THE EQUILIBRIUM STATES FOR SEMIGROUPS OF RATIONAL MAPS

HIROKI SUMI AND MARIUSZ URBAŃSKI

Abstract. We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)-conformal measure, where P(ψ) denotes the topological pressure of ψ.

1. Introduction

In this paper, we frequently use the notation from [13]. A “rational semigroup” G is a semigroup generated by a family of non-constant rational maps g : \overline{C I} → \overline{C I}, where \overline{C I} denotes the Riemann sphere, with the semigroup operation being functional composition. For a rational semigroup G, we set F (G) := \{z ∈ \overline{C I} | G is normal in a neighborhood of z\} and J(G) := \overline{C I} \ F (G). F (G) is called the Fatou set of G and J(G) is called the Julia set of G. If G is generated by a family {f_{i}}_{i}, then we write G = ⟨f_{1}, f_{2}, …⟩.

The research on the dynamics of rational semigroups was initiated by Hinkkanen and Martin ([8]), who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups, and by F. Ren’s group ([21]), who studied such semigroups from the perspective of random complex dynamics. For further studies on the dynamics of rational semigroups, see [13, 14, 15, 16, 17, 18, 12, 19].

The theory of the dynamics of rational semigroups is deeply related to that of the fractal geometry. In fact, if G = ⟨f_{1}, …, f_{s}⟩ is a finitely generated rational semigroup, then the Julia set J(G) of G has the “backward self-similarity”, i.e.,

Key words and phrases. rational semigroup, skew product, equilibrium state.

Date: August 19, 2008. Published in Monatsh. Math., 156 (2009), no. 4, 371– 390. 2000 Mathematics Subject Classification. Primary: 37F10, Secondary: 37A35.

The research of the second author was supported in part by the NSF Grant DMS 0400481. The first author thanks University of North Texas for kind hospitality, during his stay there.

J(G) = f_{1}^{−1}(J(G)) ∪ ⋯ ∪ f_{s}^{−1}(J(G))

(See [13]). Hence, the behavior of the (backward) dynamics of finitely generated rational semigroups can be regarded as that of the “backward iterated function systems.” For example, the Sierpiński gasket can be regarded as the Julia set of a rational semigroup.

The research on the dynamics of rational semigroups is also directly related to that on the random dynamics of holomorphic maps. The first study on the random dynamics of holomorphic maps was by Fornaess and Sibony ([7]), and much research has followed. (See [1, 2, 3, 4].) In fact, it is very natural and important to combine both the theory of rational semigroups and that of random complex dynamics (see [13, 14, 16, 17, 18]).

We use throughout spherical derivatives and, for each meromorphic function φ, we denote by |φ′(z)| the norm of the derivative with respect to the spherical metric. We denote by CV (φ) the set of critical values of φ. The symbol A is used to denote the spherical area. We set κ = A(B(0, 1))/4. Obviously, there exists a constant C_{sa} ≥ 1 such that for each 0 < R ≤ diam(\overline{C I})/2, C_{sa}^{−1} ≤ A(B(z, R))/κ(2R)^{2} ≤ C_{sa}.

Let G = ⟨f_{1}, …, f_{s}⟩ be a finitely generated rational semigroup. Then, we use the following notation. Let Σ_{s} := {1, …, s}^{ℕ} be the space of one-sided sequences of s-symbols endowed with the product topology. This is a compact metric space. Let f : Σ_{s} × \overline{C I} → Σ_{s} × \overline{C I} be the skew product map associated with {f_{1}, …, f_{s}} given by the formula, f(ω, z) = (σ(ω), f_{ω_{1}}(z)), where (ω, z) ∈ Σ_{s} × \overline{C I}, ω = (ω_{1}, ω_{2}, …), and σ : Σ_{s} → Σ_{s} denotes the shift map. We denote by π_{1} : Σ_{s} × \overline{C I} → Σ_{s} the projection onto Σ_{s} and π_{2} : Σ_{s} × \overline{C I} → \overline{C I} the projection onto \overline{C I}. That is, π_{1}(ω, z) = ω and π_{2}(ω, z) = z. Under the canonical identification π_{1}^{−1}\{ω\} ≅ \overline{C I}, each fiber π_{1}^{−1}\{ω\} is a Riemann surface which is isomorphic to \overline{C I}.

Let Crit(f) := ⋃_{ω∈Σ_{s}}{v ∈ π_{1}^{−1}{ω} | v is a critical point of f|_{π_{1}^{−1}{ω}} → π_{1}^{−1}\{σ(ω)\}\} (⊂ Σ_{s} × \overline{C I}) be the set of critical points of f. For each n ∈ ℕ and (ω, z) ∈ Σ_{s} × \overline{C I}, we set (f^{n})′(ω, z) := (f_{ω_{n}} ∘ ⋯ ∘ f_{ω_{1}})′(z).

For each ω ∈ Σ_{s} we define

J_{ω} := \{z ∈ \overline{C I} | \{f_{ω_{n}}∘⋯∘f_{ω_{1}}\}_{n∈ℕ} is not normal in any neighborhood of z\}

and we then set

J(f) := \overline{∪_{w∈Σ_{s}}\{ω\} × J_{ω}},

where the closure is taken in the product space Σ_{s} × \overline{C I}. By definition, J(f) is compact. Furthermore, by Proposition 3.2 in [13], J(f) is completely invariant under f, f is an open map on J(f), (f, J(f)) is topologically exact under a mild condition, and J(f) is equal to the closure of the set of repelling periodic points of f provided that ♯J(G) ≥ 3, where we say that a periodic point (ω, z) of f with period n is repelling if |(f^{n})′(ω, z)| > 1. Furthermore, π_{2}(J(f)) = J(G). We set

e_{j} := deg(f_{j}) and d := max_{j=1,...,s}(2e_{j} − 2).

Throughout the paper, we assume the following.

(E1) There exists an element g ∈ G with deg(g) ≥ 2. Furthermore, for the semigroup H := \{h^{−1} | h ∈ Aut\overline{C I} ∩ G\}, we have J(G) ⊂ F (H). (If H is empty, we put F (H) := \overline{C I}.)

(E2) ♯(Crit (f) ∩ J(f)) < ∞

(E3) There is no super attracting cycle of f in J(f). That is, if (ω, z) ∈ J(f) is a periodic point with period n, then (f^{n})′(ω, z) ≠ 0.

Any finitely generated rational semigroup G = ⟨f_{1}, …, f_{s}⟩ satisfying all the conditions (E1)-(E3) will be called an E-semigroup of rational maps.

Examples.

• If g is a rational map with deg(g) ≥ 2, then ⟨g⟩ is an E-semigroup of rational maps.

• Let G = ⟨f_{1}, …, f_{s}⟩ be a finitely generated rational semigroup such that deg(f_{j}) ≥ 2 for each j = 1, …, s. If CV(f_{j})∩J(G) = ∅ for each j = 1, …, s, then G is an E-semigroup of rational maps.

• Let g_{1} and g_{2} be two polynomials with deg(g_{j}) ≥ 2 (j = 1, 2). Suppose that

\overline{∪_{n=0}^{∞}g_{1}^{n}(CV (g_{1}))} ⊂ A_{∞}(g_{2}) and \overline{∪_{n=0}^{∞}g_{2}^{n}(CV (g_{2}))} ⊂ A_{∞}(g_{1}),

where A_{∞}(·) denotes the basin of infinity. Let m ∈ ℕ be a sufficiently large number so that

g_{1}^{m}(\overline{∪_{n=0}^{∞}g_{2}^{n}(CV (g_{2}))}) ⊂ U_{∞} and g_{2}^{m}(\overline{∪_{n=0}^{∞}g_{1}^{n}(CV (g_{1}))}) ⊂ U_{∞},

where U_{∞} denotes the connected component of F (⟨g_{1}, g_{2}⟩) with ∞ ∈ U_{∞}. Then, the semigroup ⟨g_{1}^{m}, g_{2}^{m}⟩ is an E-semigroup of rational maps.

The dynamics of the skew product map f : Σ_{s} × \overline{C I} → Σ_{s} × \overline{C I} is directly related to the dynamics of the semigroup G = ⟨f_{1}, …, f_{s}⟩. For example, we use the dynamics of f to analyze the dimension of Julia set J(G) of G (see [13, 14, 15, 16, 19]).

Our main concern in this paper is the existence and uniqueness of equilibrium states for all E-semigroups of rational maps and a large class of Hölder continuous potentials. We do not assume any kind of expanding property. If T : X → X is a continuous mapping of a compact metric space X, then one can define the topological pressure P(g) for every real-valued continuous function g on X (see [20] for instance) in purely topological terms. The link with measurable dynamics is given by the Variational Principle (see [20] again) saying that the topological pressure P(g) is equal to the supremum of all numbers h_{μ}(T) + ∫ gdμ, μ being Borel probability T -invariant measures on X. Any invariant measure μ satisfying equality

P(g) = h_{μ}(T) + ∫ gdμ

is called an equilibrium state. All equilibrium states have a profound physical meaning (see [11]), their existence as well as uniqueness is of primary importance and is in general not clear at all. A general sufficient condition for the existence is expansiveness, a weaker one, holding for all rational maps on the sphere (see [9]), is asymptotic h-expansiveness. This covers the case of hyperbolic maps studied in [15]. Our goal in this paper is to do both, existence and uniqueness of equilibrium states, for E-semigroup of rational maps and a large class of Hölder continuous potentials. Thus, the main purpose of this paper is to prove the following result.

Theorem 1.1. (Main result) Let G = ⟨f_{1}, …, f_{s}⟩ be an E-semigroup of rational maps. Let f : Σ_{s} × \overline{C I} → Σ_{s} × \overline{C I} be the skew product map associated with {f_{1}, …, f_{s}}. Let ψ : J(f) → ℝ be a Hölder continuous function. Moreover, let P^{p}(ψ) be the pointwise pressure of ψ with respect to the dynamics of f : J(f) → J(f) (see the definition (4.2)). Suppose that P^{p}(ψ) > sup(ψ) + log s. Then, all of the following statements hold.

(1) Regarding the dynamics of f : J(f) → J(f), there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)- conformal measure in the sense of [5], where P(ψ) denotes the pressure of (f|_{J(f)}, ψ).

(2) Moreover, P(ψ) = P^{p}(ψ) and for each point x ∈ J(f), we have that \frac{1}{n} log L_{ψ}^{n}11(x) → P(ψ) = P^{p}(ψ) as n → ∞, where L_{ψ} denotes the Perron-Frobenius operator associated with the potential ψ (see (4.1)) and 11 ≡ 1.

Remark 1.2. If ψ : J(f) → ℝ is a Hölder continuous function such that sup(ψ)−inf(ψ) < log(∑_{j=1}^{s} e_{j})−log s, then P^{p}(ψ) > sup(ψ)+log s.

Remark 1.3. Under a very mild condition, the topological entropy h(f) of f : Σ_{s} ×\overline{C I} → Σ_{s} ×\overline{C I} is equal to log(∑_{j=1}^{s} e_{j}) and there exists a unique maximal entropy measure for f : Σ_{s} × \overline{C I} → Σ_{s} × \overline{C I} (See [13]).

The proof of the main result is given in the following several sections. In this proof we utilize some arguments from [5] to show the existence of a conformal measure. Then, using the normality of a family of inverse branches of elements of the semigroup (see section 2), we analyze the Perron-Frobenius operator in detail. Developing the techniques worked out in [6] and [13], we show the existence of an equilibrium state and the uniqueness of a conformal measure as well as an equilibrium state.

2. Distortion Theorems

Let us recall the following well-known version of Koebe’s Distortion Theorem concerning spherical derivatives.

Theorem 2.1. For every u ∈ (0, diam(\overline{C I})/2) there exists a function k_{u} : [0, 1) → (0, +∞), continuous at 0, with k_{u}(0) = 1 and the following property. If ξ ∈ \overline{C I}, R > 0, and H : B(ξ, R) → \overline{C I} is a meromorphic univalent function such that \overline{C I}\ H(B(ξ, R)) contains a ball of radius u, then for every t ∈ [0, 1) and all z, w ∈ B(ξ, tR),

k_{u}^{−1}(t) ≤ \frac{|H′(w)|}{|H′(z)|} ≤ k_{u}(t).

As an immediate consequence of this theorem, combined with Lemma 4.5 in [13], we get the following.

Lemma 2.2. Let G = ⟨f_{1}, …, f_{m}⟩ be a finitely generated rational semigroup satisfying the condition (E1). Then, there exists a number R_{0} > 0 and a function k_{G} : [0, 1) × (0, R_{0}] → [1, ∞) such that for each x ∈ J(G), 0 < R ≤ R_{0} and 0 ≤ t < 1, the family F_{x,R} := {φ : B(x, R) → \overline{C I} | φ is a well-defined inverse branch of h, h ∈ G\} satisfies that for each H ∈ F_{x,R}, w, z ∈ B(x, tR), we have

k_{G}(t, R)^{−1} ≤ \frac{|H′(w)|}{|H′(z)|} ≤ k_{G}(t, R).

Proof. By Lemma 4.5 in [13], we have that there exists a number R_{0} > 0 such that for each x ∈ J(G) and each 0 < R ≤ R_{0}, the family F_{x,R} is normal in B(x, R). Now, suppose the statement of our lemma is false. Then, there exist a 0 < t < 1, a sequence (x_{n}) in J(G), a sequence (w_{n}) in \overline{C I}, a sequence (z_{n}) in \overline{C I}, a number 0 < R ≤ R_{0}, and a sequence (φ_{n}) of meromorphic functions, such that for each n ∈ ℕ, we have w_{n}, z_{n} ∈ B(x_{n}, tR), φ_{n} ∈ F_{x_{n},R}, and \frac{|φ′_{n}(w_{n})|}{|φ′_{n}(z_{n})|} ≥ n. We may assume x_{n} → x_{∞} ∈ J(G), w_{n} → w_{∞} ∈ \overline{C I}, and z_{n} → z_{∞} ∈ \overline{C I}. Let u be a number with t < u < 1. Then there exists an n_{0} ∈ ℕ such that each φ_{n} (n ≥ n_{0}) is defined on B(x_{∞}, uR) and the family {φ_{n}}_{n≥n_{0}} is normal in B(x_{∞}, uR). Then we may assume that φ_{n} tends to a meromorphic function φ_{∞} : B(x_{∞}, uR) → \overline{C I} as n → ∞, uniformly on B(x_{∞}, uR). Since each φ_{n} is injective, we have that φ_{∞} : B(x_{∞}, uR) → \overline{C I} is either injective or constant. Hence, φ_{∞}(B(x_{∞}, uR)) ≠ \overline{C I}. Let v be a number with t < v < u. Then, there exist a number 0 < s < 1 and a point a ∈ \overline{C I} such that for each large n ≥ n_{0},

B(a, s) ⊂ \overline{C I} \ φ_{n}(B(x_{∞}, vR)).

Then, by Theorem 2.1, it causes a contradiction. We are done.

Notation. Throughout the rest of the paper, we set K_{G}(R) := k_{G}(\frac{1}{2}, R).

3. Inverse Branches

We set Σ_{s}^{∗} := ∪_{j=1}^{∞}{1, …, s}^{j} (disjoint union). For each ω ∈ Σ_{s}^{∗}, we set |ω| = j if ω ∈ {1, …, s}^{j}. By Σ̂_{s}^{+} we denote the space dual to Σ_{s}, that is Σ̂_{s}^{+} consists of infinite sequences ω = …ω_{3}ω_{2}ω_{1} of elements from the set {1, 2, …, s}. For each ω ∈ Σ̂_{s}^{+}, by ω|_{n} we denote the finite word (ω_{n}, ω_{n−1}, …, ω_{2}, ω_{1}); more generally, for b ≥ a we put ω|_{b}^{a} = (ω_{b}, ω_{b−1}, …, ω_{a}). For each finite word ω = (ω_{n}, …, ω_{1}), we set f_{ω} := f_{ω_{1}} ∘ ⋯ ∘ f_{ω_{n}}. The technical tool that allows us to develop the further machinery is the following.

Lemma 3.1. Let G = ⟨f_{1}, …, f_{s}⟩ be a rational semigroup satisfying the condition (E1). Let R_{0} be the number in Lemma 2.2. Fix an integer q ≥ 1and a real number λ ∈ (0, 1). Then for every finite set E ⊂ J(G), every ω ∈ Σ̂_{s}^{+}, every R ∈ (0, min \{1, \frac{1}{2}R_{0}, κ^{−1/2}, \frac{1}{2}dist(E, CV (f_{ω|_{q}})), 4^{−1} min\{d(z, ξ) : z, ξ ∈ E, z ≠ ξ\}\}) , every integer n ≥ 0, and every z ∈ E, there exists a subset I_{n}(z, ω) of the set of all inverse meromorphic branches of f_{ω|_{qn}} defined on B(z, 2R) and satisfying the following properties with I_{n} = ⋃_{z∈E} I_{n}(z, ω).

(a_{n}) If z ∈ E and ϕ ∈ I_{n+1}(z, ω), then f_{ω|_{q(n+1)}^{qn+1}} ∘ ϕ ∈ I_{n}(z, ω).

(b_{n}) If ϕ ∈ I_{n}, then diam(ϕ(B(z, R))) ≤ K_{G}(2R)^{2}κ^{−1/2}λ^{n/2}C_{sa}^{1/2}.

(c_{n}) ϕ(B(z, 2R)) ∩ CV (f_{ω|_{q(n+1)}^{qn+1}}) = ∅ for all z ∈ E and all ϕ ∈ I_{n}.

(d_{n}) #(J_{n} \ I_{n}) ≤ dq + λ^{−n} for all n ≥ 1, where J_{n} is the family of all compositions of all maps ϕ ∈ I_{n−1}(z, ω), z ∈ E, with all meromorphic inverse branches of f_{ω|_{qn}^{q(n−1)+1}}.

(e) I_{0} = {Id|_{B(z,2R)} | z ∈ E}.

Proof. We shall construct recursively the sets I_{n}(z, ω), n ≥ 0, such that the conditions (a_{n}), (b′_{n}), (c_{n}) and (d_{n}) (here n ≥ 1), where (b′_{n}) requires that

(b′_{n}) If ϕ ∈ I_{n}, then A(ϕ(B(z, R))) ≤ λ^{n}. The base of induction, the family I_{0} consists of all the identity maps defined on the balls B(z, 2R), z ∈ E. The condition (b′_{0}) is satisfied since A(\overline{C I}) = 1 and (c_{0}) is satisfied because of the choice of the radius R. Now assume that for some n ≥ 0 the subsets I_{n}(z, ω), z ∈ E, have been constructed so that the conditions (b′_{n}) and (c_{n}) are satisfied. The inductive step is to construct the subsets I_{n+1}(z, ω), z ∈ E, so that the conditions (a_{n+1}), (b′_{n+1}), (c_{n+1}) and (d_{n+1}) are satisfied. This will complete our recursive construction. In view of (c_{n}) all the meromorphic inverse branches of f_{ω|_{q(n+1)}^{qn+1}} are well defined on all the sets ϕ(B(z, 2R)), z ∈ E, ϕ ∈ I_{n}(z, ω). Their compositions with corresponding elements ϕ ∈ I_{n}(z, ω) are said to form the subset J_{n+1}(z, ω). Note that J_{n+1} = ⋃_{z∈E} J_{n+1}(z, ω). The subset I_{n+1}(z, ω), z ∈ E, is defined to consist of all the elements ψ ∈ J_{n+1}(z, ω) for which the following two conditions are satisfied.

(i) A(ψ(B(z, R))) ≤ λ^{n+1}.

(ii) ψ(B(z, 2R)) ∩ CV (f_{ω|_{q(n+2)}^{q(n+1)+1}}) = ∅.

Thus, conditions (b′_{n+1}) and (c_{n+1}) are satisfied immediately. Condition (a_{n}) is satisfied since it holds for all ψ ∈ J_{n+1}(z, ω), z ∈ E, and I_{n+1}(z, ω) is a subset of J_{n+1}(z, ω). We are left to show that (d_{n+1}) holds. Now, if ψ ∈ J_{n+1}, say ψ ∈ J_{n+1}(z, ω), z ∈ E, but (i) is not satisfied, then A(ψ(B(z, R))) > λ^{n+1}. Since all the sets ψ(B(z, R)), ψ ∈ J_{n+1}(z, ω), z ∈ E, are mutually disjoint and since A(\overline{C I}) = 1, we conclude that the number of elements of J_{n+1} for which condition (i) fails is bounded above by 1/λ^{n+1} = λ^{−(n+1)}. Since the number of critical points of each generator of the semigroup G is bounded above by d, the cardinality of the set of critical values of f_{ω|_{q(n+2}^{q(n+1)+1}} is bounded

above by dq. Since all the sets ψ(B(z, 2R)), ψ ∈ J_{n+1}(z, ω), z ∈ E, are mutually disjoint, we thus conclude that the number of elements ψ ∈ J_{n+1} for which condition (ii) fails, is bounded above by dq. In conclusion #(J_{n+1} \ I_{n+1}) ≤ dq + λ^{−(n+1)}, meaning that (d_{n+1}) is satisfied. The recursive construction is complete. Since G satisfies (E1), it follows from Lemma 2.2 that for all n ≥ 0, all z ∈ E, and all ϕ ∈ I_{n}(z, ω), we have

B(ϕ(z), K_{G}(2R)^{−1}|ϕ′(z)|R) ⊂ ϕ(B(z, R)) ⊂ B(ϕ(z), K_{G}(2R)|ϕ′(z)|R).

Hence, making use of (b′_{n}), we get that

diam^{2}(ϕ(B(z, R))) ≤ (2K_{G}(2R)|ϕ′(z)|R)^{2}
= K_{G}(2R)^{4}κ^{−1}(2κ^{1/2}K_{G}(2R)^{−1}|ϕ′(z)|R)^{2}

= K_{G}(2R)^{4}κ^{−1}C_{sa}A(B(ϕ(z), K_{G}(2R)^{−1}|ϕ′(z)|R))

≤ K_{G}(2R)^{4}κ^{−1}C_{sa}A(ϕ(B(z, R)))
≤ K_{G}(2R)^{4}κ^{−1}C_{sa}λ^{n}.

Thus, diam(ϕ(B(z, R))) ≤ K_{G}(2R)^{2}κ^{−1/2}C_{sa}^{1/2}λ^{n/2}. We are done.

In order to simplify the notation, put R(E, ω|_{q}) :=

min \{1, \frac{1}{2}R_{0}, κ^{−1/2}, \frac{1}{2}dist(E, CV (f_{ω|_{q}})), 4^{−1} min\{d(z, ξ) : z, ξ ∈ E, z ≠ ξ\}\} .

Corollary 3.2. Suppose that G = ⟨f_{1}, …, f_{s}⟩ is an E-semigroup of rational maps. Fix an integer q ≥ 1and a real number λ ∈ (0, 1). Then for every z ∈ J(G), there exists a number R = R_{q}(z) > 0, a number R′_{q}(z) > 0, and a number D ≥ 1, where D does not depend on q, λ, z, such that for every ω ∈ Σ̂_{s}^{+} and every integer n ≥ 1, there exists W_{n}(z, ω), Z_{n}(z, ω), a subset of the set of all connected components of f_{ω|_{qn}}^{−1}(B(z, R)), with the following properties.

(A_{n}) If V ∈ W_{n+1}(z, ω), then f_{ω|_{q(n+1)}^{qn+1}}(V) ∈ W_{n}(z, ω).

(B_{n}) If V ∈ W_{n}(z, ω), then diam(V) ≤ K_{G}(R′_{q}(z))^{4}κ^{−1/2}λ^{n/2}C_{sa}^{1/2}. (C_{n}) Z_{n}(z, ω) is the family of all connected components of the sets

f_{ω|_{qn}^{q(n−1)+1}}^{−1}(V), V ∈ W_{n−1}(z, ω), and #(Z_{n}(z, ω) \ W_{n}(z, ω)) ≤

dq + λ^{−n}, and

(D_{n}) For every V ∈ Z_{n}(z, ω), the map f_{ω|_{qn}}|_{V} : V → \overline{C I} is at most

D-to-1. (E_{1}) W_{1}(z, ω) = Z_{1}(z, ω) = {connected components V of f_{ω|_{q}}^{−1}(B(z, R))}.

Note also that in fact W_{n}(z, ω) depends only on z and ω|_{qn}, so we can and will in the forthcoming sections write W_{n}(z, ω|_{qn})

Proof. Let z ∈ J(G) be a point. Since J(G) = π_{2}(J(f)), there exists a point ω ∈ Σ_{s} such that (ω, z) ∈ J(f). Then, from the assumption E2, E3, there exists p ≥ 1 independent of q (but depending on (ω, z)) such that

( ⋃_{r∈ℕ}(f^{pq+r})^{−1}(ω, z)) ∩ Crit(f) = ∅.

Then, we obtain

⋃_{ρ∈Σ_{s}^{∗}} ⋃_{|τ|=pq} f_{ρ}^{−1}(f_{τ}^{−1}(z)) ∩ Crit(f_{ρ}) = ∅.

In particular

( ⋃_{|τ|=pq} f_{τ}^{−1}(z)) ∩ CV (f_{ρ}) = ∅  (3.1)

for all ρ ∈ Σ_{s}^{∗}. Set E = ⋃_{|τ|=pq} f_{τ}^{−1}(z). It follows from (3.1) that

R̂_{q}(z) = \frac{1}{2}λ^{p/2} · min\{R(E, ρ) : ρ ∈ \{1, 2, …, s\}^{q}\} > 0.

Now, there is R_{q}(z) > 0 so small that the following two conditions are satisfied.

(a) For each τ ∈ {1, 2, …, s}^{pq} each connected component of f_{τ}^{−1}(B(z, R_{q}(z))) is contained in exactly one ball B(ξ, R̂_{q}(z)), where ξ ∈ E.

(b) For each γ ∈ Σ_{s}^{∗} with |γ| ≤ pq, each connected component of f_{γ}^{−1}(B(z, R_{q}(z))) has the diameter bounded above by κ^{−1/2}λ^{|γ|/2}.

Now, for every 1 ≤ k ≤ p and every ω ∈ Σ̂_{s}^{+}, define W_{k}(z, ω) and Z_{k}(z, ω) to be the family of all connected components of f_{ω|_{qk}}^{−1}(B(z, R_{q}(z))). The conditions (A_{k}), (B_{k}), and (C_{k}) are obviously satisfied for all 1 ≤ k ≤ p. Now, for every ω ∈ Σ̂_{s}^{+}, every n ≥ p + 1 and every ξ ∈ f_{ω|_{pq}}^{−1}(z) ⊂ E, consider the inverse branch ϕ : B(ξ, 2λ^{−p/2}R̂_{q}(z)) → \overline{C I} ∈ I_{n−p}(ξ, ω|_{∞}^{pq}). It follows from Lemma 3.1(b_{n−p}) and Lemma 2.2 that

\begin{aligned}2K_{G}(2λ^{− \frac{p}{2}} R̂_{q}(z))^{−1}λ^{−p/2}|ϕ′(ξ)|R̂_{q}(z) & ≤ diam(ϕ(B(ξ, λ^{−p/2}R̂_{q}(z)))) \\ & ≤ K_{G}(2λ^{− \frac{p}{2}} R̂_{q}(z))^{2}κ^{−1/2}λ^{\frac{n−p}{2}} C_{sa}^{1/2}. \end{aligned}

Thus |ϕ′(ξ)| ≤ \frac{1}{2}K_{G}(2λ^{− \frac{p}{2}} R̂_{q}(z))^{3}κ^{−1/2}λ^{n/2}R̂_{q}(z)^{−1}C_{sa}^{1/2} , and therefore, using Lemma 2.2 again and the definition of R̂_{q}(z), we get that

diam(ϕ(B(ξ, R̂_{q}(z)))) ≤ K_{G}(2λ^{− \frac{p}{2}} R̂_{q}(z))^{4}κ^{−1/2}λ^{n/2}C_{sa}^{1/2}.

So, looking also at Lemma 3.1(d_{n}) and (c_{n}), we complete the proof of items (A_{n}), (B_{n}) and (C_{n}) by defining for every n ≥ p + 1 the family W_{n}(z, ω) to consist of all the sets of the form ϕ(V_{ξ}), where ξ ∈ f_{ω|_{pq}}^{−1}(z), V_{ξ} is a connected component of f_{ω|_{pq}}^{−1}(B(z, R_{q}(z))) contained in B(ξ, R̂_{q}(z)), and ϕ ∈ I_{n−p}(ξ, ω_{∞}^{pq}). Decreasing R_{q}(z) appropriately, the items (D_{n}) follow now from this construction and Lemma 3.1.

4. Perron-Frobenius Operators and Gibbs States

From now on throughout the entire paper assume that ψ : J(f) → IR is a Hölder continuous function. Given n ≥ 1, ω ∈ {1, …, s}^{n}, and a continuous function g : J(f) → IR, define

L_{ψ,ω}^{(n)}g(τ, z) = ∑_{x∈f_{ω}^{−1}(z)} exp(S_{n}ψ(ωτ, x))g(ωτ, x),

where here and in the sequel the summation is taken with multiplicities of all critical points of f_{ω}, and S_{n}ψ := ∑_{j=0}^{n−1} ψ∘f^{j}. Since sup{#(f_{ω}^{−1}(z)) : z ∈ C I} < ∞, L_{ψ,ω}^{(n)} is a bounded linear operator acting on the Banach space C(J(f)) of continuous functions on J(f) endowed with the supremum norm. Set

L_{ψ}^{(n)} := ∑_{|ω|=n} L_{ψ,ω}^{(n)}, L_{ψ} := L_{ψ}^{(1)}.  (4.1)

Then L_{ψ}^{(n)} and L_{ψ} also act continuously on the Banach space C(J(f)). We call L_{ψ} : C(J(f)) → C(J(f)) the Perron-Frobenius operator associated with the potential ψ. Note that

L_{ψ}^{(n)}g(x) = ∑_{y∈f^{−n}(x)} exp(S_{n}ψ(y))g(y) = L_{ψ}^{n}g(x).

Define the pointwise pressure P^{p}(ψ) of the function ψ by the following formula.

P_{x}(ψ) := lim sup_{n→∞} \frac{1}{n} log L_{ψ}^{n}11(x), x ∈ J(f), and
P^{p}(ψ) := sup{P_{x}(ψ) : x ∈ J(f)},

(4.2)

where 11(x) := 1. Throughout the entire paper we work with the assumption that

P^{p}(ψ) > sup(ψ) + log s.  (4.3)

In particular, we can fix a point b ∈ J(f) such that

η := exp(sup(ψ) + log s − P_{b}(ψ)) < 1.  (4.4)

Fix λ ∈ (0, 1). There then exists q = q(λ) ≥ 1 such that

γ_{q}^{−1} := dq + λ^{−1} ≤ (1 + 2D)^{−1}η^{−q}.  (4.5)

Let L_{ψ}^{∗} : C(J(f))^{∗} → C(J(f))^{∗} be the operator conjugate to L_{ψ}, i.e., L_{ψ}^{∗}ν(g) = ν(L_{ψ}g). Using (4.4) and the fact that the map f : J(f) → J(f) is open, as an immediate consequence of Theorem 3.9 and Proposition 2.2 in [5], we get the following.

Lemma 4.1. If G = ⟨f_{1}, …, f_{s}⟩ is an E-semigroup of rational maps, ψ : J(f) → IR is a Hölder continuous potential satisfying (4.3) and b ∈ J(f) is selected so that (4.4) holds, then there exists a Borel atomless probability measure m_{ψ} on J(f) such that L_{ψ}^{∗}m_{ψ} = e^{P_{b}(ψ)}m_{ψ}.

The measure m_{ψ} is called exp(P_{b}(ψ) − ψ)-conformal for f. Obviously (L_{ψ}^{∗})^{n}m_{ψ} = e^{P_{b}(ψ)n}m_{ψ} for all n ≥ 0, and this equivalently means that

m_{ψ}(f^{n}(A)) = ∫_{A} exp(P_{b}(ψ)n − S_{n}ψ)dm_{ψ}

for every Borel set A ⊂ J(f) for which the restriction f^{n}|_{A} is injective.

Remark 4.2. Note that all forthcoming considerations depend only on the above relation and not on the particular way the measure m_{ψ} was constructed.

From now on throughout the paper put

\overline{ψ} = ψ − P_{b}(ψ).

Now set

L_{ω}^{(n)} = L_{\overline{ψ},ω}^{(n)}, L^{n} = Ln ; \overline{ψ}, and L = L\overline{ψ}.

Now for every z ∈ J(G), every n ≥ 1, every ω ∈ {1, …, s}^{qn}, and every g ∈ C(J(f)), define the function G_{z,ω}^{n}g : π_{2}^{−1}(B(z, R_{q}(z))) → IR by setting

G_{z,ω}^{n}g(τ, ξ) = ∑_{V ∈W_{n}(z,ω)} ∑_{x∈f_{ω}^{−1}(ξ)∩V} exp(S_{qn}\overline{ψ}(ωτ, x))g(ωτ, x), (4.6)

where R_{q}(z) and W_{n}(z, w) come from Corollary 3.2. Since for every V ∈ W_{n}(z, ω) the map f_{ω} : V → B(z, R_{q}(z)) is a branched covering, for every ξ ∈ B(z, R_{q}(z)) there is a bijection ξ̂: f_{ω}^{−1}(z) ∩ V → f_{ω}^{−1}(ξ) ∩ V, where all critical points of f_{ω} in f_{ω}^{−1}(z) and f_{ω}^{−1}(ξ) are counted with multiplicities. So, (4.6) can be rewritten in the following form.

G_{z,ω}^{n}g(τ, ξ) = ∑_{V ∈W_{n}(z,ω)} ∑_{x∈f_{ω}^{−1}(z)∩V} exp(S_{qn}\overline{ψ}(ωτ, ξ̂(x)))g(ωτ, ξ̂(x)).

Since the function ψ : J(f) → IR is Hölder continuous, it follows from Corollary 3.2(B_{n}) that there exists a constant H > 0 such that (with x ∈ f_{ω}^{−1}(z) ∩ V),

|S_{qn}\overline{ψ}(ωτ, ξ̂(x)) − S_{qn}\overline{ψ}(ωθ, x)| ≤ H

for all τ, θ ∈ Σ_{s}, or equivalently,

e^{−H} exp(S_{qn}\overline{ψ}(ωθ, x)) ≤ exp(S_{qn}\overline{ψ}(ωτ, ξ̂(x))) ≤ e^{H} exp(S_{qn}\overline{ψ}(ωθ, x)). (4.7)

In consequence

e^{−H}G_{z,ω}^{n}11(θ, z) ≤ G_{z,ω}^{n}11(τ, ξ) ≤ e^{H}G_{z,ω}^{n}11(θ, z).  (4.8)

Set

G_{z}^{n} = ∑_{|ω|=qn} G_{z,ω}^{n}.

It then follows from (4.8) that

e^{−H}G_{z}^{n}11(θ, z) ≤ ∑_{|ω|=qn} G_{z,ω}^{n}11(τ_{ω}, ξ_{ω}) ≤ e^{H}G_{z}^{n}11(θ, z)  (4.9)

for all n ≥ 0, all (θ, z) ∈ J(f) and all (τ_{ω}, ξ_{ω}) ∈ π_{2}^{−1}(B(z, R_{q}(z))). Since J(G) is a compact set, there exist finitely many points, say z_{1}, z_{2}, …, z_{u} ∈ J(G) such that ⋃_{j=1}^{u} B(z_{j}, R_{q}(z_{j})) ⊃ J(G). Put

M_{j,ω}^{n}g = L_{ω}^{(qn)}|_{π_{2}^{−1}(B(z_{j},R_{q}(z_{j})))}, M_{j}^{n}g = ∑_{|ω|=qn} M_{j,ω}^{n}g, G_{j,ω}^{n} = G_{z_{j},ω}^{n}, G_{j}^{n} = G_{z_{j}}^{n},

THE EQUILIBRIUM STATES FOR SEMIGROUPS OF RATIONAL MAPS 13

and

\begin{matrix}|||G_{j}^{n}g|||_{∞} = ∑_{|ω|=qn} ||G_{j,ω}^{n}g||_{∞}, |||M_{j}^{n}g|||_{∞} = ∑_{|ω|=qn} ||M_{j,ω}^{n}g||_{∞}, \\ |||L^{n}g|||_{∞} = ∑_{|ω|=n} ||L_{ω}^{(n)}g||_{∞}. \end{matrix}

We shall prove the following.

Lemma 4.3. It holds

0 < Q_{ψ} := max_{1≤j≤u} sup_{n≥0}{|||G_{j}^{n}11|||_{∞}} < ∞.

Proof. Since the map f : J(f) → J(f) is topologically exact and since {B(z_{j}, R_{q}(z_{j}))}_{j=1}^{u} is an open cover of J(f), there exists k ≥ 1 such that for all j = 1, 2, …, u,

f^{kq}(π_{2}^{−1}(B(z_{j}, R_{q}(z_{j})))) ⊃ J(f).  (4.10)

By Lemma 4.1, L^{∗n}m_{ψ} = m_{ψ} for all n ≥ 0, and consequently ∫ L^{n}11dm_{ψ} = ∫ 11dm_{ψ} = 1. Fix n ≥ 0. There then exists w_{0} ∈ J(f) such that

L^{q(k+n)}11(w_{0}) ≤ 1.  (4.11)

Now fix an arbitrary 1 ≤ j ≤ u, an arbitrary ω ∈ {1, …, s}^{qn}, and an arbitrary x_{ω} ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))). By (4.10) there exists y_{j} ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))) such that f^{kq}(y_{j}) = w_{0}. Applying (4.9) with z = z_{j}, we get that

∑_{|ω|=qn} G_{j,ω}^{n}11(x_{ω}) ≤ e^{2H}G_{j}^{n}11(y_{j}).

Also, by (4.11), we obtain

\begin{aligned}G_{j}^{n}11(y_{j}) & ≤ L^{qn}11(y_{j}) ≤ || exp(S_{kq}(−\overline{ψ}))||_{∞}L^{q(k+n)}11(w_{0}) \\ & ≤ || exp(S_{kq}(−\overline{ψ}))||_{∞}. \end{aligned}

Thus, ∑_{|ω|=qn} G_{j,ω}^{n}11(x_{ω}) ≤ e^{2H}|| exp(S_{kq}(−\overline{ψ}))||_{∞}, and we are done by taking supremum over all x_{ω} ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))).

Now we are in position to prove the required upper bound on the iterates of the Perron-Frobenius operator L.

Lemma 4.4. For every n ≥ 1, we have that

\begin{aligned}|||L^{qn}11|||_{∞} & ≤ Q_{ψ} + D ∑_{k=1}^{n}(η^{q}γ_{q}^{−1})^{k}|||L^{q(n−k)}11|||_{∞} \\ & ≤ Q_{ψ} + D ∑_{k=1}^{n}(1 + 2D)^{−k}|||L^{q(n−k)}11|||_{∞}. \end{aligned}

Proof. Fix n ≥ 1, ω = (ω_{qn}, ω_{qn−1}, …, ω_{1}) ∈ {1, …, s}^{qn}, and (τ, x) ∈ J(f). There then exists 1 ≤ j ≤ u such that (τ, x) ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))). For each a, b ∈ ℕ with a ≤ b ≤ qn, we set ω|_{b}^{a} := (ω_{b}, …, ω_{a}). Moreover, for each l ∈ ℕ with l ≤ qn, we set ω|_{l} = (ω_{l}, …, ω_{1}).

One can now represent L_{ω}^{(qn)}11(τ, x) in the following way.

M_{j,ω}^{n}11(τ, x)

= ∑_{k=1}^{n} ∑_{V ∈Z_{k}(z_{j},ω)\W_{k}(z_{j},ω)} ∑_{y∈V ∩f_{ω|_{qk}}^{−1} (x)} exp(S_{qk}\overline{ψ}(ω|_{qk}τ, y))L_{ω|_{qn}^{qk+1}}^{(q(n−k))}11(ω|_{qk}τ, y)

+ G_{j,ω}^{n}11(τ, x)
≤ ∑_{k=1}^{n} ∑_{V ∈Z_{k}(z_{j},ω)\W_{k}(z_{j},ω)} ∑_{y∈V ∩f_{ω|_{qk}}^{−1} (x)} exp(qk sup(\overline{ψ}))L_{ω|_{qn}^{qk+1}}^{(q(n−k))}11(ω|_{qk}τ, y)

+ ||G_{j,ω}^{n}11||_{∞}.

(4.12)

Applying now Corollary 3.2 (C_{k}) and (D_{k}), we estimate further as follows.

M_{j,ω}^{n}11(τ, x) ≤ ∑_{k=1}^{n} D(dq+λ^{−k}) exp(qk sup(\overline{ψ}))∥L_{ω|_{qn}^{qk+1}}^{(q(n−k))}11∥_{∞}+||G_{j,ω}^{n}11||_{∞}.

Since dq + λ^{−k} ≤ γ_{q}^{−k} (see (4.5)), we thus get

M_{j,ω}^{n}11(τ, x) ≤ D ∑_{k=1}^{n}(exp(q sup(\overline{ψ}))γ_{q}^{−1})^{k}||L_{ω|_{qn}^{qk+1}}^{(q(n−k))}11||_{∞} + ||G_{j,ω}^{n}11||_{∞}.

Taking supremum over all (τ, x) ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))), we thus get

||M_{j,ω}^{n}11||_{∞} ≤ D ∑_{k=1}^{n}(exp(q sup(\overline{ψ}))γ_{q}^{−1})^{k}||L_{ω|_{qn}^{qk+1}}^{(q(n−k))}11||_{∞} + ||G_{j,ω}^{n}11||_{∞}.

So, summing over all words ∈ {1, …, s}^{qn}, we obtain using Lemma 4.3, the following.

|||M_{j}^{n}11|||_{∞} ≤ D ∑_{k=1}^{n}(exp(q sup(\overline{ψ}))γ_{q}^{−1})^{k}s^{qk}|||L^{q(n−k)}11|||_{∞} + |||G_{j}^{n}11|||_{∞}

≤ D ∑_{k=1}^{n} exp(qk(sup(\overline{ψ}) + log s))(γ_{q}^{−1})^{k}|||L^{q(n−k)}11|||_{∞} + Q_{ψ}

= D ∑_{k=1}^{n}(η^{q}γ_{q}^{−1})^{k}|||L^{q(n−k)}11|||_{∞} + Q_{ψ}.

(4.13)

Since this inequality holds for all j = 1, …, u, we thus get

|||L^{qn}11|||_{∞} ≤ D ∑_{k=1}^{n}(η^{q}γ_{q}^{−1})^{k}|||L^{q(n−k)}11|||_{∞} + Q_{ψ}.

The first inequality to be proved is thus established. In order to derive the second one from it, invoke (4.5).

Lemma 4.5. There exists a constant \overline{Q}_{ψ} > 0 such that for all n ≥ 0, we have

|||L^{n}11|||_{∞} ≤ \overline{Q}_{ψ}.

Proof. We first shall prove by induction that

|||L^{qn}11|||_{∞} ≤ 2Q_{ψ}  (4.14)

for all integers n ≥ 0. Since ||11||_{∞} = 1, this formula holds for n = 0. So, fix n ≥ 1 and suppose that (4.14) is true for all 0 ≤ k ≤ n − 1. It then follows from Lemma 4.4 that

\begin{aligned}|||L^{qn}11|||_{∞} & ≤ Q_{ψ} + D ∑_{k=1}^{n}(1 + 2D)^{−k}2Q_{ψ} = Q_{ψ} (1 + 2D ∑_{k=1}^{n}(1 + 2D)^{−k}) \\ & ≤ 2Q_{ψ}, \end{aligned}

and (4.14) is proved. Since |||L^{i+j}11|||_{∞} ≤ |||L^{i}11|||_{∞}|||L^{j}11|||_{∞}, we are done by setting \overline{Q}_{ψ} = 2Q_{ψ} max\{|||L^{k}11|||_{∞} : k = 0, 1, …, q − 1\}.

Lemma 4.6. There exists a constant \underline{Q}_{ψ} > 0 such that for all n ≥ 0, we have

inf_{y∈J(f)} L^{n}11(y) ≥ \underline{Q}_{ψ}.

Proof. Taking q ≥ 1 sufficiently large, we can make the product η^{q}γ_{q}^{−1} (see (4.5)) as small as we wish. It therefore follows from (4.13) and Lemma 4.5 that for q ≥ 1 large enough, for all n ≥ 0 and all 1 ≤ j ≤ u, we have

|||M_{j}^{n}11|||_{∞} ≤ \frac{1}{2} + |||G_{j}^{n}11|||_{∞}.  (4.15)

Since, by Lemma 4.1, ∫ L^{qn}11dm_{ψ} = ∫ 11dm_{ψ} = 1, there thus exists x ∈ J(f) such that L^{qn}11(x) ≥ 1. Since {π_{2}^{−1}(B(z_{j}, R_{q}(z_{j})))}_{j=1}^{u} is a cover of J(f), there exists 1 ≤ i ≤ u such that x ∈ π_{2}^{−1}(B(z_{i}, R_{q}(z_{i}))). It follows from (4.15) that |||G_{i}^{n}(11)|||_{∞} ≥ 1/2. Applying now (4.9) we see that

G_{i}^{n}11(w) ≥ (2e^{2H})^{−1}  (4.16)

for all w ∈ π_{2}^{−1}(B(z_{i}, R_{q}(z_{i}))). Take now an arbitrary point y ∈ J(f). With k ≥ 1, as in the proof of Lemma 4.3, it follows from (4.10) that there exists \overline{y} ∈ π_{2}^{−1}(B(z_{i}, R_{q}(z_{i}))) such that f^{kq}(\overline{y}) = y. So, using (4.16) and the definition of the Perron-Frobenius operator, we obtain

L^{q(n+k)}11(y) ≥ exp(S_{kq}(\overline{ψ}(\overline{y})))L^{qn}11(\overline{y}) ≥ exp(inf(S_{kq}(\overline{ψ})))G_{i}^{n}11(\overline{y})
≥ (2e^{2H})^{−1} exp(inf(S_{kq}(\overline{ψ})))

Since inf(L^{a+b}11) ≥ inf(L^{a}11) inf(L^{b}11), we are therefore done by taking

\underline{Q}_{ψ} :=

min \{(2e^{2H})^{−1} exp(inf(S_{kq}(\overline{ψ}))), 1\} · min \{inf(L^{u}11) : u = 0, 1, …, kq − 1\} .

Now repeating verbatim the proof of Lemma 20 from [6], using Lemma 4.5 and Lemma 4.6, we get the following.

Lemma 4.7. Let h̃(x) := lim inf_{n→∞} \frac{1}{n}L^{n}11(x), h_{1}(x) := inf_{n≥0} L^{n}h̃(x) and h(x) := _{\frac{h_{1}(x)}{∫_{J(f)} h_{1} dm_{ψ}}} for each x ∈ J(f). Then h : J(f) → ℝ is a

Borel measurable function such that the following hold.

(a)

\begin{matrix}\underline{Q} & ψ \\ \overline{Q} & ψ \end{matrix} ≤ h(x) ≤ \begin{matrix}\overline{Q} & ψ \\ \underline{Q} & ψ \end{matrix} for every x ∈ J(f).

(b) Lh(x) = h(x) for every x ∈ J(f).

(c) ∫ hdm_{ψ} = 1.

Remark 4.8. Note that up to a normalized factor ( to make (c) hold ) the function h is independent of the conformal measure m_{ψ}.

As an immediate consequence of this lemma and Proposition 2.2 in [5], we have the following.

Theorem 4.9. The Borel probability measure μ_{ψ} = hm_{ψ} is f-invariant, equivalent to m_{ψ}, and \frac{dμ_{ψ}}{dm_{ψ}} ∈ [\underline{Q}_{ψ}/\overline{Q}_{ψ}, \overline{Q}_{ψ}/\underline{Q}_{ψ}].

5. Equilibrium States

Our objective in this section is to show that the measure μ_{ψ} produced in Theorem 4.9 is a unique equilibrium state for the potential ψ and that it is ergodic. Let P(ψ) be the ordinary topological pressure of the potential ψ. If μ is a Borel probability f-invariant measure on J(f), denote by J_{μ} its Jacobian with respect to the map f (see page 108 in [10]). We start with the following.

Lemma 5.1. P(ψ) ≥ P_{b}(ψ).

Proof. Since h_{μ_{ψ}}(f) ≥ ∫_{J(f)} log J_{μ_{ψ}}dμ_{ψ} (this is true for every finiteto-one endomorphisms and every probability invariant measure. See Lemma 10.5 and Theorem 5.14 in [10]) and since J_{μ_{ψ}} = \frac{h∘f}{h} exp(P_{b}(ψ)− ψ) everywhere, it follows from Theorem 4.9 and the Variational Principle that

P(ψ) ≥ h_{μ_{ψ}}(f) + ∫ ψdμ_{ψ} ≥ ∫ (P_{b}(ψ) − ψ + ψ)dμ_{ψ} = P_{b}(ψ). (5.1)

We are done.

Now, let μ be a Borel f-invariant ergodic measure on J(f) such that h_{μ}(f|σ) > 0, where h_{μ}(f|σ) denotes the relative entropy of (f, μ) with respect to σ, and let T_{μ} : L_{∞}(μ) → L_{∞}(μ) be the Perron-Frobenius operator associated to the measure μ. It is defined by the formula

T_{μ}g(x) = ∑_{y∈f^{−1}(x)} J_{μ}^{−1}(y)g(y).  (5.2)

Since Lh = h everywhere throughout J(f), using Lemma 6.9 from [13], we get that

1 = ∫ 11dμ = ∫ \frac{Lh}{h}dμ = ∫ T_{μ} (\frac{h exp(\overline{ψ})}{J_{μ}^{−1} · h ∘ f} ) dμ = ∫ \frac{h exp(\overline{ψ})}{J_{μ}^{−1} · h ∘ f} dμ

≥ 1 + ∫ log (\frac{h exp(\overline{ψ})}{J_{μ}^{−1} · h ∘ f} ) dμ

= 1 + ∫ log hdμ − ∫ log(h ∘ f)dμ + ∫ \overline{ψ}dμ + ∫ log J_{μ}dμ

= 1 + ∫ ψdμ − P_{b}(ψ) + h_{μ}(f).

(5.3)

Seeking contradiction suppose that P(ψ) > P_{b}(ψ). By the Variational Principle there exists a Borel probability f-invariant ergodic measure μ such that

h_{μ}(f) + ∫ ψdμ > P(ψ) − min{P(ψ) − P_{b}(ψ), P(ψ) − sup(ψ) − log s}.

Note that, because of (4.3), P(ψ) − sup(ψ) − log s > 0, and therefore

h_{μ}(f) > P(ψ) − (P(ψ) − sup(ψ) − log s) − ∫ ψdμ
= log s + sup(ψ) − ∫ ψdμ
≥ log s,

which implies h_{μ}(f|σ) > 0. Hence, (5.3) applies, and we can continue it to get that 1 ≥ 1+∫ ψdμ−P_{b}(ψ)+h_{μ}(f) > 1+P(ψ)−(P(ψ)−P_{b}(ψ))− P_{b}(ψ) = 1. This contradiction along with (5.1) and Lemma 5.1 give the following.

Proposition 5.2. P(ψ) = P_{b}(ψ) and μ_{ψ} is an equilibrium state for ψ.

Lemma 5.3. For each point x ∈ J(f), we have \frac{1}{n} log L_{ψ}^{n}11(x) → P(ψ) as n → ∞.

Proof. By Proposition 5.2, L_{ψ} = (exp(P(ψ))) · L. Combining it with Lemma 4.5 and Lemma 4.6, we obtain that the statement of our lemma holds.

Now suppose that μ is an arbitrary ergodic equilibrium state for ψ. Then h_{μ}(f) − log s = P(ψ) − ∫ ψdμ − log s ≥ P(ψ) − sup(ψ) − log s > 0. Hence h_{μ}(f|σ) > 0. In view of the proposition above, the last component in (5.3) is equal to 1. Consequently, the only inequality in this formula becomes an equality, and so log (\frac{h exp(\overline{ψ})}{J_{μ}^{−1}·h∘f}) = 0 μ-a.e. We thus get the following.

Lemma 5.4. If μ is an ergodic arbitrary equilibrium state for ψ, then J_{μ} = \frac{h∘f}{h} exp(P(ψ) − ψ) μ-a.e.

Now we shall prove the following.

Lemma 5.5. If a Borel f-invariant probability measure μ satisfies J_{μ} = \frac{h∘f}{h} exp(P(ψ)−ψ) μ-a.e., and if a Borel f-invariant probability measure m satisfies J_{m} = \frac{h∘f}{h} exp(P(ψ) − ψ) everywhere, then μ is absolutely continuous with respect to m.

Proof. It suffices to prove that for every ε > 0 there exists δ > 0 such that if g : J(f) → (0, 1] is a continuous function and ∫ gdm ≤ δ, then ∫ gdμ ≤ ε. Taking q = q(ϵ) ≥ 1 large enough, taking points z_{j} (j = 1, …, u) in J(G) such that ∪_{j=1}^{u}B(z_{j}, R_{q}(z_{j})) ⊃ J(G), using (4.5), Lemma 4.5, and redoing the considerations between (4.12) and (4.13) with the function 11 replaced by the function g, and ∥G_{j,ω}^{n}11∥_{∞} replaced by G_{j,ω}^{n}g(τ, x), after applying (5.2), we get for every 1 ≤ j ≤ u, every n = l · q ≥ 1 (l ∈ ℕ) and every x ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))), that

T_{μ}^{qn}g(x) ≤ \frac{ε}{2} + CG_{j}^{n}g(x) and T_{m}^{qn}g(x) ≥ C^{−1}G_{j}^{n}g(x), (5.4)

where C = \overline{Q}_{ψ}/\underline{Q}_{ψ} is a positive constant which does not depend on l, j, x, g. We take a partition of the set J(G) into mutually disjoint sets {Y_{j}^{i} : 1 ≤ j ≤ u, 1 ≤ i ≤ i(j)} each of which has a positive m ∘ π_{2}^{−1} measure and ⋃_{i=1}^{i(j)} Y_{j}^{i} ⊂ B(z_{j}, R_{q}(z_{j})) for all j = 1, 2, …, u. (Note that we can take such a partition, since the support of m is equal to J(f), which follows from the topological exactness of f : J(f) → J(f) and the condition “J_{m} = \frac{h∘f}{h} exp(P(ψ) − ψ) everywhere”.) Put

α = sup \{\frac{μ ∘ π_{2}^{−1}(Y_{j}^{i})}{m ∘ π_{2}^{−1}(Y_{j}^{i})} : 1 ≤ j ≤ u, 1 ≤ i ≤ i(j)\} .

Since g is everywhere positive, there exist ζ > 0 and k ≥ 1 such that

\frac{1}{2} ≤ \frac{g(θ, y)}{g(ρ, z)} ≤ 2  (5.5)

for all (θ, y), (ρ, z) ∈ J(f) with d(y, z) < ζ and θ|_{k} = ρ|_{k}. Fix now n = l · q ≥ 1 so large that for each 1 ≤ j ≤ u, K_{G}(R′_{q}(z_{j}))^{4}κ^{−1}λ^{n/2}C_{sa}^{1/2} < ζ (see item (B_{n}) of Corollary 3.2). Using then item (B_{n}) of this corollary and (5.5), the application of (4.7) gives for all n = l · q ≥ k, all ω ∈ {1, …, s}^{n}, and all ξ, x ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))), that

\frac{1}{2}e^{−2H}G_{j,ω}^{n}g(x) ≤ G_{j,ω}^{n}g(ξ) ≤ 2e^{2H}G_{j,ω}^{n}g(x).

Hence summing over all ω ∈ {1, …, s}^{n}, we get

\frac{1}{2}e^{−2H}G_{j}^{n}g(x) ≤ G_{j}^{n}g(ξ) ≤ 2e^{2H}G_{j}^{n}g(x)  (5.6)

for all n = l · q ≥ k, all 1 ≤ j ≤ u and all ξ, x ∈ π_{2}^{−1}(B(z_{j}, R_{q}(z_{j}))). Let X_{j}^{i} := π_{2}^{−1}(Y_{j}^{i}). Now applying (5.4), (5.6), and the definition of α, we get

∫ gdμ = ∫ T_{μ}^{qn}gdμ ≤ \frac{ε}{2} + ∑_{j=1}^{u} C ∑_{i=1}^{i(j)} ∫_{X_{j}^{i}} G_{j}^{n}gdμ

≤ \frac{ε}{2} + ∑_{j=1}^{u} C ∑_{i=1}^{i(j)} μ(X_{j}^{i}) sup_{X_{j}^{i}}(G_{j}^{n}g)

≤ \frac{ε}{2} + ∑_{j=1}^{u} C ∑_{i=1}^{i(j)} 2e^{2H}μ(X_{j}^{i}) inf_{X_{j}^{i}}(G_{j}^{n}g)

≤ \frac{ε}{2} + 2e^{2H} ∑_{j=1}^{u} C ∑_{i=1}^{i(j)} \frac{μ(X_{j}^{i})}{m(X_{j}^{i})} ∫_{X_{j}^{i}} G_{j}^{n}gdm

≤ \frac{ε}{2} + 2e^{2H}α ∑_{j=1}^{u} C^{2} ∑_{i=1}^{i(j)} ∫_{X_{j}^{i}} T_{m}^{qn}gdm

= \frac{ε}{2} + 2e^{2H}αC^{2} ∫ T_{m}^{qn}gdm = \frac{ε}{2} + 2e^{2H}αC^{2} ∫ gdm.

So, taking δ = (4αC^{2}e^{2H})^{−1}ε, finishes the proof.

Lemma 5.6. If μ is an ergodic equilibrium state, and if a Borel finvariant probability measure m satisfies J_{m} = \frac{h∘f}{h} exp(P(ψ) − ψ) everywhere, then μ is absolutely continuous with respect to m.

Proof. By Lemma 5.4 and Lemma 5.5, we obtain the statement.

We conclude the paper with the following.

Theorem 5.7. The measure μ_{ψ} forms a unique (ergodic) equilibrium state for ψ and m_{ψ} is a unique exp(P (ψ) − ψ)-conformal measure.

Proof. For every exp(P(ψ)−ψ)-conformal measure ν let h = h^{ν} be the density function obtained in Lemma 4.7 with m_{ψ} replaced by ν. Note that by Remark 4.8, they differ by a positive multiplicative constant. Suppose that there exist two exp(P(ψ) − ψ)-conformal measures m_{1} and m_{2} that are not equivalent. Then there exists a Borel set A ⊂ J(f) such that m_{1}(A) > 0 and m_{2}(A) = 0. Hence, on the one hand, h^{m_{1}}m_{1}(A) > 0 since, by Lemma 4.7, h^{m_{1}} > 0 everywhere, and, on the other hand, h^{m_{1}}m_{1}(A) = 0 since each measure τ_{i} = h^{m_{i}}m_{i} (i = 1, 2) satisfies J_{τ_{i}} = \frac{h^{m_{i}}∘f}{h^{m_{i}}} exp(P(ψ) − ψ) everywhere and hence these two measures are equivalent by Lemma 5.5. So, any two exp(P(ψ) − ψ)- conformal measures are equivalent. Now suppose that there are two different ergodic equilibrium states μ_{1} and μ_{2} for ψ. Then they are mutually singular and there is a completely invariant Borel set A ⊂ J(f) (f(A) = A = f^{−1}(A)) such that μ_{1}(A) = 1 and μ_{2}(A) = 0 (which implies that μ_{2}(J(f) \ A) = 1). Then, by Lemma 5.6, m_{ψ}(A) > 0 and m_{ψ}(J(f) \ A) > 0. Hence, both Borel probability measures m_{ψ}^{1} and m_{ψ}^{2} on J(f), respectively defined by the formulas

m_{ψ}^{1}(F) = \frac{m_{ψ}(F ∩ A)}{m_{ψ}(A)} and m_{ψ}^{2}(F) = \frac{m_{ψ}(F ∩ (J(f) \ A))}{m_{ψ}(J(f) \ A)},

are exp(P(ψ) − ψ)-conformal. Since they are singular, we get a contradiction, and the uniqueness of equilibrium state for ψ : J(f) → IR is established. So, coming back to conformal measures, if ν_{1} and ν_{2} are two exp(P(ψ) − ψ)-conformal measures, then h^{ν_{1}}ν_{1} = h^{ν_{2}}ν_{2}. Since the ratio h^{ν_{2}}/h^{ν_{1}} is constant, so is the ratio ν_{2}/ν_{1}. Since, in addition, both these measures ν_{2} and ν_{1} are probabilistic, they are equal. We are done.

Proof of the main result Theorem 1.1: Combining Theorem 5.7 and Lemma 5.3, the statement of the theorem follows.

References

[1] R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form z^{2} + c_{n}, Pacific J. Math., 198 (2001), no. 2, 347–372.

[2] M. Büger, Self-similarity of Julia sets of the composition of polynomials, Ergod. Th. and Dynam. Sys. 17 (1997), 1289–1297.

[3] M. Büger, On the composition of polynomials of the form z^{2} + c_{n}, Math. Ann. 310 (1998), no. 4, 661–683.

[4] R. Brück, M. Büger and S. Reitz, Random iterations of polynomials of the form z^{2} + c_{n}: Connectedness of Julia sets, Ergod. Th. and Dynam. Sys., 19, (1999), No.5, 1221–1231.

[5] M. Denker, M. Urbański, On the existence of conformal measures, Trans. A.M.S. 328 (1991), 563-587.

[6] M. Denker, M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), 103-134.

[7] J. E. Fornaess and N. Sibony, Random iterations of rational functions, Ergod. Th. and Dynam. Sys. 11(1991), 687–708.

[8] A. Hinkkanen and G.J.Martin, The Dynamics of Semigroups of Rational Functions I, Proc. London Math. Soc. (3)73(1996), 358-384.

[9] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. Dynam. Sys. 3 (1983), 351-386.

[10] W. Parry, Entropy And Generators In Ergodic Theory, Mathematics Lecture Note Series, 1969, Benjamin Inc.

[11] D. Ruelle, Thermodynamic formalism, Encyclopedia of Math. and Appl., vol. 5, Addison - Wesley, Reading Mass., 1976.

[12] R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups, preprint, http://arxiv.org/abs/0708.3187

[13] H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity 13 (2000), 995-1019.

[14] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergod. Th. and Dynam. Sys.(2001), 21, 563-603.

[15] H. Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Mathematical Journal, 28, No.2, 2005, pp390-422.

[16] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergod. Th. and Dynam. Sys. (2006), 26, 893-922.

[17] H. Sumi, Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane, Appl. Math. Comput. 187 (2007), no. 1, 489– 500. (Proceedings paper.)

[18] H. Sumi, Dynamics of postcritically bounded polynomial semigroups, preprint 2007, http://arxiv.org/abs/math/0703591.

[19] H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups, preprint 2007, http://arxiv.org/abs/0707.2447.

[20] P. Walters, An Introduction To Ergodic Theory, Springer-Verlag, 1982.

[21] W. Zhou, F.Ren, The Julia sets of the random iteration of rational functions, Chinese Bulletin, 37(12), 1992, 969-971.

Hiroki Sumi; Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

sumi@math.sci.osaka-u.ac.jp, http://www.math.sci.osaka-u.ac.jp/∼sumi/

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