we can apply Ruelle’s inequality (that is h_{μ}(f) ≤ 2 max{0, χ_{μ}}) to deduce χ_{μ} > 0.

Next we show that P_{V} (t, ϕ) > \overline{ϕ} implies

P_{HV}(t, ϕ) = P_{V}(t, ϕ).  (2.6)

Clearly, we have P_{HV}(t, ϕ) ≤ P_{V}(t, ϕ). Similar as above, let ϵ > 0 be chosen sufficiently small such that P_{V} (t ϕ) − ϵ > \overline{ϕ}. It follows that there exists a f-invariant Borel probability measure μ such that h_{μ}(f) − tχ_{μ} + ∫ ϕdμ ≥ P_{V}(t, ϕ) − ϵ > \overline{ϕ}. Hence, we are left to show that χ_{μ} > 0, which follows in exactly the same way as in the previous step.

Next we show that for 0 ≤ t < δ(ϕ) we have

P_{DU}(t, ϕ) > \overline{ϕ} and P_{DU}(t, ϕ) ≥ log λ(t, ϕ).  (2.7)

For this we remark that, by a result in [12], for each c ∈ Crit(f) there exists x_{c} ∈ ω(c) such that

lim sup_{n→∞} |(f^{n})′(x_{c})| > 0.  (2.8)

For every n ∈ IN, we define

V_{n} := ⋃_{c∈J(f)∩Crit(f)} B(x_{c}, 1/n).

We shall now prove formula (2.7) in two step. First, we show that P_{DU}(t, ϕ) > \overline{ϕ} implies P_{DU}(t, ϕ) ≥ log λ(t, ϕ), and secondly, using the construction of the first step, we show how to deduce P_{DU}(t, ϕ) > \overline{ϕ}. This will then complete the proof of the theorem.

Step 1. Assume that P_{DU}(t, ϕ) > \overline{ϕ}, and let k ∈ IN be fixed. Let E be defined as in the proof of Lemma 5.1 in [3], and consider the sets E_{n} := f|_{K(V_{k})}^{−n}. We then let

c_{k}(t) := lim sup_{n→∞} \frac{1}{n} log ∑_{x∈E_{n}} |(f^{n})′(x)|^{−t} exp(S_{n}ϕ(x)).

Since the map f|_{K(V_{k})} : K(V_{k}) → K(V_{k}) has no critical points, it follows that K(V_{k}) is an (n, ρ)-separated set, for

ρ := inf_{y∈K(V_{k})} {min{|z − x| : x, z ∈ (f|_{K(V_{k})})^{−1} (y) and x ≠ z}} > 0.

Hence, we have that

c_{k}(t) ≤ P_{top} (f|_{K(V_{k})}, −t log |f′| + ϕ) .  (2.9)

We remark the set E can be chosen such that (c.f. the proof of Lemma 5.1 in [3])

c_{k}(t) ≥ P_{top} (f|_{K(V_{k})}, −t log |f′| + ϕ) − \frac{1}{k}.  (2.10)

Next, recall from [3] that a Borel set A ⊂ C I is called special if f|_{A} is injective. The following lemma has been obtained in [3] (c.f. Lemma 3.1, Lemma 3.2 and the proof of Lemma 5.3; c.f. also [4]).

Lemma 2.10. For every t ≥ 0, there exists a Borel probability measure m_{k} suported on K(V_{k}) such that

(a) m_{k}(f(A)) ≥ ∫_{A} e^{c_{k}(t)}|f′|^{t}e^{−ϕ}dm_{k} for every special set A ⊂ J(f) and

(b) m_{k}(f(A)) = ∫_{A} e^{c_{k}(t)}|f′|^{t}e^{−ϕ}dm_{k} for every special set A ⊂ J(f) \ \overline{V_{k}}.

Now first observe that by combining (2.9) and (2.10), we have that

lim_{k→∞} c_{k}(t) = P_{DU}(t, ϕ).  (2.11)

Hence, with m referring to some weak limit of the sequence of measures m_{k} of the previous lemma, we have

m(f(A)) ≥ ∫_{A} e^{P_{DU}(t,ϕ)}|f′|^{t}e^{−ϕ}dm

for each special set A ⊂ J(f), and also

m(f(A)) = ∫_{A} e^{P_{DU}(t,ϕ)}|f′|^{t}e^{−ϕ}dm  (2.12)

for every special set A ⊂ J(f) \ {x_{c} : c ∈ J(f) ∩ Crit(f)}. Now note that our assumption P_{DU}(t, ϕ) > \overline{ϕ} implies that there exists κ > 0 and q ∈ IN such that P_{DU}(t, ϕ) > κ + \frac{1}{q} sup S_{q}(ϕ). Fix c ∈ J(f) ∩ Crit(f). By (2.8) we have that lim sup_{n→∞} |(f^{n})′(x_{c})| > 0. Now, if we would have that m(x_{c}) > 0, then it would follow that

lim sup_{n→∞} m(f^{n}(x_{c})) ≥ lim sup_{n→∞} m(f^{qn}(x_{c}))

≥ m(x_{c} lim sup_{n→∞} exp(qnP_{DU}(t, ϕ))|(f^{qn})′(x_{c})| exp(− sup(S_{qn}ϕ))

≥ m(x_{c}) lim sup_{n→∞} exp(qnP_{DU}(t, ϕ) − n sup(S_{q}ϕ)))|(f^{qn})′(x_{c})|

≥ m(x_{c}) lim sup_{n→∞} e^{κqn}|(f^{qn})′(x_{c}) = ∞,

which is contradiction. Hence, we have m(f^{j}(x_{c})) = 0, for every j ≥ 0, and therefore (2.12) holds for every special set A ⊂ J(f). This clearly gives that P_{DU}(t, ϕ) ≥ log λ(t, ϕ).

Step 2. We now assume that P_{DU}(t, ϕ) ≤ \overline{ϕ}. By [13] we have that P_{DU}(0, ϕ) ≥ P_{H}(0, ϕ) = P_{top}(ϕ), and consequently P_{DU}(0, ϕ) = P_{top}(ϕ). Since P_{top}(ϕ) > \overline{ϕ}, it follows that there exists u ∈ IN such that P_{top} (f|_{K(V_{u})}, ϕ) > \overline{ϕ}. Let ϵ > 0 be fixed such that P_{top} (f|_{K(V_{u})}, ϕ) > \overline{ϕ} + ϵ. Then there exist two sequences \{t_{n}\}_{n=1}^{∞} and \{k_{n}\}_{n=1}^{∞} such that t_{n} ≤ t for all n and lim_{n→∞} t_{n} = s ≤ t, and

\overline{ϕ} + ϵ < P_{top} (f|_{K(V_{kn})}, −t_{n} log |f′| + ϕ) ≤ ϕ + ϵ + \frac{1}{n}.

By replacing c_{k}(t) by c_{k_{n}}(t_{n}), and noting that similarly as in Step 1 we have that lim_{n→∞} c_{k_{n}}(t_{n}) = \overline{ϕ} + ϵ. Hence we can repeat the construction in Step 1, and in this way we obtain a Borel probability measure m on J(f) for which

m(f(A)) ≥ ∫_{A} e^{\overline{ϕ}+ϵ−ϕ}|f′|^{s}dm

for every special set A ⊂ J(f) and

m(f(A)) = ∫_{A} e^{\overline{ϕ}+ϵ−ϕ}|f′|^{s}dm  (2.13)

for every special set A ⊂ J(f) \ \{x_{c} : c ∈ J(f) ∩ Crit(f)\}. Since n(\overline{ϕ} + ϵ) − S_{n}ϕ > 0, for some n ∈ IN, we obtain analogously as in the previous step that (2.13) holds for every special set A ⊂ J(f). This means that a (e^{\overline{ϕ}+ϵ}, t, ϕ)-conformal measure exists. Therefore, if t ≥ 0 and α_{n} is such that lim α_{n} = α, then every accumulation point of a sequence of (α_{n}, t, ϕ)-conformal measures is necessarily a (α, t, ϕ)-conformal measure. This implies that there must exist a (e\overline{ϕ}, t, ϕ)-conformal measure on J(f). By lemma 2.8 and Lemma 2.7, it therefore follows that δ(ϕ) ≤ t < δ(ϕ), which is a contradiction and hence finishes the proof of (2.7).

For the remainder, observe that if t < δ(f) then by lemma 2.8

P_{P}(t, ϕ) ≤ log λ(t, ϕ).  (2.14)

Combining (2.4), (2.5), (2.7) and (2.14), we obtain that if t < δ(f) then

P_{P}(t, ϕ) ≥ P_{H}(t, ϕ) = P_{HV} (t, ϕ) ≥ P_{DU}(t, ϕ) ≥ log λ(t, ϕ) ≥ P_{P}(t, ϕ).  (2.15)

In here, the second inequality uses the fact that P_{DU}(t, ϕ) > \overline{ϕ} (which follows, since by the first part of (2.7), we have that t < δ(ϕ) implies that P_{DU}(t, ϕ) > \overline{ϕ}). Clearly, all the inequality signs in (2.15) are now in fact equality signs. Hence the proof follows from (2.15) and (2.7), noting that if t < δ(f) then P_{V} (t, ϕ) ≥ P_{DU}(t, ϕ) and P_{DU}(t, ϕ) > \overline{ϕ}.

3. Invariant Gibbs states

In this section we give a detailed discussion of various types of conformal measures. In particular, this includes the conformal measure m_{t,ϕ} which we already introduced in the previous section. Note that the analysis in this section extends the results obtained in [15] (section 4).

Lemma 3.1. For a parabolically semihyperbolic GPL-map f we have, for each t ∈ [0, δ(ϕ)),

m_{t,ϕ} (\overline{⋃_{n≥1} f^{n}(Crit(f))} ∪ Ω) = 0.

Proof. Put \overline{PC(f)} := \overline{⋃_{n∈IN} f^{n}(Crit(f))}. Combining (b) and (c) in Definition 2.1 and the fact that the sets U_{i} have pairwise disjoint closures, we obtain that there exists δ > 0 such that if z ∈ \overline{PC(f)}, then for every n ≥ 0 there exists a well-defined holomorphic inverse branch f_{z}^{−n}(B(f^{n}(z), 16Kδ) → U_{r} of f^{n} sending f^{n}(z) to z. Choose a sequence {n_{j}}_{j=1}^{∞} such that lim_{j→∞} f^{n_{j}}(z) = y for some y ∈ \overline{PC(f)}. By passing to a subsequence, if necessary, we may assume that |f^{n_{j}}(z) − y| < δ, for all j ∈ IN. By Lemma 2.2, we have that \overline{PC(f)} is a compact nowhere dense subset of J(f), which gives

m_{t,ϕ}(B(f^{n_{j}}(z), 2δ) \ \overline{PC(f)}) ≥ m_{t,ϕ}(B(y, δ) \ \overline{PC(f)}) > 0.

Using Koebe’s Distortion Theorem and the forward invariance of the set \overline{PC(f)}, it now follows that

B(z, K2δ|(f^{n_{j}})′(z)|^{−1}) \ \overline{PC(f)} ⊃ f_{z}^{−n_{j}} (B(f^{n_{j}}(z), 2δ) \ \overline{PC(f)}).

Applying Koebe’s Distortion Theorem once more, along with Lemma 2.5, we obtain

\frac{m_{t,ϕ}(f_{z}^{−n_{j}} (B(f^{n_{j}}(z), 2δ) \ \overline{PC(f)}))}{m_{t,ϕ}(f_{z}^{−n_{j}} (B(f^{n_{j}}(z), 8Kδ))}

≥ \frac{C_{θ}^{−1} exp(S_{n_{j}}ϕ(z))K^{−t}|(f^{n_{j}})′(z)|^{−t}e^{−P(t,ϕ)n_{j}}m_{t,ϕ}(B(f^{n_{j}}(z), 2δ) \ \overline{PC(f)}))}{C_{θ} exp(S_{n_{j}}ϕ(z))K^{t}|(f^{n_{j}})′(z)|^{−t}e^{−P(t,ϕ)n_{j}}m_{t,ϕ}(B(f^{n_{j}}(z), 8Kδ))}

= C_{θ}^{−2}K^{−2t} \frac{m_{t,ϕ}(B(f^{n_{j}}(z), 2δ) \ \overline{PC(f)}))}{m_{t,ϕ}(B(f^{n_{j}}(z), 8Kδ))}

≥ C_{θ}^{−2}K^{−2t} \frac{m_{t,ϕ}(B(y, δ) \ \overline{PC(f)})}{m_{t,ϕ}(B(y, (8K + 1)δ))} > 0.

By the \frac{1}{4}-Koebe’s Distortion Theorem, we have f_{z}^{−n_{j}} (B(f^{n_{j}}(z), 8Kδ) ⊃ B(z, K2δ|(f^{n_{j}})′(z)|^{−1}). Hence, we have

\frac{m_{t,ϕ}(B(z, K2δ|(f^{n_{j}})′(z)|^{−1}) \ \overline{PC(f)})}{m_{t,ϕ}(B(z, K2δ|(f^{n_{j}})′(z)|^{−1}))}

≥ \frac{m_{t,ϕ}(f_{z}^{−n_{j}} (B(f^{n_{j}}(z), 2δ) \ \overline{PC(f)}))}{m_{t,ϕ}(f_{z}^{−n_{j}} (B(f^{n_{j}}(z), 8Kδ))} · \frac{m_{t,ϕ}(f_{z}^{−n_{j}} (B(f^{n_{j}}(z), 8Kδ))}{m_{t,ϕ}(B(z, K2δ|(f^{n_{j}})′(z)|^{−1}))}

≥ C_{θ}^{−2}K^{−2t} \frac{m_{t,ϕ}(B(y, δ) \ \overline{PC(f)})}{m_{t,ϕ}(B(y, (8K + 1)δ))} · \frac{m_{t,ϕ}(B(z, K2δ|(f^{n_{j}})′(z)|^{−1}))}{m_{t,ϕ}(B(z, K2δ|(f^{n_{j}})′(z)|^{−1}))}

= \frac{m_{t,ϕ}(B(y, δ) \ \overline{PC(f)})}{m_{t,ϕ}(B(y, (8K + 1)δ))} > 0.

Therefore the Lebesgue’s Density Theorem gives that m_{t,ϕ}(\overline{PC(f)}) = 0. Finally, let ω ∈ Ω be arbitrary. We then have that m_{t,ϕ}(ω) = m_{t,ϕ}(f^{n}(ω)) = exp(nP(t, ϕ) − S_{n}ϕ(ω)), for each n ∈ IN, and since lim sup_{n→∞}(nP(t, ϕ) − S_{n}ϕ(ω)) = ∞, it follows that m_{t,ϕ}(ω) = 0.

For the next lemma we remark that by a standard normal family argument we have that there exist u ∈ IN and κ^{∗} > 1 such that |(f^{u})′(z)| > κ^{∗} for all z ∈ ω(Crit(J(f))). Therefore, there exist κ > 1 such that, for all j ≥ 0, n ∈ IN and every c ∈ Crit(f),

|(f^{j})′(f^{n}(c))| ≫ κ^{j}.  (3.1)

Lemma 3.2. For each δ > 0 sufficiently small, and for all s ∈ IN and c ∈ Crit(f)), we have, where q(c) ≥ 2 refers to the order of the critical point c,

m_{t,ϕ}(B(c, (δ|(f^{s})′(f(c))|^{−1})^{1/q(c)})) ≪ |(f^{s})′(f(c))|^{−t/q(c)}e^{−P(t,ϕ)s} exp(S_{s}ϕ(c)).

Therefore, we in particular have that m_{t,ϕ}(c) = 0.

Proof. Let {m_{k}}_{k=1}^{∞} and {c_{k}(t)}_{k=1}^{∞} respectively be the sequence of measures and numbers obtained in Lemma 2.10. Fix δ ∈ (0, dist(U_{r}, ∂U)/2) and c ∈ Crit(f)). Also, define λ_{n}(c) := |(f^{n})′(f(c))|, for each n ∈ IN, as well as the annulus A(w, r, R) := {z ∈ C I : r ≤ |z − w| < R} with centre at w ∈ C I, inner radius r and outer radius R. By Koebe’s distortion theorem and by Lemma 2.5, it follows that, for all l, n ∈ IN,

m_{l}(B(f(c), δλ_{n}(c)^{−1})) ≍ λ_{n}(c)^{−t}e^{−c_{k}(t)n} exp(S_{n}ϕ(f(c))).

Using this observation and the fact that |(f_{c}^{−1})′(z)| ≍ |z − f(c)|^{−(1−1/q(c))}, for z ≠ f(c) such that z is close to f(c) (here, f_{c}^{−1} refers to an inverse branch of f which maps z close to c and which is defined in some neighbourhood of z), it follows, for each s ∈ IN,

m_{l}(B(c, (δλ_{s}(c)^{−1})^{1/q(c)})) = ∑_{j=1}^{∞} m_{l}(A(c, (δλ_{s(j+1)}(c)^{−1})^{1/q(c)}, (δλ_{sj}(c)^{−1})^{1/q(c)}))

≍ ∑_{j=1}^{∞} m_{l}(f_{c}^{−1}(A(f(c), δλ_{s(j+1)}(c)^{−1}, δλ_{sj}(c)^{−1})))

≍ ∑_{j=1}^{∞} λ_{sj}(c)^{(1− \frac{1}{q(c)})t}e^{−c_{l}(t)}e^{ϕ(c)}m_{l}(A(f(c), δλ_{s(j+1)}(c)^{−1}, δλ_{sj}(c)^{−1}))

≤ e^{−c_{l}(t)} ∑_{j=1}^{∞} λ_{sj}(c)^{(1− \frac{1}{q(c)})t}e^{ϕ(c)}m_{l}(B(f(c), δλ_{sj}(c)^{−1}))

≍ ∑_{j=1}^{∞} λ_{sj}(c)^{(1− \frac{1}{q(c)})t}λ_{sj}(c)^{−t}e^{ϕ(c)} exp(S_{sj}ϕ(f(c)))e^{−c_{l}(t)sj}

= ∑_{j=1}^{∞} λ_{sj}(c)^{−t/q(c)}e^{−c_{l}(t)sj} exp(S_{sj}ϕ(c))

= λ_{s}(c)^{−t/q(c)}e^{−c_{l}(t)s} exp(S_{s}ϕ(c))

· (1 + ∑_{j=2}^{∞} (\frac{λ_{sj}(c)}{λ_{s}(c)})^{−t/q(c)} exp(S_{(j−1)s}ϕ(f^{s}(c)) − c_{l}(t)s(j − 1))) .

Now, we have that

(\frac{λ_{sj}(c)}{λ_{s}(c)}) = |(f^{s(j−1)})′(f^{s}(c))|

and by (3.1), these numbers are uniformly bounded away from zero. Therefore, we have (λ_{sj}(c)/λ_{s}(c))^{−t/q(c)} ≪ 1, for all s, j ∈ IN. Since lim_{l→0} c_{l}(t) = P(t, ϕ), it follows that c_{l}(t) ≥ \overline{ϕ} + ϵ, for some ϵ > 0 and for all l ∈ IN large enough. Consequently, using lemma 2.6, we deduce, for all l ∈ IN sufficiently large,

m_{l}(B(c, (δλ_{sk}(c)^{−1})^{1/q(c)})) ≪ λ_{s}(c)^{−t/q(c)}e^{−P(t,ϕ)s} exp(S_{s}ϕ(c)),

and hence,

m_{t,ϕ}(B(c, (δλ_{sk}(c)^{−1})^{1/q(c)})) ≪ λ_{s}(c)^{−t/q(c)}e^{−P(t,ϕ)s} exp(S_{s}ϕ(c)).

Lemma 3.3. For t ∈ [0, δ(ϕ)), the measure m_{t,ϕ} has no atoms.

Proof. Suppose that m_{t,ϕ}(z) > 0, for some z ∈ J(f). Using Lemma 3.1 and Lemma 3.2, it then follows that z ∉ ⋃_{n≥0} f^{−n}(Ω ∪ Crit(f)). We shall prove that there exists δ > 0 and a sequence {n_{j}}_{j=1}^{∞} depending on z such that there are well-defined holomorphic inverse branches f_{z}^{−n_{j}} : B(f^{n_{j}}(z), 4δ) → U of f^{n_{j}} which map f^{n_{j}}(z) to z. Clearly, such branches exist if f^{n}(z) ∈ U_{o} ∪ U_{p}, for infinitely many n. For the remaining cases note that there then exists q ≥ 0 such that f^{n}(z) ∈ U_{r}, for all n ≥ q. Therefore, there exits δ > 0 such that the holomorphic inverse branches f_{q}^{−(n−q)} : B(f^{n}(z), 2δ) → U of f^{n−q}, which map f^{n}(z) to f^{q}(z), are well defined, for all n ≥ q. Since z ∉ ⋃_{j≥0} Crit(f^{j}), there exists γ > 0 such that the holomorphic inverse branch f_{z}^{−q} : B(f^{q}(z), γ) → U of f^{q}, which maps f^{q}(z) to z, is well defined. Since lim_{n→∞} diam(f_{q}^{−(n−q)}(B(f^{n}(z), 2δ)) = 0 (as z ∈ J(f)), the compositions f_{z}^{−q} ∘ f_{q}^{−(n−q)} : B(f^{n}(z), 2δ) → U are well defined, for all n ≥ q. This shows that in any case we have the claimed existence of inverse branches. Let us emphasize that we just saw that lim_{j→∞} diam(f_{z}^{−n_{j}}(B(f^{n_{j}}(z), 2δ)) = 0, which immediately implies that lim_{j→∞} |(f^{n_{j}})′(z)| = ∞. Since P(t, ϕ) > \overline{ϕ}, it follows that lim_{j→∞} = ∞,

?????????????????????????????????

which is a contradiction and hence finishes the proof.

For f-invariant Gibbs states we now observe the following.

Theorem 3.4. Let m_{t,ϕ} be an (exp(P(t, ϕ), t, ϕ)-conformal Gibbs state for a non-recurrent GPL-map f such that m_{t,ϕ} (\overline{⋃_{n≥1} f^{n}(Crit(f))} ∪ Ω) = 0. Then up to a multiplicative constant there exists a unique f-invariant, σ-finite measure μ_{t,ϕ} (called an invariant Gibbs state of the potential −t log |f′| + ϕ) which is conservative and ergodic, and which is equivalent to m_{t,ϕ}.

The idea of the proof of Theorem 3.4 is to apply a general method of [8], which gives a sufficient condition from which the existence of σ-finite absolutely continuous invariant measure can be deduced. We now recall this result, and we shall also give a brief outline of how this result is obtained. Let X be a σ-compact metric space, m a Borel probability measure on X which is positive on open sets, and let T : X → X be a measurable map such that m is quasi-invariant with respect to T (that is, the measure m ∘ T^{−1} is absolutely continuous with respect to m). Moreover, let A = {A_{n} : n ≥ 0} be a countable partition consisting of σ-compact subsets of X of positive m-measure, such that m(X \ ⋃_{n≥0} A_{n}) = 0. Recall that in this situation A is called irreducible, if we have that for all m, n ∈ IN there exists k ≥ 0 such that

m(T^{−k}(A_{m}) ∩ A_{n}) > 0.  (3.2)

The following gives the result of Martens (c.f. Proposition 2.6 and Theorem 2.9 of [8]).

Theorem 3.5. Let X, T, m be as above. Suppose that T is conservative and ergodic with respect to m, and let A = {A_{n} : n ≥ 0} be an irreducible partition. If we have that for every n ∈ IN there exists K_{n} ≥ 1 such that, for all k ≥ 0 and all Borel sets A ⊂ A_{n},

K_{n}^{−1} \frac{m(A)}{m(A_{n})} ≤ \frac{m(T^{−k}(A))}{m(T^{−k}(A_{n}))} ≤ K_{n} \frac{m(A)}{m(A_{n})},  (3.3)

then there exists a σ-finite T -ivariant measure μ which is equivalent to m. Moreover, μ is conservative and ergodic, as well as unique up to a multiplicative constant.

Since in our application of this result we will not only require the statement of Theorem 3.5 but also the method with which the invariant measure in there is derived, we now give the sketch of the proof of this result of Martens.

Proof of Theorem 3.5 (sketch). Following Martens, one considers the following sequences of measures

S_{k}m := ∑_{i=0}^{k−1} m ∘ T^{−i} and Q_{k}m := \frac{S_{k}m}{S_{k}m(A_{0})}.

It is shown in [8] that each weak limit μ of the sequence Q_{k}(m) fulfills the preliminaries of Theorem 3.5 (where a sequence {ν_{k} : k ∈ IN} of measures on X is said to converge weakly if the measures ν_{k} converge weakly on A_{n}, for all n ∈ IN). Moreover, it is shown in [8] that the sequence Q_{k}m converges and that we have, for every Borel set F ⊂ X,

μ(F) = lim_{n→∞} Q_{k}m(F).

Clearly, we have that μ(A) ≤ 1 < ∞. Using (3.2) and (3.3), one then obtains the following two lemmata (c.f. Lemma 2.4 in [8]).

Lemma 3.6. For each n ≥ 0 we have that 0 < μ(A_{n}) < ∞ and that the Radon-Nikodym derivative \frac{dμ}{dm} is bounded on A_{n}.

Lemma 3.7. For all i, j ≥ 0 there exists a constant κ > 0 such that, for all n ∈ IN and for all Borel sets D ⊂ A_{i} and E ⊂ A_{j},

\frac{S_{n}m(D)}{S_{n}m(E)} ≤ κ\frac{m(D)}{m(E)}.

We now return to the situation of a generalized polynomial-like map f. For the proof of the ergodicity and conservativity of the measure m_{t,ϕ} we refer to [17] (Theorem 4.1). Therefore, in order to be able to apply Theorem 3.5, we only need to construct an irreducible partition A which has the property (3.3). For this, let Y := J(f) \ (\overline{⋃_{n∈IN} f^{n}(Crit(f))} ∪ Ω), and consider, for each y ∈ Y, a ball B(y, r(y)) such that m(∂B(y, r(y))) = 0 and 0 < r(y) < (1/2)dist(y, \overline{⋃_{n∈IN} f^{n}(Crit(f))} ∪ Ω). Clearly, by associating to each y ∈ Y a fixed ball of this type, this gives a cover of Y. Since Y is a separable metric space, one can reduce this cover to a countable, locally finite cover of Y, denoted by {Ã_{n} : n ≥ 0} (here, locally finite means that each point x ∈ Y has an open neighborhood intersecting at most finitely many elements of the cover). The partition A = {A_{n} : n ≥ 0} is then defined by induction as follows.

A_{0} := Ã_{0} and for n ∈ IN, let A_{n} := Ã_{n} \ ⋃_{k=0}^{n−1} \overline{Ã_{k}}.

Clearly, by construction we have that the elements of A are pairwise disjoint, and

⋃_{n∈IN} A_{n} ⊃ J(f) \ (\overline{⋃_{n∈IN} f^{n}(Crit(f))} ∪ Ω) \ ⋃_{n≥0} ∂Ã_{n}.

Using the assumption of Theorem 3.4, it follows that m_{t,ϕ}(⋃_{n≥0} A_{n}) = 1. Now, the fact that (3.3) holds in the situation here is an immediate consequence of combining Koebe’s Distortion Theorem and the observation that by Lemma 2.5 we have exp(S_{n}ψ(y))/ exp(S_{n}ψ(x)) ≪ 1, for all n ∈ IN and all x, y ∈ f_{∗}^{−n}(A_{k}) (here A_{k} refers to some arbitrary element of the partition A, and f_{∗}^{−n} to some arbitrary holomorphic inverse branch of f^{n} defined on A_{k}). Finally, the fact that A is irreducible follows, since the A_{n} are open sets and the map f : J(f) → J(f) is topologically exact.

The aim now is to provide a sufficient condition which guarantees that the σ-finite measure μ_{t,ϕ} is in fact a finite measure. For the following recall that the T -invariant measure μ_{t,ϕ} (see Theorem 3.4) is called of finite condensation at x ∈ J(f) if and only if there exists an open neighborhood V of x such that μ_{t,ϕ}(V) < ∞. Otherwise μ_{t,ϕ} is said to be of infinite condensation at x. We shall now see that the points of infinite condensation of μ_{t,ϕ} are necessarily parabolic fixed points.

Theorem 3.8. Let f be a parabolically semi-hyperbolic GPL- map. For t ∈ [0, δ(ϕ)), we have that Ω(f) contains the set of points of infinite condensation of μ_{t,ϕ}.

Proof. Put m := m_{t,ϕ}. Since the conformal measure m is positive on non-empty open sets, it follows that inf{m(B(x, r)) : x ∈ J(f)} > 0, for every r > 0. Even more, there exists θ_{0}(r) ∈ (0, r) such that

M(r) = inf{m(B(x, r) \ B(x, θ_{0}(r)) : x ∈ J(f)} > 0.  (3.4)

Recall from the beginning of the proof of Lemma 3.1 that there exists

δ ∈ (0, dist(Ω(f), ⋃_{n≥0} f^{n}(Crit(f)))

such that for every c ∈ Crit(J(f)), k ∈ IN and n ≥ 0 we have that the holomorphic inverse branch f_{f^{n}(c)}^{−k} : B(f^{n+k}(c), 4δ) → C I which maps f^{n+k}(c) to f^{n}(c) is well-defined. It follows from (3.1) that we have, for all u sufficiently large, c ∈ Crit(J(f)), k ≥ 0 and 0 ≤ i ≤ u − 1,

f_{f^{i+ku}}^{−u}(B(f^{i+(k+1)u}(c), 2δ)) ⊂ B(f^{i+ku}(c), θ_{0}(δ)).  (3.5)

We define, for c ∈ Crit(J(f)), 0 ≤ j ≤ u − 1 and i ≥ 0,

\begin{aligned}R_{i,j}(c) & := f_{f^{i}(c)}^{−ju}(B(f^{i+ju}(c), δ)) \ f_{f^{i}(c)}^{−(j+1)u}(B(f^{i+(j+1)u}(c), δ)) \\ & = f_{f^{i}(c)}^{−ju}(B(f^{i+ju}(c), δ) \ f_{f^{i+ju}(c)}^{−u}(B(f^{(i+(j+1)u}(c), δ)). \end{aligned}  (3.6)

By (3.1) and Koebe’s distortion theorem, we have that |S_{ju}ϕ(x) − S_{ju}ϕ(y)| ≪ 1, for all x, y ∈ R_{i,j}(c). Thus, applying (3.4), (3.5) and once more Koebe’s distortion theorem, we conclude

m(R_{i,j}(c)) ≍
≍ e^{−P(t,ϕ)ju}|(f^{ju})′(f^{i}(c))|^{−t} exp(S_{ju}ϕ(f^{i}(c)))
· m(B(f^{iju}(c), 2δ) \ f_{f^{i+ju}(c)}^{−u}(B(f^{(i+(j+1)u}(c), 2δ))
≍ e^{−P(t,ϕ)ju}|(f^{ju})′(f^{i}(c))|^{−t} exp(S_{ju}ϕ(f^{i}(c))).

(3.7)

Now let x ∈ \overline{⋃_{n≥0} f^{n}(Crit(J(f)))} be fixed. Clearly, since f is parabolically semi-hyperbolic, the latter set is disjoint from Ω ∪ Crit(f). Since Crit(J(f)) ∩ ω(Crit(J(f))) = ∅, we deduce from [16] (Lemma 2.13) that there exists 0 < γ < δ/2 such that if n ∈ IN and y ∈ f^{−n}(x), then there exists at most one 0 ≤ k ≤ n − 1 such that f^{k}(C_{n}(y, B(x, 4γ))) ∩ Crit(f) ≠ ∅ consists of at exactly one point, which will be denoted by c. Without loss of generality we may assume that the element A_{0} of the partition A is contained in B(x, γ). If we now assume that C_{n}(y, B(x, 2γ)) ∩ Crit(f^{n}) = ∅, then Koebe’s distortion theorem and Lemma 2.5 gives that

\frac{m(C_{n}(y, B(x, γ)))}{m_{t}(C_{n}(y, B(x, γ)) ∩ f^{−n}(A_{0}))} ≪ \frac{m_{t}(B(x, γ))}{m(A_{0})} ≍ 1.  (3.8)

On the other hand, if C_{n}(y, B(x, 2γ)) ∩ Crit(f^{n}) ≠ ∅, then there exists 0 ≤ k ≤ n − 1 such that c ∈ f^{k}(C_{n}(y, B(x, 2γ))) and

(f^{k}(C_{n}(y, B(x, 4γ))) \ f^{k}(C_{n}(y, B(x, 2γ)))) ∩ Crit(f^{n−k}) = ∅.  (3.9)

We have that

|(f^{i})′(z)| ≍ |z − c|^{q(c)−1},  (3.10)

for all 0 ≤ i ≤ u − 1 and all z ∈ C_{i}(c, B(f^{i}(c), 2δ)) (note that q(c) is the order of c also for the function f^{i}). Let us write n − k = su + r, for s ≥ 0 and 0 ≤ r ≤ u − 1. Using (3.7), (3.10) and the fact

f^{k}(C_{n}(y, B(x, γ))) ⊂ C_{r+su}(c, B(f^{r+su}(c), δ)),

it follows that

m(f^{k}(C_{n}(y, B(x, γ))))

≪ ∑_{j≥s} |(f^{ju})′(f^{r}(c))|^{−t}e^{−P(t,ϕ)(r+ju)} exp(S_{r+ju}ϕ(c)) (|(f^{ju})′(f^{r}(c))|^{−1})^{(\frac{1}{q(c)} −1)t}

≍ ∑_{j≥s} e^{−P(t,ϕ)(r+ju)}|(f^{ju})′(f^{r}(c))|^{− \frac{t}{q(c)}} exp(S_{r+ju}ϕ(c)).

(3.11)

Since A_{0} ⊂ B(x, γ) ⊂ B(f^{r+su}(c), δ), using Koebe’s distortion theorem and Lemma 2.5, we obtain

m(f^{k}(C_{n}(y, B(x, γ))) ∩ f^{−(n−k)}(A_{0})) ≍
≍ m(A_{0})|(f^{su})′(f^{r}(c))|^{−t}e^{−P(t,ϕ)(r+su)} exp(S_{r+su}ϕ(c))(|(f^{su})′(f^{r}(c))|^{−1})^{(\frac{1}{q(c)} −1)t}
≍ e^{−P(t,ϕ)(r+su)}|(f^{su})′(f^{r}(c))|^{− \frac{t}{q(c)}} exp(S_{r+su}ϕ(c)).

(3.12)

Therefore, using (3.11), (3.12), (3.1) and Lemma 2.6, we conclude, with S(c) referring to some finite number which only depends on c,

\frac{m(f^{k}(C_{n}(y, B(x, γ))))}{m(f^{k}(C_{n}(y, B(x, γ))) ∩ f^{−(n−k)}(A_{0}))}  (3.13)

≪ ∑_{j≥0} e^{−P(t,ϕ)(r+ju)}|(f^{ju})′(f^{r+su}(c))|^{− \frac{t}{q(c)}} exp(S_{ju}ϕ(f^{r+su}(c))) ≤ S(c)

By (3.9) we have that Mod(f^{k}(C_{n}(y, B(x, 4γ))) \ f^{k}(C_{n}(y, B(x, 2γ)))) ≥ (log 2)/q(c). Hence, applying Koebe’s distortion theorem and (3.13), we obtain

\frac{m(C_{n}(y, B(x, γ)))}{m(C_{n}(y, B(x, γ))) ∩ f^{−n}(A_{0}))}

≍ \frac{|(f^{k})′(y)|^{−t}e^{−P(t,ϕ)k} exp(S_{k}ϕ(y))}{|(f^{k})′(y)|^{−t}e^{−P(t,ϕ)k} exp(S_{k}ϕ(y))} · \frac{m(f^{k}(C_{n}(y, B(x, γ))))}{m(f^{k}(C_{n}(y, B(x, γ))) ∩ f^{−(n−k)}(A_{0}))}

≪ S(c).

Therefore, we have

\frac{m(f^{−n}(B(x, γ))}{m(f^{−n}(A_{0})}) ≪ max\{S(c) : c ∈ Crit(J(f))\},

which implies Q_{n}(B(x, γ)) ≪ max{S(c) : c ∈ Crit(J(f))}, for all n ∈ IN. It now follows that μ_{t,ϕ}(B(x, γ)) < ∞.

The main result in this section is the following.

Theorem 3.9. Let f be a parabolically semi-hyperbolic GPL-map. If t ∈ [0, δ(ϕ)), then the invariant Gibbs state μ_{t,ϕ} is finite. Furthermore, by normalizing μ_{t,ϕ} such it becomes a probability measure, we obtain an equilibrium state for the potential −t log |f′|+ϕ, in the sense that it maximizes the supremum appearing in the definition (P2) of variational pressure.

Proof. Since t ∈ [0, δ(ϕ)), we have that Theorem 3.8 is applicable. Hence, the invariant measure μ_{t,ϕ} exists and it is finite on compact subsets of J(f)\Ω. Let ω ∈ Ω be fixed. Without loss of generality we may assume that the element A_{0} of the partition A is a fundamental domain of some repelling sector with respect to the relation ‘∼’ (where we let x ∼ y, for x and y in this sector such that x and y are sufficiently close to ω, if and only if f_{ω}^{−n}(y) = x or f_{ω}^{−n}(x) = y). Fix x ∈ A_{0} and put x_{k} = f_{ω}^{−k}(x), k ≥ 0. Let B_{j} = f_{ω}^{−j}(A_{0}), j ≥ 0. We have

m_{t,ϕ}(B_{j}) ≍ e^{−P(t,ϕ)j} exp(S_{j}ϕ(x_{j}))(j + 1)^{− \frac{p(ω)+1}{p(ω)} t}.  (3.14)

Since ω ∈ Ω\\overline{⋃_{n∈IN} f^{n}(Crit(f) ∩ J(f))}, in view of Lemma 3.8, for every y ∈ f^{−1}(ω)\\{ω\} there exists an open neighborhood U_{y} of y such that μ_{t,ϕ}(U_{y}) < ∞ and U_{y} ⊂ Ã_{j} for some j ≥ 0. Take B to be a ball in J(f) (either closed or open) centered at ω and with radius so small that f_{y}^{−1}(B) ⊂ U_{y} for all y ∈ f^{−1}(ω) \ {ω}, where f_{y}^{−1} : B → C I is the local holomorphic inverse branch of f sending ω to y. Without loss of generality we may assume that A_{0} ⊂ B. Now, fix y ∈ f^{−1}(ω) \ {ω} and z_{j} ∈ f^{−j}(y), for j ≥ 0. Let 2U_{y} be the ball centered at y of radius twice the radius of U_{y}. Using (c) in Definition 2.1, it follows that 2U_{y} ∩ ⋃_{n≥1} f^{n}(Crit(f)) = ∅, for U_{y} sufficiently small. Letting m := m_{t,ϕ}, by Lemma 2.5 and Koebe’s Distortion Theorem, we then have, for every Borel set A ⊂ U_{y},

m(f_{z_{j}}^{−j}(A)) ≍ e^{−P(t,ϕ)j}|(f^{j}), (z_{j})|^{−t} exp(S_{j}ϕ(z_{j}))m(A).

Hence it follows, for k ≥ 0,

m(f_{z_{j}}^{−j}(f_{y}^{−1}(B_{k}))) ≍ e^{−P(t,ϕ)j}|(f^{j}), (z_{j})|^{−t} exp(S_{j}ϕ(z_{j}))m(f_{y}^{−1}(B_{k}))

≍ e^{−P(t,ϕ)j}|(f^{j}), (z_{j})|^{−t} exp(S_{j}ϕ(z_{j}))m(B_{k})

≍ m(f_{z_{j}}^{−j}(B_{y}))\frac{m(B_{k})}{m(B_{y})} ≍ m(f_{z_{j}}^{−j}(B_{y}))m(B_{k}).

Summing over all z_{j} ∈ f^{−j}(y), we get

m(f^{−j}(f_{y}^{−1}(B_{k}))) ≍ m(f^{−j}(B_{y}))m(B_{k}).

Hence, for i ∈ IN fixed, we can sum up over all 0 ≤ j ≤ i − 1, which gives for the measure S_{i}, introduced in the proof of Theorem 3.5,

S_{i}m(f_{y}^{−1}(B_{k})) ≍ S_{i}m(B_{y})m(B_{k}).  (3.15)

Since we have, for arbitrary j ≥ 0 and n ∈ IN,

S_{n}m(B_{j}) = m(f_{ω}^{−(n−1)}(B_{j})) + ∑_{y∈f^{−1}(ω)\{ω}} ∑_{k=0}^{n−2} S_{n−(k+1}m(f_{y}^{−1}(f_{ω}^{−k}(B_{j}))),

we can apply (3.15), Lemma 3.7 and (3.14), which then gives, that for all j ≥ 0 and n ∈ IN,

Q_{n}m(B_{j}) = \frac{S_{n}m(B_{j})}{S_{n}m(A_{0})}

≍ \frac{m(f_{ω}^{−(n−1)}(B_{j})}{S_{n}m(A_{0})} + ∑_{y∈f^{−1}(ω)\\{ω\}} ∑_{k=0}^{n−2} S_{n−(k+1}m(U_{y})m(B_{j+k})

≪ \frac{1}{S_{n}m(A_{0})} + ∑_{y∈f^{−1}(ω)\\{ω\}} ∑_{k=0}^{n−2} \frac{S_{n−(k+1}m(U_{y})}{S_{n−(k+1}m(A_{0})} · \frac{S_{n−(k+1}m(A_{0})}{S_{n}m(A_{0})}m(B_{j+k})

≪ \frac{1}{S_{n}m(A_{0})} + ∑_{y∈f^{−1}(ω)\\{ω\}} ∑_{k=0}^{n−2} m(B_{j+k})

≪ \frac{1}{S_{n}m(A_{0})} + deg(f) ∑_{k=0}^{n−2} exp(S_{j+k}ϕ(x_{j+k}) − P(t, ϕ)(j + k))(j + k + 1)^{− \frac{p(ω)+1}{p(ω)} t}

≪ \frac{1}{S_{n}m(A_{0})} + deg(f) ∑_{k=0}^{n−2} exp(S_{j+k}ϕ(x_{j+k}) − P(t, ϕ)(j + k)).

(3.16)

Now let ϵ > 0 be fixed such that that P(t, ϕ) > \overline{ϕ} + 2ϵ. By Lemma 2.6, there exists q ∈ IN such that sup\{S_{q}ϕ\} ≤ (\overline{ϕ} + ϵ)q < qP(t, ϕ) − qϵ. For ease of exposition we assume that q = 1. We can then continue the estimate in (3.16) as follows.

Q_{n}m(B_{j}) ≪ \frac{1}{S_{n}m(A_{0})} + deg(f) ∑_{k=0}^{n−2} e^{−ϵ(j+k)} ≪ \frac{1}{S_{n}m(A_{0})} + e^{−ϵj}.

By letting n tend to infinity, we obtain that μ_{t,ϕ}(B_{j}) ≪ e^{−ϵj}. If we sum this up over all j ≥ 0, then it follows that μ_{t,ϕ} (⋃_{j≥0} B_{j}) ≪ ∑_{j=0}^{∞} e^{−ϵj} < ∞. Finally, summing up over all repelling sectors of ω (note, there are only finitely many such sectors), we derive μ_{t,ϕ}(V_{ω}) < ∞, for every sufficiently small neighbourhood V_{ω} of ω. Therefore, ω has to be a point of finite condensation of μ_{t,ϕ}, and using Theorem 3.8, it follows that the f-invariant measure μ_{t,ϕ} is finite. It remains to show that μ_{t,ϕ} is an equilibrium state for the potential −t log |f′| + ϕ. Without loss of generality we may assume that μ_{t,ϕ} is a probability measure. Let μ := μ_{t,ϕ} and ρ := dμ/dm, and let J be the Jacobian given by

J := \frac{dμ ∘ f}{dμ} = \frac{ρ ∘ f}{ρ} exp(t log |f′| − ϕ + P(t, ϕ)).

Since μ(f(A)) ≥ μ(A) for any Borel set A ⊂ J(f), we always have that J ≥ 1. Also, since ∫ ρdm = 1 and ρ is non-negative, we see that ∫ ρdμ > 0. Hence, in view of Birkhoff’s Ergodic Theorem and Theorem 3.4, there exists z ∈ J(f) such that (note that log J ≥ 0, and that log |f′| is bounded from above)

ρ(f^{n}(z)) > \frac{1}{2} ∫ ρdm > 0 for infinitely many n ≥ 0,  (3.17)

lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1} log(J(f^{j}(z))) = ∫ log Jdμ,  (3.18)

lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1}(t log |f′| − ϕ) ∘ f^{j}(z) = ∫ (t log |f′| − ϕ)dμ.  (3.19)

Since ∫ log |f′|dμ ≥ 0 (c.f. [12]) and since ∫ log Jdμ = h_{μ}(f) ≤ h_{top}(f) < ∞, we have that log J and log |f′| are integrable. By (3.17) we have that lim sup_{n→∞} \frac{1}{n} log ρ(f^{n}(z)) ≥ 0. Combining this along with (3.18) and (3.19), we get

h_{μ}(f) + ∫ (−t log |f′| + ϕ)dμ ≥ ∫ log Jdμ + ∫ (−t log |f′| + ϕ)dμ =

= lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1} log(J(f^{j}(z))) + ∫ (−t log |f′| + ϕ)dμ

= lim_{n→∞} (\frac{1}{n} ∑_{j=0}^{n−1} (log (\frac{ρ ∘ f}{ρ} ) + t log |f′| − ϕ + P(t, ϕ)) (f^{j}(z))) + ∫ (−t log |f′| + ϕ)dμ

= P(t, ϕ) + lim_{n→∞} (\frac{1}{n}(log ρ ∘ f^{n}(z) − log ρ(z)) + \frac{1}{n} ∑_{j=0}^{n−1}(t log |f′| − ϕ + P(t, ϕ))(f^{j}(z)))

+ ∫ (−t log |f′| + ϕ)dμ

≥ P(t, ϕ) + lim sup_{n→∞} \frac{log ρ ∘ f^{n}(z)}{n} + lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1}(t log |f′| − ϕ)(f^{j}(z)) + ∫ (−t log |f′| + ϕ)dμ

≥ P(t, ϕ) + 0 + ∫ (t log |f′| − ϕ)dμ + ∫ (−t log |f′| + ϕ)dμ

= P(t, ϕ).

4. Real analyticity of the topological pressure

In this section we consider analytic properties of the pressure P (t, ϕ) seen as a function in t, and of the pressure P (t, qϕ) seen as a function in q (for certain fixed t). We remark that our analysis here is based on and generalizes the work in [15]. As always, let f be a parabolically semi-hyperbolic GPL-map. In order to introduce some auxiliary ‘critical parameters’, recall that for c ∈ Crit(J(f)), the order q(c) of c is determined by the local behaviour of f around c. That is, for z sufficiently close to c we have for the Taylor expansion of f that

f(z) = f(c) + b_{0}(z − c)^{q(c)} + …. (LBC)

Then the critical parameters χ_{0}, χ(c), χ_{q} and χ are defined as follows.

χ_{0} := inf\{lim sup_{n→∞} \frac{1}{n} log |(f^{n})′(z)| : z ∈ Ω ∪ ω(Crit(f))\},

χ(c) := lim inf_{k→∞} \frac{1}{k} log inf_{n≥1}\{|(f^{k})′(f^{n}(c))|\},

χ_{q} := min \{\frac{χ(c)}{q(c)} : c ∈ Crit(f)\} and χ := min\{χ_{q}, χ_{0}\}.

We begin with showing that for t in a certain range, the measure m_{t,ϕ} vanishes on the limsupset of the inverse images of the regular part. For this it is sufficient to have the following.

Lemma 4.1. If t ∈ [0, δ(ϕ)), then there exists 0 < ρ < 1 such that for all n ∈ IN we have

m_{t,ϕ} ( ⋂_{j=0}^{n} f^{−j}(U_{r})) ≪ ρ^{n}.

Proof. Put m = m_{t,ϕ}. Fix q ∈ IN, and consider the set

U_{r}^{(q)} := U_{r} ∩ f^{−1}(U_{r}) ∩ … ∩ f^{−q}(U_{r}).

Since the map f : U_{j} → U is univalent for each j ∈ I_{r}, it follows by induction that there exist finitely many, say k_{q}, holomorphic inverse branches of f^{q}, denoted by f_{1}^{−q} : U → U_{r}, …, f_{k_{q}}^{−q} : U → U_{r}, such that

U_{r}^{(q)} = ⋃_{j=1}^{k_{q}} f_{j}^{−q}(U_{r}).  (4.1)

Hence, for any arbitrary set A ⊂ U_{r} it follows that

U_{r}^{(q)} ∩ f^{−q}(A) = ⋃_{j=1}^{k_{q}} f_{j}^{−q}(A),  (4.2)

and by conformality of m we have for each j ∈ {1, 2, …, k_{q}} that

m(f_{j}^{−q}(A)) ≤ m(A)e^{−P(t,ϕ)q} sup_{z∈A}{|(f_{j}^{−q})′(z)|}^{t} sup{exp S_{q}ϕ(z) : z ∈ f_{j}^{−q}(A)}

≤ m(A)e^{−P(t,ϕ)q} sup_{z∈U_{r}}{|(f_{j}^{−q})′(z)|}^{t} sup{exp S_{q}ϕ(z) : z ∈ f_{j}^{−q}(U_{r})}, (4.3)

as well as

m(f_{j}^{−q}(U_{r})) ≥ m(U_{r})e^{−P(t,ϕ)q} inf_{z∈U_{r}}{|(f_{j}^{−q})′(z)|}^{t} inf{exp S_{q}ϕ(z) : z ∈ f_{j}^{−q}(U_{r})}. (4.4)

Now, applying Koebe’s distortion theorem on U_{r}, we see that there exists a constant K ≥ 1 such that

sup_{z∈U_{r}}{|(f_{j}^{−q})′(z)|} ≤ K inf_{z∈U_{r}}{|(f_{j}^{−q})′(z)|}.

It also follows from Lemma 2.5 that

sup{exp S_{q}ϕ(z) : z ∈ f_{j}^{−q}(U_{r})} ≤ C_{θ} inf{exp S_{q}ϕ(z) : z ∈ f_{j}^{−q}(U_{r})}

with some universal constant C_{ϕ}. Therefore, (4.3) and (4.4) imply that

m(f_{j}^{−q}(A)) ≤ \frac{K^{t}C_{θ}}{m(U_{r})}m_{t}(A)m(f_{j}^{−q}(U_{r})).

Combining this estimate with (4.1) and (4.2), it follows that

m(U_{r}^{(q)} ∩ f^{−q}(A)) ≤ \frac{K^{t}C_{θ}}{m_{t}(U_{r})}m(U_{r}^{(q)})m_{t}(A).  (4.5)

Let U_{r}^{(∞)} := ⋂_{j≥0} f^{−j}(U_{r}) = ⋂_{q∈IN} U_{r}^{(q)}, and observe that f^{−1}(U_{r}^{(∞)}) ⊃ U_{r}^{(∞)}. By ergodicity of μ_{t}, we hence have that μ_{t}(U_{r}^{(∞)}) ∈ {0, 1}. Now, since μ_{t,ϕ}(U_{o}) > 0, and since U_{r} ⊂ U \ U_{o}, we have μ_{t,ϕ}(U_{r}) < 1, which then implies that μ_{t,ϕ}(U_{r}^{∞}) = 0. Since {U_{r}^{(q)}}_{q=1}^{∞} is a descending sequence of sets, we conclude that lim_{q→∞} μ_{t,ϕ}(U_{r}^{(q)}) = 0, and hence that lim_{q→∞} m(U_{r}^{(q)}) = 0. Therefore, we can choose q ∈ IN sufficiently large such that K^{t}C_{ϕ}m(U_{r}^{(q)})/m(U_{r}) ≤ 1/2. Inserting this observation into (4.5), we obtain that for any arbitrary A ⊂ U_{r} we have that

m(U_{r}^{(q)} ∩ f^{−q}(A)) ≤ \frac{1}{2} m(A).  (4.6)

In order to finish the proof, we use (4.6) and observe that for every k ∈ IN we have that

m ( ⋂_{j=0}^{qk} f^{−j}(U_{r})) = m (U_{r}^{(q)} ∩ f^{−q} ( ⋂_{j=0}^{q(k−1)} f^{−j}(U_{r}))) ≤ \frac{1}{2}m ( ⋂_{j=0}^{q(k−1)} f^{−j}(U_{r})) .

By way of induction, this gives that

m ( ⋂_{j=0}^{qk} f^{−j}(U_{r})) ≤ (\frac{1}{2})^{k} ,

which also holds for k = 0. Now let n ∈ IN be given, and write n = qk + r, for 0 ≤ r < q and k ≥ 0. It follows that

m ( ⋂_{j=0}^{n} f^{−j}(U_{r})) ≤ m ( ⋂_{j=0}^{qk} f^{−j}(U_{r})) ≤ (\frac{1}{2})^{k} ≤ (\frac{1}{2})^{\frac{n}{q} −1} = 2 ((\frac{1}{2})^{\frac{1}{q}} )^{n} .

As an immediate consequence we derive the following corollary, which shows that the sets J(f) and J_{Φ_{f}} coincide m_{t,ϕ}-almost everywhere on U_{o}.

Corollary 4.2. If t ∈ [0, δ(ϕ)), then m_{t,ϕ}(J_{Φ_{f}} ∩ U_{o}) = m_{t,ϕ}(U_{o}) > 0.

Proof. Recall that by Proposition 2.3 we have J_{Φ_{f}} ∩ U_{o} = J(f) ∩ U_{o} \ ⋃_{n≥0} f^{−n}(Ω ∪ ⋂_{k≥0} f^{−k} (U_{r})). Also, Proposition 3.1 implies that m_{t,ϕ} has no atoms. Finally, by Lemma 4.1 we have that m_{t,ϕ}(⋂_{k≥0} f^{−k}(U_{r})) = 0. Combining these three observations, the statement of the corollary follows.

Lemma 4.3. If t ∈ [0, δ(ϕ)), then there exists l ∈ IN such that, for each Borel set A ⊂ U,

m_{t,ϕ}(f^{−1}(A)) ≪ (m_{t,ϕ}(A))^{1/l}.

Proof. Put m = m_{t,ϕ}. Using the conformality of m, it follows that the assertion holds for all Borel sets A ⊂ U such that A ∩ ⋃_{c∈Crit(J(f))} B(f(c), δ) = ∅, for some fixed positive δ. Hence, from now on let a Borel set A ⊂ B(f(c), δ) be fixed, for some c ∈ Crit(J(f)), with m(A) > 0 and where δ < dist(U_{r}, ∂U)/2 is chosen sufficiently small (which will be specified during the proof). Let f_{c}^{−1}(A) be the intersection of f^{−1}(A) with the component of f^{−1}(B(f(c), δ)) which contains c. Also, for n ∈ IN we define

λ_{n}(c) := |(f^{n})′(f(c))|,

and let A(w, r, R) := {z ∈ C I : r ≤ |z − w| < R} denote the annulus centred at w ∈ C I of inner radius r and outer radius R. The structure of the proof is as follows. We shall show that u is a finite number, and by combining this with Lemma 3.2, we obtain

(i) m(f_{c}^{−1}(A)) ≪ λ_{su}(c)^{−t/q(c)}e^{−P(t,ϕ)su} exp(S_{su}ϕ(c)).

Finally, we prove the following two facts, which then finishes the proof of the proposition.

(ii) λ_{su}(c)^{−t/q(c)}e^{−P(t,ϕ)su} exp(S_{su}ϕ(c)) ≤ (λ_{su}(c)^{−t}e^{−P(t,ϕ)su} exp(S_{su}ϕ(c))))^{1/l} for some l ∈ IN and for all s sufficiently large.

(iii) λ_{su}(c)^{−t}e^{−P(t,ϕ)su} exp(S_{su}ϕ(c)) ≪ m(A ∩ A(f(c), δλ_{s(u+1)}(c)^{−1}, δ)) ( ≤ m(A) ).

For (i), we combine Lemma 3.2 and the finiteness of u to obtain

m(f_{c}^{−1}(A)) = m(f_{c}^{−1}(A ∩ \overline{B(f(c), δλ_{su}(c)^{−1})})) + m(f_{c}^{−1}(A ∩ A(f(c), δλ_{su}(c)^{−1}), δ)))
≤ m(f_{c}^{−1}(\overline{B(f(c), δλ_{su}(c)^{−1})})) + m(f_{c}^{−1}(A ∩ A(f(c), δλ_{su}(c)^{−1}), δ)))
≪ m (\overline{B (c, (δλ_{su}(c)^{−1})^{1/q(c)})}) +
+ (δλ_{su}(c)^{−1})^{(\frac{1}{q(c)} −1)t}e^{−P(t,ϕ)}e^{ϕ(c)}m(A ∩ A(f(c), δλ_{su}(c)^{−1}, δ))
≪ λ_{su}(c)^{−t/q(c)} exp(S_{su}ϕ(c) − P(t, ϕ)su) +
+ (δλ_{su}(c)^{−1})^{(\frac{1}{q(c)} −1)t}λ_{su}(c)^{−t} exp(S_{su}ϕ(c) − P(t, ϕ)su)
≍ λ_{su}(c)^{−t/q(c)} exp(S_{su}ϕ(c) − P(t, ϕ)su).

For (ii), recall that \frac{1}{v} log |(f^{v})′(f(c))| ≥ −q(c)(P(t, ϕ) − \overline{ϕ})/t + κ for all for all v ≥ s. Hence, by choosing l(c) sufficiently large so that

κ > \frac{(P(t, ϕ) − \overline{ϕ})q(c)(1 − q(c))}{t(l(c) − q(c))} + \frac{κ}{2} \frac{l(c) − 1}{l(c) − q(c)}

it follows that

(l(c) − q(c))\frac{log λ_{su}(c)}{su} ≥ (\frac{−q(c)(P(t, ϕ) − \overline{ϕ})}{t} + κ) (l(c) − q(c))

≥ (\frac{q(c)(P(t, ϕ) − \overline{ϕ})}{t} − \frac{κ}{2}) (1 − l(c))

= \frac{q(c)}{t} (P(t, ϕ) − \overline{ϕ} − \frac{κt}{2q(c)}) (1 − l(c))

≥ \frac{q(c)}{t} (P(t, ϕ) − \frac{S_{su}ϕ(c)}{su}) (1 − l(c)).

An elementary rearrangement then gives

λ_{su}(c)^{−t/q(c)} exp(S_{su}ϕ(c) − P(t, ϕ)su) ≤ (λ_{su}(c)^{−t} exp(S_{su}ϕ(c) − P(t, ϕ)su))^{1/l(c)}.

By defining l := max{l(c) : c ∈ Crit(J(f))}, the statement in (iii) follows. Finally for (iii), the finiteness of u gives

m(A ∩ A(f(c), δλ_{s(u+1)}(c)^{−1}, δ) > λ_{s(u+1)}(c)^{−t} exp(S_{s(u+1)}ϕ(c) − P(t, ϕ)s(u + 1))
≥ exp(−(P(t, ϕ)s + ||ϕ||_{∞})||f′||^{−st}λ_{su}(c)^{−t} exp(S_{su}ϕ(c) − P(t, ϕ)su),

which completes the proof of the lemma.

We now pass to the CGDM-system Φ_{f} associated with the GPL-map f. For this the reader is asked to recall the construction and notation given in Section 2. For each t ≥ 0, s ∈ IR and e ∈ E_{f} we define the potential g_{t,s}^{(e)} : W_{t(e)} → IR by, for x ∈ W_{t(e)},

g_{t,s}^{(e)}(x) := t log |ϕ′_{e}(x)| − sN(e) + S_{n(e)}ϕ(ϕ_{e}(x)).

We shall see that for suitably chosen s and t the family G_{t,s} := {g_{t,s}^{(e)} : e ∈ E_{f}} is a summable Hölder family of functions, where Hölder refers to the fact that for some γ > 0 we have (cf. [5], [10])

sup_{n≥1} sup_{(τ_{1},τ_{2},...)∈E_{f}} sup_{z,w∈U_{t(τn)}} |g_{t,s}^{(τ_{1})}(ϕ_{τ_{2},... ,τ_{n}}(z)) − g_{t,s}^{(τ_{1})}(ϕ_{τ_{2},... ,τ_{n}}(w))|e^{γ(n−1)} < ∞.

Lemma 4.4. For each u > 0 such that P(u, ϕ) > −χu + \overline{ϕ}, there exists δ > 0 such that G_{t,s} is a summable Hölder family of functions, for each t ∈ (u − δ, u + δ) and s > P(u, ϕ) − δ.

Proof. In [18] (Lemma 5.4) we obtained that the family {S_{n(e)}ϕ∘ϕ_{e}}_{e∈E_{f}} is Hölder continuous (in fact, in [18] we only considered iterated function systems rather than CGDM-systems; nevertheless after minor modifications the proof in [18] goes through also for CGDM-systems). Furthermore, in [15] (Lemma 4.5) we have shown that the family {t log |ϕ′_{e}(x)| − sN(e)}_{e∈E_{f}} is Hölder continuous. Therefore, by combining these two results, it follows that {g_{t,s}^{(e)}}_{e∈E_{f}} is a Hölder family. In order to prove that G_{t,s} is summable, we let Z^{(n)} := {e ∈ E_{f} : N(e) = n} and define, for n ∈ IN,

R_{n} := ⋃_{e∈Z^{(n)}} ϕ_{e}(U_{t(e)}).

If there are no parabolic elements, then we have for each n > 1 that R_{n} ⊂ f^{−1} (⋂_{j=0}^{n−2} f^{−j}(U_{r})) (for n = 1, we have R_{1} ⊂ U_{o}), and hence Lemma 4.1 and Lemma 4.3 imply that

m_{u}(R_{n}) ≤ m_{u} (f^{−1} ( ⋂_{j=0}^{n−2} f^{−j}(U_{r}))) ≪ (m_{u} ( ⋂_{j=0}^{n−2} f^{−j}(U_{r})))^{1/l} ≪ ρ^{n/l}.  (4.7)

If there are parabolic points then χ_{0} = 0, and consequently the condition P(u, ϕ) > −χu + \overline{ϕ} implies that P(u, ϕ) > \overline{ϕ}. Then note that for every e ∈ Z^{(n)} there exists 1 ≤ k ≤ n such that f^{j}(U_{i(e)}) ⊂ U_{r}, for all k ≤ j < n, and such that f^{j}(U_{i(e)}) ⊂ U_{i}, for all 1 ≤ j < k and for some i ∈ I_{p}. Let ϵ > 0 be chosen sufficiently small such that \overline{ϕ} − P(u, ϕ) < −2ϵ. By Lemma 2.6, there exists k_{ϵ} ∈ IN such that sup\{S_{k}ϕ\} ≤ (\overline{ϕ} + ϵ)k, for all k ≥ k_{ϵ}. Combining these observations, it follows that sup{S_{k}ϕ} − P(u, ϕ)k < −kϵ. Using Lemma 4.1, Lemma 4.3 and (2.1), we then obtain, for some fixed β ∈ (max{e^{−ϵ}, ρ}, 1) and for every x ∈ f(U_{i(e)}),

m_{u,ϕ}(R_{n}) ≪ (m_{u,ϕ}(f(R_{n})))^{1/l}

≪ (∑_{k=1}^{n} exp(S_{k}ϕ(x) − kP(u, ϕ)) ∑_{i∈I_{p}} k^{− \frac{p(ω_{i})+1}{p(ω_{i})} u}m_{u,ϕ} ( ⋂_{j=1}^{n−k} f^{−j}(U_{r})))^{1/l}

≪ (∑_{k=1}^{n} exp(sup\{S_{k}ϕ\} − kP(u, ϕ)) ∑_{i∈I_{p}} k^{− \frac{p_{i}+1}{p_{i}} u}m_{u,ϕ} ( ⋂_{j=1}^{n−k} f^{−j}(U_{r})))^{1/l}

≪ (e^{−ϵk}ρ^{n−k} card(I_{p}))^{1/l} ≪ β^{n/l}.

Combining this estimate and (4.7), we conclude, no matter if there are parabolic points or not, that there exists α > 0 such that, for all n ∈ IN,

m_{u}(R_{n}) ≪ e^{−αn}.  (4.8)

Using the definition of the measure m_{u,ϕ} along with Koebe’s distortion theorem and Lemma 2.5, we now have, for all n ∈ IN,

∑_{e∈Z^{(n)}} sup_{z∈\overline{U_{i(e)}}}\{|(f^{n})′(z)|^{−u}) exp(S_{n}ϕ(z) − P(u, ϕ)n)\} ≪ e^{−αn}.

Observe that {ϕ_{e} : W_{t(e)} → U_{i(e)}}_{e∈E_{f}} is a normal family of functions, and hence all its limit functions are constant functions. This implies that

Δ_{1} := sup_{e∈E_{f}} sup_{z∈\overline{U_{t(e)}}} |ϕ′_{e}(z)| < ∞.

Therefore, for fixed u > 0 there exists 0 < δ < min\{u, \frac{α}{2}, \frac{α}{4}| log Δ_{1}|^{−1}, \frac{α}{4}| log Δ_{2}|^{−1}\}, where we have put Δ_{2} := sup_{e∈E_{f}} sup_{z∈\overline{U_{i(e)}}} |f′(z)|. With this choice of δ we obtain for each t ∈ (u − δ, u + δ) and s > P(u) − δ that

∑_{e∈Z^{(n)}} sup_{z∈\overline{U_{i(e)}}}(|(f^{n})′(z)|^{−t}) exp(S_{n}ϕ(z))e^{−sn}

≤ ∑_{e∈Z^{(n)}} sup_{z∈\overline{U_{i(e)}}}(|(f^{n})′(z)|^{−u}) exp(S_{n}ϕ(z) − P(u, ϕ)n)e^{δn} max\{Δ_{1}^{δ}, Δ_{2}^{nδ}\}

≪ e^{−αn}e^{\frac{α}{2} n}e^{\frac{α}{4} n} = e^{− \frac{α}{4} n}.

For the following lemma recall that the topological pressure P associated with the family G_{t,s} is given by (cf. [5], [10], [15])

P(t, s) := lim_{n→∞} \frac{1}{n} log ∑_{(τ_{1},... ,τ_{n})∈E_{f}^{(n)}} sup_{z∈U_{t(τn)}} exp (g_{t,s}^{(τ_{n})}(z) + ∑_{i=1}^{n−1} g_{t,s}^{(τ_{i})}(ϕ_{τ_{i+1},... ,τ_{n}}(z))) ,

where we have set E_{f}^{(n)} := {(τ_{1}, …, τ_{n}) ∈ E_{f}^{n} : A_{τ_{j}τ_{j+1}} = 1 for all j = 1, 2, …, n − 1}. Also, associated with G_{t,s} there exists a unique G_{t,s}-conformal probability measure m_{t,s} supported on J_{Φ_{f}}. That is, for each n ≥ 1 and τ = (τ_{1}, …, τ_{n}) ∈ E_{f}^{(n)} we have for every Borel set A ⊂ U_{t(τ_{n})} that

m_{t,s}(ϕ_{τ}(A)) = ∫_{A} exp (g_{t,s}^{(τ_{n})}(z) + ∑_{i=1}^{n−1} g_{t,s}^{(τ_{i})}(ϕ_{τ_{i+1},... ,τ_{n}}(z)) − nP(t, s)) dm_{t,s}(z).

Lemma 4.5. For t > 0 such that P(t, ϕ) > −χt + \overline{ϕ}, we have P(t, P(t, ϕ)) = 0. Furthermore, we have that, for each n ∈ IN and τ = (τ_{1}, …, τ_{n}) ∈ E_{f}^{(n)},

m_{t,P(t,ϕ)}(ϕ_{τ}(U_{t(τ_{n})})) ≍ m_{t,ϕ}(ϕ_{τ}(U_{t(τ_{n})})),

with comparability constants not depending on n and τ. Furthermore, we have in particular m_{t,P(t,ϕ)} and m_{t,ϕ} coincide on J_{Φ_{f}} up to a positive multiplicative constant.

Proof. Put N(τ) := ∑_{j=1}^{n} N(τ_{j}). By conformality of m_{tϕ} and m_{t,s}, we have for each n ∈ IN,

m_{tϕ}(ϕ_{τ}(U_{t(τ_{n})})) = ∫_{U_{t(τ)}} |ϕ′_{τ}(z)|^{t} exp(S_{N(τ)}ϕ ∘ ϕ_{τ} − P(t, ϕ)N(τ))dm_{t}(z)
≍ ||ϕ′_{τ}||^{t} exp(sup{S_{N(τ)}ϕ ∘ ϕ_{τ}} − P(t, ϕ)N(τ))m_{t}(U_{t(τ)})

≍ e^{P(t,P(t,ϕ))n}||ϕ′_{τ}||^{t} exp(sup{S_{N(τ)}ϕ ∘ ϕ_{τ}} − P(t, ϕ)N(τ)) − P(t, P(t, ϕ))n)

≍ e^{P(t,P(t,ϕ))n}m_{t,P(t,ϕ)}(ϕ_{τ}(U_{t(τ)})).

Therefore, if P(t, P(t, ϕ)) > 0 then m_{t,P(t,ϕ)}(J_{Φ_{f}}) = 0, which contradicts m_{t,P(t,ϕ)}(J_{Φ_{f}}) = 1. On the other hand, if P(t, P(t, ϕ)) < 0 then we obtain m_{t}(J_{Φ_{f}}) = 0, which is also a contradiction. Thus, it follows that P(t, P(t, ϕ)) = 0. The reamainder of the lemma is an immediate consequence of Theorem 3.2.3 in [10].

We now obtain the following two theorems which are the main results of this section.

Theorem 4.6. Let f be a parabolically semihyperbolic GPL-map. For values t > 0 for which P(t, ϕ) > −χt + \overline{ϕ}, we have that P(t, ϕ) is real-analytic as a function in t.

Proof. Using Lemma 4.4 and applying Theorem 2.6.12 of [10] (or alternatively [5] Theorem 6.4), we have for each positive u with P(u, ϕ) > −χu + \overline{ϕ} that there exists δ > 0 such that P is real-analytic on (u − δ, u + δ) × (P(u, ϕ) − δ, P(u, ϕ) + δ) in both variables t and s. In order to prove that P is real-analytic on (u − δ, u + δ), we employ the implicit function theorem, showing that P is the unique real-analytic function which satisfies P(t, P(t)) = 0 for all t ∈ (u − δ, u + δ). For this it is now sufficient to verify that for all t ∈ (u − δ, u + δ) we have

\frac{∂P(t, s)}{∂s}|_{(t,P(t,ϕ))} exists and is strictly negative.  (4.9)

Denote the measure m_{t,P(t,ϕ)} by ν_{t}. Proposition 2.3, Lemma 4.4 and Lemma 4.5 guarantee that Theorem 3.7 of [10] is applicable. This gives that the measure ν_{t} has a lift ν̃_{t} to the symbolic space E_{f}, and that there exists a measure μ̃_{t} in the measure class of ν̃_{t} which is invariant under the shift map on the space E_{f}, and whose Radon-Nikodym derivative with respect to ν̃_{t} is bounded away from zero and infinity. We can now apply Proposition 2.6.13 of [10] (or alternatively [5], Proposition 6.5), which gives

\frac{∂P(t, s)}{∂s}|_{(t,P(t,ϕ))} = − ∫ Ndμ̃_{t}.  (4.10)

Using the estimate in (4.8) and the second part of Lemma 4.5 we then compute

∫ Ndμ̃_{t} ≍ ∫ Ndν̃_{t} = ∫ Ndν_{t} = ∑_{n∈IN} nν_{t}(R_{n}) ≍ ∑_{n∈IN} nm_{t}(R_{n}) ≪ ∑_{n∈IN} ne^{−αn} < ∞, (4.11)

where after the first equality sign we treated the function N slightly informally as being defined on the limit set J_{Φ_{f}}. Combining (4.10) and (4.11), and using the fact that the function N is strictly positive, we derive (4.9), which then completes the proof.

Finally, let us consider the family Ĝ_{q,s} := {ĝ_{q,s}^{(e)} : e ∈ E_{f}}, which is given by, for fixed t > 0,

ĝ_{q,s}^{(e)}(x) := t log |ϕ′_{e}(x)| − sN(e) + qS_{n(e)}ϕ(ϕ_{e}(x)).

If in the construction above we use this family instead of the family G_{t,s}, then the proof of Theorem 4.6 is in fact easier. Using this modified family of functions ĝ_{q,s}^{(e)}, we then obtain the following result (c.f. [18], Sect. 5).

Theorem 4.7. Let f be a parabolically semi-hyperbolic GPL-map. If t ∈ [0, δ(ϕ)), then the function q ↦ P(t, qϕ) is real-analytic in a small neighbourhood of q = 1.

5. Multifractal analysis

5.1. The general case of a parabolically semi-hyperbolic GPL-map.

In this section we derive the main results of this paper, namely we give a multifractal analysis for parabolically semi-hyperbolic GPL-maps f. Throughout let ϕ : J(f) → IR be a Hölder continuous function such that P_{top}(f, ϕ) > sup(ϕ). Recall from the introduction that we define, for every x ∈ J(f),

\overline{ρ}_{ϕ}(x) := \overline{lim_{n→∞}} \frac{S_{n}(P_{top}(f, ϕ) − ϕ(x))}{log |(f^{n})′(x)} and \underline{ρ}_{ϕ}(x) := \underline{lim}_{n→∞} \frac{S_{n}(P_{top}(f, ϕ) − ϕ(x))}{log |(f^{n})′(x)}.

If \overline{ρ}_{ϕ}(x) = \underline{ρ}_{ϕ}(x), then we let ρ_{ϕ}(x) refer to their common value. We are interested in studying the (ϕ, α)-level sets K_{ϕ}(α), given by

K_{ϕ}(α) := {x ∈ J(f) : ρ_{ϕ}(x) = α},

and in particular in the associated ϕ-spectrum k_{ϕ}, which is given by

k_{ϕ}(α) := HD(K_{ϕ}(α)).

Also, recall that by using ϕ − P_{top}(f, ϕ) instead of ϕ, we can assume without loss of generality that

P_{top}(f, ϕ) = 0 and sup(ϕ) < 0.  (5.1)

Lemma 5.1. For every q ∈ (0, 1] there exists a unique T (q) ≥ 0 such that P(T (q), qϕ) = 0.

Proof. We have that P_{top}(f, ϕ) = 0, and that the graph of the function t ↦ P(t, qϕ) lies below the graph of the function t ↦ P(t, 0) (this follows, since by assumption q > 0 and sup(ϕ) < 0). Also, by definition (P2) of variational pressure, the function t ↦ P(t, qϕ) is continuous. Combining these observations, it follows that for P(0, qϕ) ≥ 0 there exists t ≥ 0 such that P(t, qϕ) = 0. On the other hand, since sup(ϕ) < 0, we have that P(0, qϕ) ≥ P(0, ϕ) = 0, for each q ∈ (0, 1]. Therefore, it follows that P(t, qϕ) = 0, for every q ∈ [0, 1] and for some t ≥ 0 (which depends on q). Hence, in order to finish the proof it is now sufficient to show that if for some t ≥ 0 and q > 0 we have that P(t, qϕ) = 0, then this implies that, for all u > 0,

P(t + u, qϕ) < 0 and P(t − u, qϕ) > 0.  (5.2)

In order to derive this implication, we remark that if μ is an f-invariant Borel probability measure such that, for ϵ ∈ (0, −\frac{q}{2} sup(ϕ)),

− ϵ ≤ h_{μ}(f) − tχ_{μ} + q ∫ ϕdμ,  (5.3)

then it follows that

χ_{μ} ≥ −\frac{q}{4} sup(ϕ).  (5.4)

This can be seen by using (5.3) and the fact that χ_{μ} ≥ 0, which gives

h_{μ}(f) ≥ tχ_{μ} − q ∫ ϕdμ − ϵ ≥ −q sup(ϕ) − ϵ ≥ −\frac{q}{2} sup(ϕ).

Hence, by Ruelle’s inequality we have χ_{μ} ≥ \frac{1}{2}h_{μ}(f) ≥ −\frac{q}{4} sup(ϕ), which gives the inequality in (5.4). We now prove the first inequality in (5.2) by contradiction as follows. Suppose that P(t + u, qϕ) ≥ 0, and let 0 < ϵ < min \{−\frac{q}{2} sup(ϕ), −\frac{qu}{4} sup(ϕ)\} be given. By definition of the pressure function, there exists an f-invariant Borel probability measure ν such that P(t + u, qϕ) − ϵ ≤ h_{ν}(f) − (t + u)χ_{ν} + q ∫ ϕdν. Since P(t + u, qϕ) ≥ 0 and P(t, qϕ) = 0, this implies that

0 ≥ h_{ν}(f) − tχν + q ∫ ϕdν ≥ uχ_{n}u − ϵ ≥ −ϵ.  (5.5)

Hence, (5.3) is satisfied, and consequently (5.4) holds. Now, combining (5.4) and (5.5), we obtain 0 ≥ −\frac{qu}{4} sup(ϕ) − ϵ > 0, which is a contradiction and hence gives the first inequality in (5.2). In order to prove the second inequality in (5.2), note that again by definition of pressure and since P(t, qϕ) = 0, there exists an f-invariant Borel probability measure μ satisfying (5.3) for every 0 < ϵ < min \{−\frac{q}{2} sup(ϕ), −\frac{qu}{4} sup(ϕ)\}. Hence, applying (5.4), we get

\begin{aligned}P(t − u, qϕ) & ≥ h_{μ}(f) − (t − u)χ_{μ} + q ∫ ϕdμ = (h_{μ}(f) − tχ_{μ} + q ∫ ϕdμ) + uχ_{μ} \\ & ≥ −ϵ + uχ_{μ} ≥ −ϵ + uχ_{μ} ≥ −ϵ − \frac{qu}{4} sup(ϕ) > 0. \end{aligned}

This latter estimate gives the second inequality in (5.2), and hence completes the proof of the lemma.

Lemma 5.2. For q ∈ (0, 1] we have that

\frac{∂P}{∂q}(q, T (q)) = ∫ ϕdμ_{q} and \frac{∂P}{∂t}(q, T (q)) = − ∫ log |f′|dμ_{q}.

Proof. Let u ∈ IR be fixed, and consider the equilibrium state μ_{q} for the potential −T (q) log |f′| + qϕ. By definition of the variational pressure P(T (q), (q + u)ϕ), we have

\begin{matrix}P(T (q) & , (q + u)ϕ) − P(T (q), q) ≥ \\ ≥ −T (q)χ_{μ_{q}} + (q + u) ∫ ϕdμ_{q} + h_{μ_{q}}(f) − (−T (q)χ_{μ_{q}} + q ∫ ϕdμ_{q} + h_{μ_{q}}(f)) \\ = u ∫ ϕdμ_{q}. \end{matrix} (5.6)

Also, by Lemma 5.1 we have that 0 = P(T (q), qϕ) > sup(qϕ) and that T (q) ≥ 0, which together with the definition of δ(ϕ) implies that T (q) ∈ [0, δ(qϕ)). Hence, we are now in the position to apply Theorem 4.7, which gives that the function s ↦ P(T (q), sϕ) is real-analytic on a neighbourhood of s = q. Since this latter function is convex (by definition (P2) of variational pressure), (5.6) gives that that \frac{∂P}{∂q}(q, T (q)) = ∫ ϕdμ_{q}, and hence the first assertion of the lemma follows. The proof of the second formula of the lemma is analogous and will be omitted.

We now come to the first main result of this paper.

Theorem 5.3. Let f be a parabolically semi-hyperbolic GPL-map and let ϕ : J(f) → IR be a Hölder continuous potential such that sup(ϕ) < P_{top}(f, ϕ) = 0. Then the following holds.

(a) For μ_{ϕ}-a.e. x ∈ J(f), we have that ρ_{ϕ}(x) exists and

ρ_{ϕ}(x) = \frac{− ∫ ϕdμ_{ϕ}}{∫ log |f′|dμ_{ϕ}} .

(b) For q ∈ (0, 1), the function q ↦ T (q) is real-analytic and T′(q) < 0.

(c) For each q ∈ (0, 1), we have that k_{ϕ}(−T′(q)) = T (q) − qT′(q).

Proof. The statement in (a) is an immediate consequence of Birkhoff’s Ergodic Theorem. For (b), note that by differentiating the equation P(T (q), qϕ) = 0 and using Lemma 5.2, we obtain

0 = \frac{∂P}{∂t}T′(q) + \frac{∂P}{∂q} = −T′(q) (− ∫ log |f′|dμ_{q}) + ∫ ϕdμ_{q},

and therefore

T′(q) = \frac{∫ ϕdμ_{q}}{∫ log |f′|dμ_{q}} < 0.  (5.7)

In order to prove (c), we first give the estimate of the function k_{ϕ}(−T′(q)) from below. By Birkhoff’s Ergodic Theorem there exists a Borel set X ⊂ J(f) such that μ_{q}(X) = 1 and such that, for every x ∈ X,

lim_{n→∞} \frac{1}{n} log |(f^{n})′(x)| = ∫ log |f′|dμ_{q} and lim_{n→∞} \frac{1}{n}S_{n}ϕ(x) = ∫ ϕdμ_{q}.

Hence, using (5.7) we obtain

lim_{n→∞} \frac{−S_{n}ϕ(x)}{log |(f^{n})′(x)|} = −\frac{∫ ϕdμ_{q}}{∫ log |f′|dμ_{q}} = −T′(q),

which implies that X ⊂ K_{ϕ}(−T′(q)). Thus, using (5.7) and the fact that P(T (q), qϕ) = 0, we get

\begin{aligned}k_{ϕ}(−T′(q)) & = HD(K_{ϕ}(−T′(q))) ≥ HD(X) ≥ HD(μ_{q}) = \frac{h_{μ_{q}}(f)}{χ_{μ_{q}}} \\ & = \frac{T (q)χ_{μ_{q}} − q ∫ ϕdμ_{q}}{χ_{μ_{q}}} = T (q) − q \frac{∫ ϕdμ_{q}}{χ_{μ_{q}}} = T (q) − qT′(q). \end{aligned}

This gives the required lower bound for k_{ϕ}. For the upper bound, let us fix an element

x ∈ K_{ϕ}(−T′(q)) \ ⋃_{n=0}^{∞} f^{−n}(Ω(f) ∪ Crit(f)).

Using [16] (Proposition 6.1), there exists ρ(x) > 0 and an unbounded increasing sequence {k_{n}}_{n=1}^{∞} such that, for each n ∈ IN,

Comp(x, f^{k_{n}}(x), f^{k_{n}}, 2ρ(x)) = ∅ and f^{k_{n}}(x) ∉ B(Ω, ρ(x)).

In here, Comp(x, f^{j}(x), f^{j}, r) refers to the connected component of f^{−j}(B(f^{j}(x), r)) which contains x. By Koebe’s Distortion Theorem, we have that

B(x, K|(f^{k_{n}})′(x)|^{−1}ρ(x)) ⊃ f_{x}^{−k_{n}}(B(f^{k_{n}}(x), ρ(x))),  (5.8)

where K ≥ 1 denotes the ‘Koebe constant’ for the scale 1/2, and f_{x}^{−k_{n}} : B(f^{k_{n}}(x), ρ(x)) → C I refers to the holomorphic inverse branch of f^{k_{n}} which maps f^{k_{n}}(x) to x. For ease of notation, we put m_{q} := m_{T (q),qϕ}. Using Lemma 2.5, (5.8) and once more Koebe’s Distortion Theorem, it follows that

\begin{aligned}m_{q}(B(x, K|(f^{k_{n}})′(x)|^{−1}ρ(x))) & ≥ ∫_{B(f^{kn}(x),ρ(x))} |(f^{k_{n}})′(z)|^{T (q)} exp(qS_{k_{n}}ϕ(f_{x}^{−k_{n}}(z)))dm_{q}(z) \\ & ≥ e^{−qC_{ρ(x)}}m(B(f^{k_{n}}(x), ρ(x)))|(f^{k_{n}})′(x)|^{−T (q)} exp(qS_{k_{n}}ϕ(x)). \end{aligned}

Hence, if we let r_{n} := K|(f^{k_{n}})′(x)|^{−1}ρ(x), it follows that, for every x ∈ K_{ϕ}(−T′(q)),

lim inf_{n→∞} \frac{log m_{q}(B(x, r_{n}))}{log r_{n}} ≤ lim_{n→∞} \frac{−T (q) log |(f^{k_{n}})′(x)| + qS_{k_{n}}ϕ(x)}{− log |(f^{k_{n}})′(x)|} = T (q) − qT′(q).

Note that (b) and (c) in Theorem 5.3 show that if the function T′(q) is locally invertible at at least one point in (0, 1), then we have that the multifractal ϕ-spectrum k_{ϕ} is real-analytic on a proper interval. Hence, our target now is to show that T′(q) is in fact locally invertible at at least one point in (0, 1). For this we require the following lemma.

Lemma 5.4. If the set {q ∈ (0, 1) : T′′(q) = 0} has an accumulation point in (0, γ), then we have that for all q ∈ (0, 1) the measures μ_{q} coincide, and that they are equivalent to the h-conformal measure ν_{h}.

???????????????????????????????WHAT IS γ in the Lemma ????????????????????????? Proof. By Theorem 5.3 (b) we have that the function T : (0, 1) → [0, ∞) is real-analytic. Hence, since the set (T′′)^{−1}(0) has an accumulation point in (0, 1), we conclude that T is affine, that is there exist α, β ∈ IR such that

T (q) = αq + β.

Since for each q ∈ (0, 1] we have P(T (q), qϕ) = 0, and since μ_{q} is an equilibrium state for the potential −T (q) log |f′| + qϕ, (5.7) implies that

h ≥ HD(μ_{q}) = \frac{h_{μ_{q}}}{χ_{μ_{q}}} = \frac{T (q)χ_{μ_{q}} − q ∫ ϕdμ_{q}}{χ_{μ_{q}}} = T (q) − qT′(q) = αq + β − αq = β. (5.9)

Since χ_{μ_{q}} ≤ log ||f′|| < ∞, and since by Przytycki’s result ????????????????????????????????????????????????????WHICH RESULT OF FELIKS’s? we have for each of the f-invariant measure μ_{q} that χ_{μ_{q}} ≥ 0, it follows that the function (t, q) ↦ P_{V} (t, qϕ) is continuous in both variables, for t ≥ 0 and q ∈ IR. Therefore,

P_{V} (β, 0) = lim_{q→0^{+}} P(T (q), qϕ) = 0.

Combining this with a result in [15] (Theorem 2.1), it follows that β ≥ h, which then gives, by using (5.9), that HD(μ_{q}) = h for all q ∈ (0, γ). Now note that we have h_{μ_{q}} − T (q)χ_{μ_{q}} + q ∫ ϕdμ_{q} = P(T (q), qϕ) = 0, for every q ∈ (0, 1]. Hence, we obtain h_{μ_{q}} = T (q)χ_{μ_{q}} − q ∫ ϕdμ_{q} ≥ −q ∫ ϕdμ_{q} > 0, which implies, using Ruelle’s inequality, that χ_{μ_{q}} > 0. It therefore follows by a result in [6] (Theorem B; the theorem is stated in the context of rational maps, nevertheless the proof can be adapted to GPL-maps) that for each q ∈ (0, γ) the measure μ_{q} is equivalent to the h-conformal measure ν_{h}. In particular, all the measures μ_{q} are mutually equivalent, and since they are ergodic, they must coincide.

For the following, we recall a notation of [10]. We let g_{1} : E_{f} → IR denote the amalgamated function of the family G_{0,0}, which is given by

g_{1}(ω) = S_{n(ω_{1})}ϕ(π(ω)).

Similarly, we let g_{2} : E_{f} → IR denote the amalgamated function, which is given by

g_{2}(ω) = −h log |(f^{n(ω_{1})})′(π(ω)|.

Also, recall that one says that g_{1} and g_{2} are cohomologous up to constant in the class of bounded Hölder continuous functions on E_{f}, if and only if there are a ∈ IR and a bounded Hölder continuous function u : E_{f} → IR such that

g_{2} − g_{1} = a + u − u ∘ σ.  (5.10)

We require the following lemma.

Lemma 5.5. If the measure μ_{0,ϕ} is equivalent to the h-conformal measure ν_{h}, then the functions g_{1} and g_{2} are cohomologous up to a constant in the class of bounded Hölder continuous functions on E_{f}.

Proof. Recall that P_{top}(f, ϕ) = 0. Let μ_{G_{0.0}} be the Φ_{f}-invariant version of the measure m_{0,0} ((c.f. formula (3.10) in [10]). Note that these two measures are both supported on J_{Φ_{f}}. We now show that μ_{G_{0.0}} is equivalent to m_{0,0}. For this observe that Lemma 4.5 implies that m_{0,0} coincides with m_{0,ϕ}|_{J_{Φ_{f}}}. Also, by Theorem 3.4, we have that m_{0,ϕ}|_{J_{Φ_{f}}} is equivalent to the h-conformal measure m̂_{h} on J_{Φ_{f}} for the system Φ_{f}. Finally, m̂_{h} is equivalent to μ̂_{h}, which is the Φ_{f}-invariant version of the measure m̂_{h}. Therefore, μ_{G_{0.0}} and μ̂_{h} are equivalent. By the result in [10] (formula (3.10)), we have that the Gibbs states μ̃_{g_{1}} and μ̃_{g_{2}}, which are both supported on E_{f}, are equivalent. Again by a result in [10] (Theorem 2.2.4), these measures are ergodic with respect to the shift map σ : E_{f} → E_{f}, and hence they must coincide. Therefore, by applying Theorem 2.2.7 in [10], the lemma follows.

For the following lemma, recall that a critical point c of a GPL-map f is called exceptional if f^{−n}(c) ⊂ Crit(f^{n}), for every n ∈ IN. Clearly, since there are only finitely many critical points and since these cannot form periodic cycles, each exceptional critical point must be eventually periodic.

Lemma 5.6. If f has a parabolic point or if J(f) contains a non-exceptional critical point, then g_{1} and g_{2} are not cohomologous up to any constant in the class of bounded Hölder continuous functions on E_{f}.

Proof. Suppose that that g_{1} and g_{2} are cohomologous up to a constant in the class of bounded Hölder continuous functions on E_{f}. Then there exist a ∈ IR and a bounded Hölder continuous function u : E_{f} → IR such that

g_{2} − g_{1} = a + u − u ∘ σ.  (5.11)

Let us assume that Ω ≠ ∅, and let i ∈ I_{p} be fixed. Note that both sets I_{o} and I_{r} are nonempty, and hence we can fix some j ∈ I_{o} and k ∈ I_{r}. Then, for every n ∈ IN, there exists e_{n} ∈ E_{f} such that f_{k}^{−1} ∘ f_{i}^{−n} = ϕ_{e_{n}}. For ω^{(n)} ∈ E_{f} such that ω_{1}^{(n)} = e_{n}, we have by (5.11) that

S_{n+1}ϕ(π(ω^{(n)})) + h log |(f^{n+1})′(π(ω^{(n)}))| = a + u(π(ω^{(n)})) − u(σ(π(ω^{(n)}))),

or equivalently,

S_{n+1}ϕ(π(ω^{(n)})) − h(− log |(f′(π(ω))| + log |(f_{i}^{−n})′(f^{n+1}(π(ω)))|).

Applying (2.1), we hence have

a + 2||u||_{∞} ≤ sup(S_{n+1}ϕ) + h log ||f′|| − h log(inf(|(f_{i}^{−n})′|_{\overline{U}_{j}} )

≤ (n + 1) sup(f) + h log ||f′|| + h (log(C_{\overline{U}_{j}}) + \frac{p_{i} + 1}{p_{i}} log n) .

Since sup(ϕ) < 0, we see that the right-hand side of the latter inequality gets arbitrarily small, and hence we have a contradiction. We now consider the case in which J(f) contains a non-exceptional point critical point c. Assume that c is chosen such that {f^{n}(c) : n ∈ IN} ∩ Crit(f) = ∅. Note that for each n ≥ 0 there exists a unique index i_{n} ∈ I such that f^{n}(c) ∈ U_{i_{n}}, and by Definition 2.1 (c), we have that i_{n} ∈ I_{r}, for all n ∈ IN. Consider the inverse branches, for n ≥ 1,

f_{∗}^{−n} = f_{i_{1}}^{−1} ∘ f_{i_{2}}^{−1} ∘ … f_{i_{1}}^{−i_{n}} : U ↦ U_{i_{1}}.

Since \overline{U}_{r} ⊂ U (by Definition 2.1 (b)) and since f(c) ∈ J(f), it follows by a standard normal families argument that

lim_{n→∞} diam(f_{∗}^{−n}(U)) = 0.  (5.12)

Now, let B(c, r_{1}) be a sufficiently small ball centered at the critical point c. Since f(c) ∈ f_{∗}^{−n}(U), it follows from (5.12) that f_{∗}^{−n}(U) ⊂ f(B(c, r_{1})), for all n ≥ k_{1}, with k_{1} ∈ IN sufficiently large. For j ∈ I_{o}, let w ∈ U_{j} be fixed, and choose w_{n} ∈ B(c, r_{1}) such that f(w_{n}) = f_{∗}^{−n}(w). Since j ∈ I_{o}, there exists a holomorphic inverse branch f_{∗}^{−(n+1)} : U_{j} → U of f^{n+1} which maps w to w_{n}, and for which f ∘ f_{∗}^{−(n+1)} = f_{∗}^{−n}. Since B(c, r_{1}) ⊂ U_{i}, for some i ∈ I_{c} ⊂ I_{o} and every n ≥ k_{1}, there exists a_{n} ∈ E_{f} such that ϕ_{a_{n}} = f_{∗}^{−(n+1)}. Then (5.11) gives, for τ^{(n)} ∈ E_{f} such that τ_{1}^{(n)} = a_{n},

S_{n+1}ϕ(π(τ^{(n)})) + h log |(f^{n+1})′(π(τ^{(n)}))| = a + u(π(τ^{(n)})) − u(σ(π(τ^{(n)}))). (5.13)

Since c is not exceptional, there exist q ≥ 0, s ∈ IN, and y ∈ f^{−s}(f^{q}(c)) ∩ U_{j} \ Crit(f^{s}). Let us now consider the inverse branches

f_{q}^{−(n−q+1)} = f_{i_{q}}^{−1} ∘ f_{i_{q+1}}^{−1} ∘ … f_{i_{n}}^{−1} : U → U_{i_{n}}.

As above, we have that

lim_{n→∞} diam(f_{q}^{−(n−q+1)}(U)) = 0.  (5.14)

Choose B(y, r_{2}) ⊂ U_{j} sufficiently small such that f^{s}|_{B(y,2r_{2})} is 1–1, and let f_{y_{n}}^{−s} refer to the inverse of f^{s}|_{B(y,2r_{2})}. By (5.14) we have, for all n ≥ k_{2}, with k_{2} ≥ k_{1} sufficiently large,

f_{q}^{−(n−q+1)}(U) ⊂ f^{s}(B(y, r_{2})).  (5.15)

Clearly, there now exists b_{n} ∈ E_{f} such that the map f_{q,s}^{−(n−q+1+s)} = f_{y_{n}}^{−s} ∘ f_{q}^{−(n−q+1)} : U → U_{j} if restricted to U_{j} is equal to ϕ_{b_{n}}. By (5.11) we then have, for ρ^{(n)} ∈ E_{f} such that ρ_{1}^{(n)} = b_{n},

S_{n−q+1+s}ϕ(π(ρ^{(n)})) + h log |(f^{n−q+1+s})′(π(ρ^{(n)}))| = a + u(π(ρ^{(n)})) − u(σ(π(ρ^{(n)}))).(5.16)

Applying Koebe’s distortion theorem along with Definition 2.1(b), we see that, for all n ≥ max{q, k_{2}} and x, y ∈ U,

\frac{|(f_{q}^{−(n−q+1)})′(y)|}{|(f_{q}^{−(n−q+1)})′(x)|} ≪ 1.  (5.17)

Therefore, using (3.1) and Definition 2.1 (c), it follows, for all n ≥ max{q, k_{2}},

diam(f_{q}^{−(n−q+1)}(U)) ≪ κ^{−(n−u+1)}diam(U).  (5.18)

Applying (5.17), we conclude that

|h log |(f^{n−q+1})′(f^{q}(π(τ^{(n)})))| − h log |(f^{n−q+1})′(f^{f}(π(ρ^{(n)})))|.  (5.19)

?????????????????????????????????????????????????????????????????????? Recall that α > 0 denotes the Hölder exponent of ϕ, and let L > 0 be the Hölder constant of the function ϕ. Using (5.18), it now follows that

|S_{n−q+1}ϕ(f^{q}(π(τ^{(n)}))) − S_{n−q+1}ϕ((f^{s}(π(ρ^{(n)})))|

≤ ∑_{j=0}^{n−q} |ϕ(f^{j}(f^{q}(π(τ^{(n)})))) − ϕ(f^{j}(f^{s}(π(ρ^{(n)}))))|

≤ ∑_{j=0}^{n−q} L|f^{j}(f^{q}(π(τ^{(n)}))) − f^{j}(f^{s}(π(ρ^{(n)})))|^{α}

≤ L ∑_{j=0}^{n−q} (diam(f_{q+j}^{n−q−j+1}(U)))^{α}

≪ ∑_{j=0}^{n−q} K_{1}^{α}diam(U)^{α}κ^{−α(n−q−j+1)} ≪ diam(U)^{α} ∑_{i=1}^{∞} κ^{−αi}

= diam(U)^{α}κ^{−α}(1 − κ^{−α})^{−1} < ∞.

(5.20)

Furthermore, note that we have

|S_{q}ϕ(π(τ^{(n)}))| ≤ q||ϕ||_{∞} and |S_{s}ϕ(π(ρ^{(n)}))| ≤ s||ϕ||_{∞},  (5.21)

and also that, using (5.15) and Koebe’s distortion theorem, with K ≥ 1 the Koebe constant,

|h log |(f^{s})′(π(ρ^{(n)}))| ≤ h log K + h log |(f^{s})′(y)|.  (5.22)

Finally, if we combine (5.13), (5.16), (5.19), (5.20), (5.21), (5.22) and and the fact that the function u is uniformly bounded, we conclude that h log |(f′(π(τ^{(n)}))| ≤ Ĉ, for all n sufficiently large and with some constant Ĉ > 0 which does not depend on n. Since lim_{n→∞} π(τ^{(n)}) = c, this gives a contradiction and hence finishes the proof.

Finally, we can now combine Theorem 5.3, Lemma 5.4 and Lemma 5.5 which then gives the following second main result of this paper.

Theorem 5.7. Let f be a parabolically semi-hyperbolic GPL-map and let ϕ : J(f) → IR be a Hölder continuous potential such that sup(ϕ) < P_{top}(f, ϕ) = 0. If μ_{0,ϕ} is not equivalent to the h-conformal measures ν_{h}, and hence in particular if f has a parabolic point or a nonexceptional critical point, then the domain of the multifractal ϕ-spectrum k_{ϕ} contains a nondegenerated interval on which k_{ϕ} is real-analytic.

5.2. The parabolic case without critical points in the Julia set.

In this section we consider the special class of parabolically semi-hyperbolic GPL-maps for which J(f) does not contain critical points of f. Maps of this type are called parabolic GPLmaps, and we show that for them the results of the previous section have a more transparent geometric interpretation, namely in terms of the local scaling behavior of the equilibrium state μ_{ϕ}. Here ϕ refers to a Hölder continuous potential with Hölder exponent exceeding p_{max}/(p_{max} + 1) and such that P_{top}(f, ϕ) > \overline{ϕ}. Note that by passing to a suitable higher iterate of the parabolic GPL-map f, we can always assume without loss of generality that U contains no critical points of f. For a measure μ supported on J(f) and for α ∈ [0, ∞), the (μ, α)-level sets L_{μ}(α) and the multifractal μ-spectrum ℓ_{μ} are defined by

L_{μ}(α) := \{z ∈ J(f) : lim_{r→0} \frac{log μ(B(x, r))}{log r} = α\} and ℓ_{μ}(α) := HD(L_{μ}(α)).

For the equilibrium state μ_{ϕ} and its f-invariant version m_{ϕ}, we have by a result in [17] (Lemma 2.4.4) that the symmetric difference of L_{μ_{ϕ}}(α) and L_{m_{ϕ}}(α) is contained in Ω. This implies that

ℓ_{μ_{ϕ}}(α) = ℓ_{m_{ϕ}}(α).  (5.23)

The main result of this section, that is Theorem 5.10, will be an immediate consequence of Theorem 5.3 in combination with the following two lemmata.

Lemma 5.8. For each α ∈ (0, ∞), we have that K_{ϕ}(α) ⊂ L_{m_{ϕ}}(α).

Proof. Let x ∈ J(f) \ ⋃_{n≥0} f^{−n}(Ω) be fixed. Since f is a parabolic GPL-map, there exists an infinite sequence (n_{j}) of positive integers (depending on x) and δ > 0 (independent of x) such that

B(f^{n_{j}}(x), 4Kδ) ⊂ U \ B(Ω, θ) and B(f^{n_{j}}(x), 4Kδ) ∩ ⋃_{k=1}^{∞} f^{k}(Crit(f)) = ∅.

In particular, for each j we hence have a well-defined holomorphic inverse branch f^{−n_{j}} : B(f^{n_{j}}(x), 4Kδ) → C I which maps f^{n_{j}}(x) to x. If r > 0 is given, then we let n := max{n_{i} : r ≤ δ|(f^{n_{i}})′(x)|^{−1}} and define r_{n} := δ|(f^{n})′(x)|^{−1}. Now we have for some j that n = n_{j}, and also that n_{j+1} = n + k, for some k. Clearly, we have that r ≥ r_{n+k}δ|(f^{n+k})′(x)|^{−1}, and hence it follows that

\frac{log(m_{ϕ}(B(x, r_{n})))}{log r_{n+k}} ≤ \frac{log(m_{ϕ}(B(x, r)))}{log r} ≤ \frac{log(m_{ϕ}(B(x, r_{n+k})))}{log r_{n}}.  (5.24)

Using Koebe’s distortion theorem, we see that B(x, r_{n+k}) ⊃ f_{x}^{−(n+k)}(B(f^{n+k}(x), Kδ), which if combined with Lemma 2.5, gives m_{ϕ}(B(x, r_{n+k})) ≥ e^{−C_{θ}} exp(S_{n+k}ϕ(x)). Therefore,

\frac{log(m_{ϕ}(B(x, r_{n+k})))}{log r_{n}} = \frac{log(m_{ϕ}(B(x, r_{n+k})))}{log r_{n+k}} \frac{log r_{n+k}}{log r_{n}} ≤ \frac{S_{n+k}ϕ(x) − C_{θ}}{− log |(f^{n+k})′(x)| + log δ} · \frac{log r_{n+k}}{log r_{n}} .

(5.25)

Similarly, using Koebe’s \frac{1}{4}-distortion theorem, it follows that B(x, r_{n}) ⊂ f_{x}^{−n}(B(f^{n}(x), 4δ)), and by combining this and Lemma 2.5, we obtain

m_{ϕ}(B(x, r_{n})) ≤ e^{C_{θ}} exp(S_{n+k}ϕ(x)).  (5.26)

Hence, we have

\frac{log(m_{ϕ}(B(x, r_{n})))}{log r_{n+k}} = \frac{log(m_{ϕ}(B(x, r_{n})))}{log r_{n}} \frac{log r_{n}}{log r_{n+k}} ≥ \frac{S_{n+k}ϕ(x) − C_{θ}}{− log |(f^{n})′(x)| + log δ} \frac{log r_{n}}{log r_{n+k}} .

(5.27)

Now, the aim is to show that if x ∈ K_{ϕ}(α) then it follows that

lim_{n→∞} \frac{log r_{n}}{log r_{n+k_{n}}} = 1.  (5.28)

In order to prove this, we proceed as follows

lim_{n→∞} \frac{log r_{n}}{log r_{n+k}} = lim_{n→∞} \frac{log |(f^{n})′(x)|}{log |(f^{n+k})′(x)|} = lim_{n→∞} (1 − \frac{log |(f^{k})′(f^{n}(x))|}{log |(f^{n+k})′(x)|})

= 1 + lim_{n→∞} \frac{S_{n+k}(−ϕ)(x)|}{log |(f^{n+k})′(x)|} \frac{log |(f^{k})′(f^{n}(x))|}{S_{n+k}ϕ(x)} = 1 + α lim_{n→∞} \frac{log |(f^{k})′(f^{n}(x))|}{S_{n+k}ϕ(x)}.

Now note that we have, for some universal constant C ≥ 1,

− log C ≤ log |(f^{k})′(f^{n}(x))| ≤ log C + \frac{p_{max} + 1}{p_{max}} log(k + 1).  (5.29)

Using the fact that \overline{ϕ} < 0, Lemma 2.6 implies that there exist q ∈ IN and ϵ > 0 small such that S_{q}ϕ(z) ≤ −ϵq, for all z ∈ J(f). Therefore, we have S_{n+k}ϕ(x) ≤ −ϵ(n + k) + q||ϕ||_{∞}, and combining this with (5.29), we obtain

lim_{n→∞} \frac{log |(f^{k_{n}})′(f^{n}(x))|}{S_{n+k}ϕ(x)} = 0,

which proves (5.28). For x ∈ K_{ϕ}(α), we can now combine (5.28), (5.24), (5.25) and (5.27), which implies

lim_{r→0} \frac{log(m_{ϕ}(B(x, r)))}{log r} = α,

and which therefore gives that x ∈ L_{m_{ϕ}}(α).

Lemma 5.9. For each q ∈ (0, 1], we have that ℓ_{m_{ϕ}}(−T′(q)) ≤ −qT′(q) + T (q).

Proof. Let x ∈ L_{m_{ϕ}}(−T′(q)) \ ⋃_{n≥0} f^{−n}(Ω) be fixed. For r > 0, let n = n_{j} be determined as in the proof of the previous lemma. Using Koebe’s distortion theorem, we have that B(x, r_{n}) ⊃ f_{x}^{−n}(B(f^{n}(x), Kδ). Therefore, applying Lemma 2.5 and once more Koebe’s distortion theorem, we obtain for the equilibrium state m_{q} of the potential −T (q) log |f′|+qϕ,

\begin{aligned}m_{q}(B(x, r_{n})) & ≫ exp(S_{n}(qϕ)(x))|(f^{n})′(x)|^{−T (q)}m_{q}(B(f^{n}(x), Kδ)) \\ & ≫ exp^{q}(S_{n}ϕ(x))|(f^{n})′(x)|^{−T (q)} ≫ m_{ϕ}^{q}(B(x, r_{n}))|(f^{n})′(x)|^{−T (q)}. \end{aligned}

In here the second inequality sign follows since inf{m_{q}(B(z, Kδ) : z ∈ J(f)} > 0. Hence, we now have that

\underline{lim}_{r→0} \frac{log(m_{q}(B(x, r)))}{log r} ≤ \underline{lim}_{n→∞} \frac{log(m_{q}(B(x, r_{n})))}{log r_{n}}

≤ lim_{n→∞} \frac{q log(m_{ϕ}(B(x, r_{n}))) − T (q) log |(f^{n})′(x)|}{log r_{n}} = −qT′(q) + T (q).

Theorem 5.10. Let f be a parabolic GPL-map, and ϕ : J(f) → IR a Hölder continuous potential with Hölder exponent exceeding p_{max}/(p_{max} + 1) such that P_{top}(f, ϕ) > \overline{ϕ}. Then the following holds.

(a) For q ∈ (0, 1), the function q ↦ T (q) is real analytic and T′(q) < 0.

(b) For every q ∈ (0, 1), we have that ℓ_{μ_{ϕ}}(−T′(q)) = T (q) − qT′(q).

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Bernd O. Stratmann; Mathematical Institute, University of St Andrews, St Andrews

KY16 9SS, Scotland. bos@maths.st-and.ac.uk, http://www.maths.st-and.ac.uk/∼bos

Mariusz Urbański; Department of Mathematics, University of North Texas, P.O. Box

311430, Denton, TX 76203-1430, USA. urbanski@unt.edu, http://www.math.unt.edu/∼urbanski