EQUILIBRIUMS OF SOME NON-HÖLDER POTENTIALS

Huyi Hu

Abstract. We consider one-sided subshifts σ with some potential functions φ which satisfy the Hölder condition everywhere except at a fixed point and its preimages. We prove that the systems have conformal measures ν and invariant measures μ absolutely continuous with respect to ν, where μ may be finite or infinite. We show that the systems (σ, μ) are exact, and μ are weak Gibbs measures and equilibriums for φ. We also discuss uniqueness of equilibriums and phase transition.

These results can be applied to some expanding dynamical systems with an indifferent fixed point.

0. Introduction

The motivation of the paper is to understand statistical properties of physical measures for almost expanding dynamical systems with Markov partitions. We say that a piecewise smooth system is almost expanding if it is expanding everywhere except at a finite number of periodic orbits. Examples of such systems are given in Section 2, which include piecewise expanding maps on the unit interval, parabolic rational maps on Julia sets, etc. We only consider the case where the systems contain one indifferent fixed point p. Systems with more indifferent fixed points or periodic orbits can be treated similarly.

Since we assume that the systems have Markov partitions, they can be represented by a one-sided subshift of finite type, and we can work on potentials φ. With the usual metric on symbolic space, the potentials we study do not satisfy Hölder conditions at the fixed point and its preimages. Therefore, statistical properties of the systems become different from those with Hölder potentials.

We obtain existence of a conformal measure ν and an invariant measure μ for such a potential, where μ is a physical measure of the system we are interested in, and study the properties of the measures. We show that such a system (σ, μ) is exact, and therefore is ergodic if the symbolic system is topologically mixing. We prove that μ is a weak Gibbs measure, and obtain conditions under which the formula P (σ, φ) = h_{μ}(σ) + μ(φ) holds. We also study uniqueness of equilibriums

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 37C40, 37A60; Secondary 28D05.

Key words and phrases. Potential, Equilibrium, Invariant measure, Exactness, Ergodicity, Weak Gibbs measure.

Part of this work was done when the author was in the Penn State University and University of Southern California. This work was supported by NSF under grants DMS-9970646 and DMS-0240097.

and phase transition. Lastly, we give the rates of convergence, without proof, of test functions to their equilibriums under the transfer operators.

There are many literatures related to the topic. For non-Hölder potentials, existence and uniqueness of equilibriums, and rates of convergence to the equilibriums are studied by Hofbauer, Fisher-Lopes, Maume-Deschamps and others (see e.g. [Ho], [M], [KMS], [N], [FL]). In those references, potentials are usually assumed to be piecewise constant, or to be summable. The systems with indifferent fixed points are sometimes coded by a subshift with countably many states (see e.g. [S1]- [S4] and their references). Also, some ergodic properties have been studied for the systems discussed in Section 2. For piecewise expanding maps with an indifferent fixed point on the unit interval, existence and condition for finiteness of absolutely continuous invariant measures was proved by Pianigiani ([Pi]) and Thaler ([T]). When the invariant measure is finite, weak Gibbsianness, thermodynamic formalism, and phase transition have been studied by M. Yuri (see e.g. [Yu1]-[Yu4] and their references). When it is infinite, ergodic properties were studied by Zweimüller ([Z]). Parabolic rational maps on Julia sets have been studied by Denker-Urbański (see e.g. [DU2], [DU3]). These results were extended to parabolic Cantor sets by Urbański (see [U1] and its references). Rates of convergence to equilibriums and rates of decay of correlations of systems were studied in [Y2], [H1], [S4], [G], and others (see [H2] for more references).

In this paper, we try to give the simplest conditions on the potential functions and to obtain most ergodic properties for varieties of almost expanding systems. Hence, among the systems discussed in Section 2, an ergodic property found in one kind of system may also hold for others. The conditions we give are weaker since for most results we do not need Assumption(III′), i.e. we only need the lower bound of |φ(¯0) − φ(x)|. Further, the potentials we study are more general. For example, in one dimensional almost expanding systems, our potentials are not necessary to have the form −t log f′(x). We may have bounded density functions and exponential rates of convergence to the equilibriums. In this case, the behaviors of the systems are just like those with Hölder potentials. We may also have unbounded density functions, finite or infinite invariant measure. In this case, the measures of the tail of the Young’s tower may or may not converge to 0 (see Corollary A.2), and the rates of convergence to the equilibriums are only polynomial. Moreover, we prove exactness and study Gibbs property in both cases where μ are finite and infinite. We prove uniqueness of weak Gibbs state and equilibrium, and give a complete description for phase transition. Our main approach avoids the first return maps and goes down to Bowen’s method.

This paper is organised as follows. The assumptions and results are stated in Section 1. In Section 2 we apply these results to piecewise smooth almost expanding maps. In Section 3 we prove Theorem A, that mainly deals with existence of conformal measures and invariant measures. In Section 4 we discuss properties of the density functions of the invariant measures, which are stated in Corollary A.1. The measures of the tail of tower are estimated in Section 5, Section 6 is for exactness and a proof of Theorem B, while Section 7 is for Gibbs properties and a proof of Theorem C. The last section, deals with equilibriums and uniqueness.

1. Assumptions, statements of results and notations

Let Σ^{+} = ∏_{0}^{∞}{0, 1, ⋯, r^{∗} − 1} and σ : Σ^{+} → Σ^{+} be the left shift.

If A is an r^{∗} × r^{∗} matrix of 0’s and 1’s, let Σ_{A}^{+} = {x ∈ Σ^{+} : A_{x_{i}x_{i+1}} = 1 ∀i ≥ 0}. It is well known that σΣ_{A}^{+} = Σ_{A}^{+}. We assume that σ is topologically mixing. We also assume A_{00} = 1, so, ¯0 = 000 ⋯ is a fixed point of σ.

For convenience we assume that A_{11} = 1 so that \overline{1} = 111 ⋯ is another fixed point. We can check that the results are still true without the assumption.

We say that w is an n-word if w = w_{0}w_{1} ⋯ w_{n−1} and A_{w_{i}w_{i+1}} = 1 ∀0 ≤ i < n−1. The word uw is the word u followed by the word w.

Given an n-word w = w_{0}w_{1} ⋯ w_{n−1}, we define

R_{w} = {x ∈ Σ_{A}^{+} : x_{i} = w_{i}, ∀0 ≤ i ≤ n − 1}.

This set is called an n-cylinder, or simply a cylinder.

Let ξ be the partition of Σ_{A}^{+} into {R_{s} : s = 0, 1 ⋯ r^{∗} − 1}, and

ξ_{n} = ⋁_{i=0}^{n−1} σ^{−i}ξ.  (1.1)

For any n-word w, we have R_{w} ∈ ξ_{n}. We simply write w ∈ ξ_{n} instead.

For k ≥ 0, we denote O_{k} = R_{0^{k}}, P_{k} = O_{k}\O_{k+1}, and Q_{k} = Σ_{A}^{+}\O_{k+1}. In other words, O_{k}, P_{k} and Q_{k} are sets of the points that start with at least, exact, and at most k zeros, respectively. Also, we denote P′_{k} = P_{k−1} ∪ P_{k} ∪ P_{k+1}.

Take κ ∈ (0, 1) and γ > 0. Let K_{0} be the largest number k such that κ^{k} ≥ (k + 1)^{−(1+γ)}. Define a metric on Σ_{A}^{+} inductively by the following rules:

i) d(x, y) = 1 if x_{0} ≠ y_{0}.

ii) d(x, y) = κd(σx, σy) if x_{0} = y_{0} and x, y ∈ Q_{K_{0}}.

iii) d(x, y) = (k + 1)^{−(γ+1)}d(σ^{k}x, σ^{k}y) if x ∈ P_{k}, y ∈ P′_{k}, k > K_{0}.

iv) d(x, y) = ∑_{i=k}^{k+l−1} d(x^{(i)}, x^{(i+1)}), if x ∈ P_{k} and y ∈ P_{k+l}, k ≥ K_{0}, where

x^{(k)} = x, x^{(k+l)} = y, and x^{(i)} ∈ P_{i} for i = k + 1, ⋯, k + l − 1. With this metric, the left shift σ : Σ_{A}^{+} → Σ_{A}^{+} is uniformly expanding with a rate

κ^{−1} on Q_{K_{0}}. The expanding rate of σ on P_{k} converges to 1 if k → ∞.

By the metric we can see that if x = 0^{k}x̃ ∈ P_{k}, y = 0^{k}ỹ ∈ P′_{k} such that x̃_{0} ≠ ỹ_{0}, then d(x, y) = max{κ^{k}, (k + 1)^{−(α+1)}}. Hence, there exists C_{γ} ≥ 1 such that

diam O_{k} ≤ C_{γ}k^{−γ}  ∀k ≥ 0,  (1.2)

where the diameter of a set S is defined by diam S = sup{d(x, y) : x, y ∈ S}.

We assume that the potential function φ satisfies the following.

Assumption A.

(I) φ is a continuous function on Σ_{A}^{+}; (II) ∃ θ ∈ (0, 1], α ∈ [0, θ(1 + γ)) and C_{φ} > 0 such that

|φ(x) − φ(y)| ≤ C_{φ} max{K_{0}^{α−1}, k^{α−1}}d(x, y)^{θ} ∀x ∈ P_{k}, y ∈ P′_{k}; (III) ∃ β > −1, K_{1} > 0 such that for all k ≥ K_{1},

φ(¯0) − φ(x) ≥ \frac{β + 1}{k + 1} − \frac{C_{δ}}{(k + 1)^{1+δ}} ∀x ∈ P_{k}

for some constants δ > 0 and C_{δ} > 0 independent of k and x. Sometimes we also assume (III′) ∃ β′ ≥ β, K′_{1} > 0 such that for all k ≥ K′_{1},

φ(¯0) − φ(x) ≤ \frac{β′ + 1}{k + 1} + \frac{C_{δ}}{(k + 1)^{1+δ}} ∀x ∈ P_{k}.

We may assume δ ≤ min{1, γθ} since we can always reduce δ.

We will also assume φ(¯0) = 0 since otherwise we can use φ(x) − φ(¯0) instead.

Remark 1.1. With the standard metric d̃(x, y) = 2^{−k}, where k = min{i : x_{i} ≠ y_{i}} for x = {x_{i}} and y = {y_{i}}, φ is not a Hölder function because of Assumption A(III). However, under the metric we define, φ satisfies the Hölder condition.

Remark 1.2. If we denote

var_{k}(φ) = max{|φ(x) − φ(y)| : x_{i} = y_{i}, ∀i = 0, ⋯, k − 1},  (1.3)

then by Assumption A(III) var_{k}(φ) ≥ Ck^{−1} for some C > 0. So the potential φ

does not have summable variations since ∑_{k=0}^{∞} var_{k}(φ) is not summable.

Let C^{0}(S) denote the set of continuous real functions on the set S. Define the Perron-Frobenius Operator L_{φ} from C^{0}(Σ_{A}^{+}\{¯0}) or C^{0}(Σ_{A}^{+}) to itself by

L_{φ}g(x) = ∑_{y∈σ^{−1}x} e^{φ(y)}g(y).

Denote by L_{φ}^{∗} the dual operator of L_{φ} on M(Σ_{A}^{+}).

Denote by φ̃ the first return map with respect to P_{0}, and by φ̃ the corresponding potential, that is, for any x ∈ P_{0}, σ̃x = σ^{n}x and φ̃(x) = S_{n}φ(x) where n = n(x) is the smallest positive integer such that σ^{n}x ∈ P_{0}. The corresponding Perron-Frobenius Operator L̃φ̃ is given by

L̃φ̃g(x) = ∑_{σ̃y=x} e^{φ̃(y)}g(y) = ∑_{j=1}^{∞} ∑_{s≠0} e^{S_{j}φ(s0^{j−1}x)}g(s0^{j−1}x).  (1.4)

Denote D_{k} = max{k, K_{0}}. For J ≥ 0, we define

\overline{G}_{J} = \{g ∈ C^{0}(Σ_{A}^{+}\\{¯0\}) : g > 0, g(y) ≤ g(x)e^{JD_{k}^{α}d(x,y)^{θ}} ∀x ∈ P_{k}, y ∈ P′_{k}, ∀k ≥ 0\}.

Denote by M(Σ_{A}^{+}) the set of Borel probability measures on Σ_{A}^{+}.

Theorem A (Existence of the invariant measures). Suppose φ satisfies either Assumption A(I)-(III) with β > 0, or Assumptions A(I)-(III) and (III′) with −1 < β ≤ β′ ≤ 0. Then there is a measure ν ∈ M(Σ_{A}^{+}), which is positive on nonempty open sets, a constant λ ≥ e^{φ(¯0)} = 1, λ > 1 if −1 < β′ ≤ 0, and a function h ∈ \overline{G}_{J_{φ}} for some J_{φ} > 0 such that L_{φ}^{∗}ν = λν, L_{φ}h = λh.

Moreover, μ(g) = ν(hg) defines a finite or infinite σ-invariant measure μ. · μ is finite if either h^{∗}(¯1) < 0, or h^{∗}(¯1) = 0 and β > 1; · μ is infinite if either h^{∗}(¯1) > 0, or h^{∗}(¯1) = 0 and Assumption A(III′) holds with 0 < β′ ≤ 1,

where

h^{∗}(x) = h(x) − ∑_{j=1}^{∞} ∑_{s≠0} e^{S_{j}φ(s0^{j−1}x)}h(s0^{j−1}x).  (1.5)

If μ is finite, then we assume that μ is a probability measure.

By Corollary C.1, we see that log λ is the topological pressure P (σ, φ) for the potential function φ. We will prove in Lemma 4.3 that the sign of h^{∗} is independent of x, and λ > 1 if and only if h^{∗} < 0.

Remark 1.3. Let σ̃ be the first return map and φ̃(x) the corresponding potential. If the topological pressure P (σ̃, φ̃) can be defined, then one should expect that P (σ̃, φ̃) and h^{∗} have the opposite sign.

Remark 1.4. Sometimes the measure ν is called an e^{log λ−φ}-conformal measure in the sense that for any Borel set E such that σ|_{E} is injective,

ν(σE) = ∫_{E} ϵ^{log λ−φ}dν.  (1.6)

For any function g defined on Σ_{A}^{+} or Σ_{A}^{+}\{¯0}, we denote

ĝ(0x) = ∑_{s≠0} e^{φ(sx)}g(sx).  (1.7)

Corollary A.1 (Properties of the density function).

i) If h^{∗}(¯1) < 0, then lim_{x→¯0} h(x) = \frac{ĥ(¯0)}{λ − 1}; otherwise lim_{x→¯0} h(x) = ∞.

ii) If h^{∗}(¯1) = 0, then for any x ≠ ¯0, lim sup_{n→∞} \frac{β}{n}h(0^{n}x) ≤ ĥ(¯0). Suppose As-

sumption A(III′) also holds; then lim inf_{n→∞} \frac{β′}{n} h(0^{n}x) ≥ ĥ(¯0). So if β = β′, then the lim sup and lim inf become limit.

iii) If h^{∗}(¯1) > 0, then there exists B_{h} > 0 such that lim inf_{n→∞} \frac{1}{n^{β+1}} h(0^{n}x) ≥ B_{h}h^{∗}(x) for any x ∈ P_{0}. Suppose Assumption A(III′) also holds; then there

exists B′_{h} > 0 such that lim sup_{n→∞} \frac{1}{n^{β′+1}} h(0^{n}x) ≤ B′_{h}h^{∗}(x) for any x ∈ P_{0}.

Further, if β = β′, then lim_{n→∞} \frac{1}{n^{β+1}} h(0^{n}x) = B_{h}^{∗}h^{∗}(x) for some B_{h}^{∗} > 0.

Since functions in \overline{G}_{J_{φ}} are undefined at ¯0, by Corollary A.1 and Lemma 4.1, we can define

h(¯0) = ĥ(¯0)(λ − 1)^{−1} if λ > 1, and h(0) = ∞ otherwise  (1.8)

It is well known that the convergence rates of the tail of Young’s tower ([Y1], [Y2]) determine the rates of convergence of test functions to the equilibrium, and the rates of decay of correlations.

Corollary A.2 (Convergence rates of the tail).

For any k ≥ 0, there exist B_{μ} = B_{μ,k}, B′_{μ} = B′_{μ,k}, C_{μ} = C_{μ,k} > 0 and C′_{μ} = C′_{μ,k} > 0 such that the limits lim_{k→∞} A_{μ,k}, where A = B, B′, C, C′ exist, and for all n ≥ k:

i) if h^{∗}(¯1) < 0, then μP_{n} ≤ B_{μ,k}λ^{−n}n^{−(β+1)} and μO_{n} ≤ C_{μ,k}λ^{−n}n^{−β}; ii) if h^{∗}(¯1) = 0, then μP_{n} ≤ B_{μ,k}n^{−β} ∀β > 0, μO_{n} ≤ C_{μ,k}n^{−(β−1)} ∀β > 1; iii) if h^{∗}(¯1) > 0, then lim_{n→∞} μP_{n} > 0 and μO = ∞.

Suppose Assumption A (III′) also holds; then i) and ii) are true if we replace B_{μ,k}, C_{μ,k} and “ ≤” by B′_{μ,k}, C′_{μ,k} and “ ≥” respectively, and for case ii) it also holds that μO_{n} = ∞ ∀0 < β′ ≤ 1.

Moreover, if β = β′, then B_{μ,k} and B′_{μ,k}, and C_{μ,k} and C′_{μ,k} can be chosen in such

a way that the following limits exist: lim_{k→∞} B_{μ,k} = lim_{k→∞} B′_{μ,k} = lim_{n→∞} λ^{n}nβ̃νP_{n} for

β̃ > 0, and lim_{k→∞} C_{μ,k} = lim_{k→∞} C′_{μ,k} = lim_{n→∞} λ^{n}n^{β̃−1}νO_{n} for β̃ > 1, where β̃ = β + 1

in case i) and β̃ = β in case ii).

Estimates for νP_{n} and νO_{n} are given in Lemma 4.5.

Define ψ(x) = φ(x) + log h(x) − log h(σx) − log λ for x ∈ Σ_{A}^{+}, where we regard log h(¯0) − log h(σ¯0) = 0 if h(¯0) = ∞.

It is easy to see that ψ(x) = − log \frac{dμ ∘ σ}{dμ} (x), while φ(x) = − log \frac{dν ∘ σ}{dν} (x). So

we can put assumption on ψ instead of φ to get statistic properties of the systems (see [H2]). We only state the results for the case h^{∗}(¯1) = 0, since this is the most interesting case.

Note that if x ∈ σ^{−1}(σP_{k}), then x has the form s0^{k−1}w, where w = w_{0}w_{1} ⋯ with w_{0} ≠ 0.

Corollary A.3 (Properties of the function ψ).

Consider the case h^{∗}(¯1) = 0.

(I) ψ is continuous on each R_{s} except at s¯0, s ≠ 0.

(II) ∃ J_{ψ} > 0 such that for all n-words w = w_{0}w_{1} ⋯ w_{n−1},

|S_{n}ψ(wx) − S_{n}ψ(wy)| ≤ J_{ψ} max{K_{0}^{α}, k^{α}}d(x, y)^{θ} ∀x ∈ P_{k}, y ∈ P′_{k},

and if w_{n−1} ≠ 0, then

|(S_{n}ψ(wx) + log h(x)) − (S_{n}ψ(wy) + log h(y))| ≤ J_{ψ}K_{0}^{α}d(x, y)^{θ} ∀x, y ∈ Σ_{A}^{+}.

(III) ∃ K_{2} > 0 such that for all k ≥ K_{2},

ψ(¯0) − ψ(x) ≥ \frac{β}{k} − \frac{C_{δ}}{k^{1+δ}} ∀x ∈ P_{k}

for some C_{δ} > 0 independent of k and x. Moreover, if Assumption A (III′) also holds and λ = 1, then (III′) ∃ K′_{2} > 0 such that for all k ≥ K′_{2},

ψ(¯0) − ψ(x) ≤ \frac{β′}{k} + \frac{C_{δ}}{k^{1+δ}} ∀x ∈ P_{k}

for some C′_{δ} > 0 independent of k and x.

Recall that a σ-invariant measure μ is ergodic if for any measurable set E, σ^{−1}E = E (mod μ) implies μE = 0 or μ(Σ_{A}^{+} \ E) = 0. A system (σ, μ) is ex-

act if the tail σ-algebra ⋂_{n=1}^{∞} σ^{−n}B is trivial, where B is the σ-algebra for the system

(σ, μ). These definitions work for both probability and infinite measures. (See [A] for the infinite measure case.) It is well known that exactness implies ergodicity. Also, if μ is a probability measure, then exactness implies mixing.

Theorem B (Ergodicity and exactness). Under the assumptions of Theorem A, the system (σ, μ) is exact. Therefore, μ is an ergodic measure.

Recall that a measure ρ is a Gibbs measure, if there exist constants P and C such that for any x, and n ≥ 0,

C^{−1} ≤ \frac{ρR_{x_{0}x_{1}···x_{n−1}}}{exp\{−nP + S_{n}φ(x)\}} ≤ C.

In our case one cannot expect that the measure μ obtained in Theorem A is a Gibbs measure, since if we take x = ¯0, then by Corollary A.2, μO_{n} may decrease polynomially, while exp{−nP + nφ(¯0)} = e^{−nP} decreases exponentially if P > 0 or equal to 1 if P = 0. However, μ is a week Gibbs measure. A measure ρ is a weak Gibbs measure, if there exists a constant P and a sequence {C_{n}} with

lim_{n→∞} \frac{1}{n} log C_{n} = 0 such that for any x, and n ≥ 0,

C_{n}^{−1} ≤ \frac{ρR_{x_{0}x_{1}···x_{n−1}}}{exp\{−nP + S_{n}φ(x)\}} ≤ C_{n}.  (1.9)

We refer to [Yu2] and [Yu3] and their references for more information about weak Gibbs measures.

We can extend the definition for σ-finite measures. An invariant measure ρ is said to be an infinite weak Gibbs measure if ρ(Σ_{A}^{+}) = ∞, and (1.9) holds for all x ∈ Σ_{A}^{+} and n > 0 provided ρR_{x_{0}x_{1}···x_{n−1}} < ∞.

Theorem C (Gibbs properties). Under the assumptions of Theorem A, μ is a weak Gibbs measure or an infinite weak Gibbs measure. Moreover, μ is the unique invariant measure satisfies the following properties: There is a real number P, and a function p(x, n) such that for every x,

\frac{1}{p(x, n)} ≤ \frac{μR_{x_{0}x_{1}···x_{n−1}}}{exp\{−nP + S_{n}φ(x)\}} ≤ p(x, n),  (1.10)

provided μR_{x_{0}x_{1}···x_{n−1}} < ∞, where p(x, n) satisfies the following:

a) lim_{n→∞} \frac{1}{n} log p(x, n) = 0 for any x ≠ ¯0;

b) lim_{k→∞} \frac{1}{k} log sup_{x∈Q_{k}∩σ^{−n+1}Q_{0}} p(x, n) = 0.

Remark 1.5. Part b) implies that for each k, we can find uniform bounds for cylinders R_{w} if R_{u} ⊂ Q_{k} and the last symbol of w is nonzero. If we think that such cylinders are “good” cylinders, then for any x ∈ Q_{k} which is not a preimage of ¯0, R_{x_{0}x_{1}···x_{n−1}} is a “good” cylinder for infinitely many n.

Corollary C.1 (The constant P). The constant P in both (1.9) and (1.10) is equal to the topological pressure P (σ, φ) and log λ, where λ is given in Theorem A.

For entropy of a σ-finite measure ρ, we follow the definition given by Krengel (see [Kr], also [Z]). For a subset Γ ⊂ Σ_{A}^{+}, we denote by σ_{Γ} the corresponding first return map, and by ρ_{Γ} the conditional measure of ρ, that is, ρ_{Γ}S = ρS/ρΓ for S ⊂ Γ. The measure theoretic entropy of ρ is defined by

h_{ρ}(σ) = ρ(Γ)h_{ρ_{Γ}}(σ_{Γ})  (1.11)

for any subset Γ of positive finite measure.

Recall that a probability measure ρ is an equilibrium state for a potential η if it satisfies

P (σ, η) = h_{ρ}(σ) + ∫ ηdρ.  (1.12)

We also denote P (η) = P (σ, η).

Theorem D (Equilibrium states). Under the assumptions of Theorem A, · μ satisfies (1.12) with η = φ if and only if h^{∗}(¯1) ≤ 0; · the Dirac measure δ¯0 satisfies (1.12) with η = φ if and only if h^{∗}(¯1) ≥ 0.

Further, the only ergodic (probability) equilibrium for φ is · μ if P (φ) > 0; · μ and δ¯0 if P (φ) = 0 and μΣ_{A}^{+} = 1; · δ¯0 if P (φ) = 0 and μΣ_{A}^{+} = ∞.

Suppose Assumption A (III′) also holds; then μ is the only possible infinite ergodic measure ρ with |ρ(φ)| < ∞ such that (1.12) holds with η = φ.

By Theorem A and Corollary C.1, if μ is an infinite measure, then P (σ, φ) = 0. So Theorem D provides conditions for the Rohlin’s formula h_{μ}(σ) = − ∫ φdμ.

Corollary D.1 (Phase transition). Suppose φ ≤ 0 satisfies Assumptions A (I)- (III) and (III′) with β = β′. Then there exists t_{0} > 0 such that

i) for 0 ≤ t < t_{0}, P (tφ) > 0 and φ has the unique equilibrium μ_{tφ}; ii) for t = t_{0}, P (tφ) = 0 and φ has exactly two equilibriums μ_{tφ} and δ¯0 if

t_{0}(β + 1) > 2, and has the unique equilibrium δ¯0 otherwise; iii) for t > t_{0}, P (tφ) = 0 and φ has the unique equilibrium δ¯0.

Corollary D.2 (Uniqueness of weak Gibbs measures). Under the assumptions of Theorem A, if μ is a probability measure, then it is the only σ-invariant weak Gibbs measure for φ.

We define L_{ψ} : C^{0}(Σ_{A}^{+}) → C^{0}(Σ_{A}^{+}) by L_{ψ}g = \frac{1}{λh}L_{φ}(hg), or, equivalently

L_{ψ}g(x) = ∑_{y∈σ^{−1}x} e^{ψ(y)}g(y) = \frac{1}{λh(x)} ∑_{y∈σ^{−1}x} e^{φ(y)}h(y)g(y).

If h^{∗}(¯1) ≤ 0, the convergence rate of a test function to its equilibrium under the operators L_{ψ} is determined by P (φ) and β. We state the results here.

Denote

G ={g ∈ C^{0}(Σ_{A}^{+}) : ∃C > 0, s.t. |g(y) − g(x)| ≤ CD_{k}^{α}d(x, y)^{θ}
∀x ∈ P_{k}, y ∈ P′_{k} ∀k ≥ 0},
G_{0} ={g ∈ G : g(¯0) ≠ μ(g)},
G_{δ} ={g ∈ G : ∃L > 0, s.t. |g(¯0) − μ(g)| ≤ L(n + 1)^{−δ} ∀n ≥ 0}.

Let F(Σ_{A}^{+}) be the set of all bounded real functions on Σ_{A}^{+}. Denote

\begin{matrix}F_{τ}^{+} = & \{g ∈ F(Σ_{A}^{+})) : ∃L > 0, s.t. |g(x) − g(¯0)| ≤ L(n + 1)^{−τ} ∀x ∈ P_{n}, n ≥ 0\} , \\ F_{τ}^{−} = & \{g ∈ F(Σ_{A}^{+})) : ∃L′ > 0, s.t. g(x) − g(¯0) ≥ L′(n + 1)^{−τ} ∀x ∈ P_{n}, n ≥ 0\} . \end{matrix}

Theorem E (Rates of convergence). Suppose φ satisfies Assumptions A(I)-(III). If h^{∗}(¯1) < 0, then there is λ̃ > 0 such that for any g ∈ G, there is A > 0 with

|L_{ψ}^{n}g(x) − μ(g)| ≤ Aλ̃^{−n} ∀x ∈ Q_{k}, n ≥ 0.

If h^{∗}(¯1) = 0 and β > 1, then for any g ∈ G_{τ}, τ ∈ [0, 1], there is A > 0 such that

|L_{ψ}^{n}g(x) − μ(g)| ≤ \frac{A}{(n + 1)^{min\{β−1+τ,β\}}} ∀x ∈ Q_{k}, n ≥ 0.

Moreover, if Assumption A (III′) also holds with β′ ≤ β + 1, then for any g ∈ G_{0}, there is A′ > 0 such that

|L_{ψ}^{n}g(x) − μ(g)| ≥ \frac{A′}{(n + 1)^{β′−1}} ∀x ∈ Q_{k}, n ≥ 0.

The above inequalities are also true if we replace |L_{ψ}^{n}g(x)−μ(g)| by ∥L_{ψ}^{n}g−μ(g)∥. If h^{∗}(¯1) = 0 and Assumptions A (III′) also holds with β = β′ ∈ (0, 1], then for

any g ∈ F_{τ}^{+}, there is A > 0 such that

|L_{ψ}^{n}g(x) − g(¯0)| ≤ \frac{A(n + 1)^{max\{0,1−β−τ\}}}{(n + 1)^{1−β}} ∀x ∈ Σ_{A}^{+}, n ≥ 0;

and for any g ∈ F_{τ}^{−}, there is A′ > 0 such that

L_{ψ}^{n}g(x) − g(¯0) ≥ \frac{A′(n + 1)^{max\{0,1−β−τ\}}}{(n + 1)^{1−β}} ∀x ∈ Q_{n}, n ≥ 1,

where (n+1)^{1−β} or (n+1)^{1−β−τ} in the inequalities should be replaced by log(n+1) if β = 1 or β + τ = 1, respectively.

We are not going to prove the theorem in this paper. For the case P (φ) > 0, we have λ > 1. By Corollary A.2, μP_{k} and μO_{n} decreases exponentially fast. Then we can apply results of Young ([Y1]) to get exponential convergence. For the case P (φ) = 0, the results and more details can be seen in [H2] and [HH] for the case β > 1 and β = β′ ≤ 1 respectively. (See also [Y2], [S4], [G] for the case β > 1.)

From this theorem, we can get corresponding results for rates of decay of correlations when the invariant measure is finite. Further, if the covariance

μ(g · g ∘ σ^{n}) − μ(g)^{2}

is summable with n, then the Central Limit Theorem holds. In our case, if β−1+τ > 1, then the Central Limit Theorem holds for any g ∈ G_{τ}.

2. Almost expanding maps: Applications

Consider a map f : X → X, where X = ℝ^{m} or ℂ̄, the Riemannian sphere. Suppose f has an invariant subset Λ, i.e. fΛ = Λ.

Assumption B.

(I) f|_{Λ} : Λ → Λ is topologically mixing.

(II) f|_{Λ} : Λ → Λ has a Markov partition into subsets {R_{i}}_{i=0}^{r^{∗}−1}.

(III) f is piecewise smooth. More precisely, for each i, f|_{int R_{i}} is a C^{2} map from int R_{i} to its image, and it can be C^{1} extended to \overline{R}_{i}.

Here a Markov partition means a finite cover \{R_{0}, ⋯, R_{r^{∗}−1}\}, R_{i} = \overline{int_{Λ} R_{i}}, of Λ with int R_{i} ∩ int R_{j} = ∅ ∀i ≠ j; and f(R_{i}) is a union of some R_{j}.

A fixed point p of f is indifferent if Df_{p} has an eigenvalue on the unit circle in ℂ. We say that the map f is expanding on an invariant set Λ with an indifferent fixed point p if f has an indifferent fixed point p ∈ Λ and f is uniformly expanding away from p. The latter means that for any open neighbourhood U of p, f is uniformly expanding on Λ\U.

If f is expanding on an invariant subset Λ, with or without indifferent fixed points, and has a Markov partition, then there is a map π : Σ_{A}^{+} → Λ given by

π({x_{i}}) = ⋂_{j=0}^{∞} f^{−i}R_{x_{j}}, where {x_{i}} ∈ Σ_{A}^{+}, such that f ∘ π = π ∘ σ.

Assume that f has an indifferent fixed point p. Then we can choose a Markov partition {R_{i}}_{i=0}^{r^{∗}−1} such that p ∈ R_{0}. Hence, π(¯0) = p. Note that the projection π : Σ_{A}^{+} → Λ induces a map π^{∗} : C^{0}(Λ) → C^{0}(Σ_{A}^{+}) by π^{∗}g = g ∘ π ∀g ∈ C^{0}(Λ).

Assume further that near p, there is a local coordinate system such that

fx = x(1 + |x|^{r}) + g(x),  (2.1)

where

|g(x)|, |x| ∥Dg(x)∥, |x|^{2} ∥D^{2}g(x)∥ = O(|x|^{1+r′})  (2.2)

for some r′ > r as x near p.

Lemma 2.1. Suppose f(x) satisfies (2.1) and (2.2). Then there is 0 < δ < 1, and an integer k_{0} ≥ k′_{0} such that for all large k,

\frac{1}{(k + k_{0})r} (1 − \frac{1}{(k + k_{0})^{δ}} ) ≤ |x|^{r} ≤ \frac{1}{(k + k′_{0})r} (1 + \frac{1}{(k + k′_{0})^{δ}} ) ∀x ∈ P_{k}. (2.3)

Proof. We may assume that Λ ⊂ I so that we can drop the norm sign | · |. We claim that if fx ≤ x+x^{1+γ} +t_{0}x^{1+r′} for some t_{0} > 0, then there is 0 < δ < 1

such that for all large n, x^{r} ≤ \frac{1}{nr} (1 − \frac{1}{n^{δ}} ) implies

(fx)^{r} ≤ \frac{1}{(n − 1)r} (1 − \frac{1}{(n − 1)^{δ}} ).  (2.4)

This implies the first inequality of (2.3). In fact, for any large k we can always find

k_{0} such that x^{r} ≥ \frac{1}{(k + k_{0})r} (1 − \frac{1}{(k + k_{0})^{δ}} ) ∀x ∈ P_{k+k_{0}}. Then we use induction.

Denote γ_{n} = γ(1 − n^{−δ})^{−1}. By the condition,

(fx)^{γ} ≤ x^{γ}(1 + x^{γ} + t_{0} · x^{γ′})^{γ} ≤ \frac{1}{nγ_{n}} (1 + \frac{1}{nγ_{n}} + \frac{t_{0}}{(nγ_{n})^{γ′/γ}} )^{γ}.

To prove the lemma we only need to show that

\frac{1}{nγ_{n}} (1 + \frac{1}{nγ_{n}} + \frac{t_{0}}{(nγ_{n})^{γ′/γ}} )^{γ} ≤ \frac{1}{(n − 1)γ_{n−1}} ,

or, equivalently,

\frac{n − 1}{n}(1 + \frac{1}{nγ} − \frac{1}{n^{1+δ}γ} + \frac{t_{0}}{(nγ_{n})^{γ′/γ}} )^{γ} ≤ \frac{γ_{n}}{γ_{n−1}} = \frac{1 − (n − 1)^{−δ}}{1 − n^{−δ}} .  (2.5)

Take δ < min{1, γ′/γ − 1}. Then (nγ_{n})^{−(γ′/γ)} is of higher order. It is easy to see that

lim_{n→∞} n^{1+δ}(\frac{n − 1}{n}(1 + \frac{1}{nγ} − \frac{1}{n^{1+δ}γ} )^{γ} − 1) = lim_{n→∞} n^{1+δ}(−\frac{γ}{n^{1+δ}γ} − \frac{γ + 1}{2n^{2}γ}) = −1

and

lim_{n→∞} n^{1+δ}(\frac{1 − (n − 1)^{−δ}}{1 − n^{−δ}} − 1) = lim_{n→∞} n · \frac{1 − (1 − n^{−1})^{−δ}}{1 − n^{−δ}} = −δ.

So we know that as n → ∞, the left side in (2.5) is like 1 − n^{−(1+δ)} and the right side is like 1 − δn^{−(1+δ)}. Since δ < 1, the right side is larger for all large n.

The second inequality in (2.3) can be proved similarly. □

2.1. Maps on the unit interval. Let f be a piecewise smooth expanding map from the unit interval I onto itself with an indifferent fixed point p = 0. Denote by f′ the derivative of f.

Theorem F. Suppose f : I → I is an expanding map with an indifferent fixed point 0 that satisfies Assumptions B(I)-(III) with Λ = I, and near 0 f has the form (2.1) with r > 0. Then for any potential φ such that π^{∗}φ satisfies Assumptions A(I)-(III) with β > 0, or Assumptions A(I)-(III) and (III′) with −1 < β ≤ β′ ≤ 0, Theorems A-E and their corollaries hold.

In particular, if we take φ(x) = − log f′(x), then φ satisfies Assumptions A(I)- (III) and (III′) with α = β = β′ = γ = r^{−1}. In this case, the measure ν obtained in Theorem A is the Lebesgue measure, the measure μ is an absolutely continuous invariant measure, and the density function h satisfies h^{∗}(¯1) = 0. Moreover, μ is finite if 0 < r < 1 and infinite if 1 ≤ r < ∞.

Remark 2.1. If we take γ = r^{−1}, then the map π : Σ_{A}^{+} → I is Lipschitz. So any Hölder potential on I satisfies the condition in the theorem.

Remark 2.2. The requirement of conditions (2.1) can be slightly relaxed. For example, the same proof can go through if we assume x + ax^{1+r} ≤ fx ≤ x + bx^{1+r} for some 0 < a ≤ b.

These systems with potential φ(x) = − log f′(x) have been studied extensively. The part concerning the existence of the absolutely continuous invariant measure μ is well known (see e.g. [Pi], [T]). It is proved that μ is a weak Gibbs measure (see e.g. [Yu2], [Yu3]) and an equilibrium (see e.g. [Yu1], [S1]).

The rates of convergence to the equilibriums and rates of decay of correlations are also well known for the case r ∈ (0, 1) ([Y2], [H1], [S4], [G], also [LiSV], [PY]).

Proof of Theorem F. We only need to show that if φ(x) = − log f′(x), then it satisfies Assumptions A(I)-(III) and (III′). Let α = β = β′ = γ = r^{−1}.

Clearly, φ is continuous.

By (2.1) and (2.2), there exist t_{1}, t_{2} > 0 such that

f′(x) ≥ 1 + (1 + r)x^{r} − t_{1}x^{r′},
f′′(x) ≤ r(1 + r)x^{−1+r} + t_{2}x^{−1+r′} ≤ Cx^{−1+r}

for some C ≥ r(1 + r).

Let k be a large integer and x ∈ P_{k}. By the first inequality in (2.3), we have x^{r} ≥ (k + k_{0})^{−1}r^{−1} and therefore

f′′(x) ≤ C((k + k_{0})r)^{α(1−r)} ≤ Cr^{α−1}(k + k_{0})^{α−1} ≤ C_{φ}k^{α−1}  (2.8)

for some C_{φ} ≥ C(2r)^{α−1} > 0 if k ≥ k_{0}. Also, note that f′(x) is bounded below by

1 + (1 + r)\frac{1}{(k + k_{0})r} (1 − \frac{1}{(k + k_{0})^{δ}} ) − t_{1}(\frac{β}{k + k_{0}} )^{βr′}(1 − \frac{1}{(k + k_{0})^{δ}} )^{βr′}.

Since βr′ ≥ 1 + δ, we can find C_{δ} ≥ C′_{δ} > 0 such that

f′(x) ≥ 1 + \frac{1 + r}{(k + k_{0})r} − \frac{C′_{δ}}{(k + k_{0})^{1+δ}} ≥ 1 + \frac{β + 1}{k} − \frac{C_{δ}}{k^{1+δ}} .  (2.9)

Hence, Assumptions A (II) and (III) follow from the definition of φ and the fact that s > log(1 + s) ≥ s − s^{2}/2.

Assumption A (III′) can be obtained similarly. □

2.2. Parabolic rational maps. A rational map f : ℂ̄ → ℂ̄ on the Riemannian sphere ℂ̄ with degree larger than or equal to 2 is parabolic if its Julia set J = J(f) contains indifferent fixed points or periodic orbits but no critical point. The equivalent condition is that restricted to J, the map is positive expansive but not expanding in the spherical metric ([DU2]). The map has Markov partitions of arbitrarily small diameter ([DU3]).

In the case that an indifferent orbit contains more than one point, we can take f^{n} to get indifferent fixed points.

We say that a measure μ on J is a measure of full Hausdorff dimension if dim_{H}(μ) = dim_{H}(J), where dim_{H}(μ) = inf{dim_{H}(Λ_{0}) : Λ_{0} ⊂ Λ, μ(Λ_{0}) = 1}.

Theorem G. Suppose f : J → J is the restriction to the Julia set of a parabolic rational map on the Riemannian sphere with an indifferent fixed point p and where f is topologically mixing. Then for any potential φ such that π^{∗}φ satisfies Assumptions A(I)-(III) with β > 0, or Assumptions A(I)-(III) and (III′) with −1 < β ≤ β′ ≤ 0, Theorems A-E and their corollaries hold.

Further, we suppose that near p, the Taylor expansion of f can be written as f(z) = z + z^{1+r}+higher order terms, and that t is the Hausdorff dimension of J. If we take φ(x) = −t log |f′(x)|, then φ satisfies Assumptions A(I)-(III) and (III′) with α = γ = r^{−1} and β = β′ = t(1+r^{−1})−1. In this case, the conformal measure ν and invariant measure μ are measures of full Hausdorff dimension, and the density function h satisfies h^{∗}(¯1) = 0. Moreover, μ is finite if 2 < t(1 + r^{−1}) < ∞, and infinite if 1 < t(1 + r^{−1}) ≤ 2.

For potentials of the form φ(x) = −t log |f′(x)|, existence of conformal measures and invariant measures of such maps was proved, and some statistical properties such as central limit theorems and the wandering rates have been established by Denker, Urbański and Aaronson (see e.g. [DU2]-[DU4], [ADU] and their references). We refer to [U2], Section 3, for complete information on what is known before.

For general rational maps f on ℂ̄ and general Hölder potential φ, it is known that if P (f, φ) > sup{|φ(x)| : x ∈ J}, then all of the main results corresponding to P (f, φ) > 0 = φ(¯0) in the above theorem hold (see [DU1], [Pr], [Ha]).

Remark 2.3. The Hausdorff dimension dim_{H}(J) of J is larger than r/(1 + r) (see e.g. [ADU]). So we always have 1 < dim_{H}(J)(1 + r^{−1}).

Remark 2.4. Note that here a conformal measure ν means an e^{log λ−φ}-conformal measure. That is, ν satisfies (1.6). If λ = 1, then ν is the same conformal measure studied by Denker and Urbeński.

Proof of Theorem G. It is obvious that Assumptions B(I)-(III) are satisfied. So we only need consider the case φ(x) = −t log f′(x).

Assumption A(I) is clear. Assumption A(II) follows from the definition of φ and the same arguments for (2.8). For Assumption A(III), by the same arguments for (2.9) we get that for x ∈ P_{k},

−φ(x) ≥ t log(1 + \frac{1 + r^{−1}}{k} − \frac{C_{δ}}{k^{1+δ}} ) = \frac{t(1 + r^{−1})}{k} − \frac{tC_{δ}}{k^{1+δ}} + O(\frac{1}{k^{2}} )

for some C_{δ} > 0. Assumption A(III′) can be obtained in a similar way.

By Corollary C.1 and Theorem A, P (φ, f) = log λ ≥ φ(0) = 0. So by Theorem D,

0 ≤ P (φ, f) = h_{μ}(f) + ∫ φdμ = h_{μ}(f) − t ∫ log f′dμ.

That is,

t ≤ \frac{h_{μ}(f)}{∫ log f′dμ}.

Since μ is ergodic, and f is a conformal map, the right side of the equality is equal to dim_{H}(μ). So we get t ≤ dim_{H}(μ) and therefore dim_{H}(J) = t ≤ dim_{H}(μ). It means that μ is a measure of full Hausdorff dimension. Since μ ≪ ν, ν is also a measure of full Hausdorff dimension. □

2.3. Parabolic Cantor sets. Let f : I → ℝ^{+} be a piecewise smooth expanding map.

Denote Λ = {x ∈ I : f^{n}x ∈ I ∀n ≥ 0}. Clearly, fΛ = Λ and 0 ∈ Λ. If f(I) = I, then Λ = I, and it becomes the same case studied in Subsection 2.1. If f(I) ⊃ I, then Λ is a Cantor set topologically.

Let t be the Hausdorff dimension of Λ.

Theorem H. Suppose f|_{Λ} : Λ → Λ is an expanding map with an indifferent fixed point 0 that satisfies Assumptions B(I)-(III), and near 0, f has the form (2.1) with r > 0. Then the same conclusions stated in Theorem G hold, including the case φ(x) = −t log f′(x), the choice of α, β, β′ and γ, and the conditions for finiteness of μ.

Parabolic Cantor sets were studied by M. Urbański (see [U1] and its reference). He obtained the existence of invariant measure and conformal measure of full Hausdorff dimension and investigated the equilibrium state and the phase transition of the systems.

Proof of Theorem H. It is the same as for the proof of Theorem G. □

2.4. Maps on higher dimensional spaces. We can generalise the results in Subsection 2.1 to a higher dimensional case. For a map f from the m-dimensional cube I^{m} to itself, we denote by det Df(x) the determinant of Df at x.

Theorem I. Suppose f : I^{m} → I^{m} is an expanding map with an indifferent fixed point 0 that satisfies Assumptions B (I)-(III) with Λ = I^{m}, and near 0, f has the form (2.1) with r > 0. Then similar results stated in Theorem F hold with φ(x) = − log | det Df(x)| and α = γ = r^{−1}, β = β′ = mr^{−1}. Also, μ is finite if 0 < r < m and infinite if m ≤ r < ∞.

Some expanding maps with indifferent fixed points in higher dimensional space are studied by M. Yuri ([Yu1]-[Yu4]), and the maps we discuss here also satisfy her assumptions, though she did not give these kinds of examples explicitly.

Remark 2.5. In these examples we require that near 0 the map f has about the same expanding rates along different radial directions. If f has two neutral directions along which f has different expanding rates, then Proposition 3.1 fails to hold and therefore Assumption A(II) cannot be true (see examples in [HV]).

Remark 2.6. We can also discuss the case that Λ is a fractal in I^{m} which has the form {(c, s) : c ∈ Γ, s ∈ S} near the fixed point, where Γ[0, ∞) is a parabolic Cantor set and and S ⊂ S^{m−1} is a fractal.

Proof of Theorem I. We only need to verify β = β′ = mr^{−1}. The rest proof is the same as for Theorem F.

In fact,

| det Df(x)| = (1 + |x|^{r})^{m−1}(1 + (1 + r)|x|^{r}) + O(|x|^{r′}) = 1 + (m + r)|x|^{r} + O(|x|^{r′}).

If x ∈ P_{k}, then by (2.3),

−φ(x) = log | det Df(x)| = log[(1 + (m + r)|x|^{r}) + O(|x|^{r′})]
≥\frac{(m + r)}{(k + k_{0})r} − \frac{C′_{δ}}{(k + k_{0})^{1+δ}} ≥ \frac{mr^{−1} + 1}{k} − \frac{C_{δ}}{k^{1+δ}} .

So we have β = mr^{−1}. β′ = mr^{−1} can be obtained similarly. □

3. The operator L_{φ}: Proof of Theorem A

Proposition 3.1. There is J_{φ} > 0 such that for all J ≥ J_{φ}, the following holds: i) For x, y ∈ Q_{k},

|φ(0x) − φ(0y)| + JD_{k+1}^{α}d(0x, 0y)^{θ} ≤ JD_{k}^{α}d(x, y)^{θ},
and if s ≠ 0, then for x, y ∈ Σ_{A}^{+},
|φ(sx) − φ(sy)| + JK_{0}^{α}d(sx, sy)^{θ} ≤ JK_{0}^{α}d(x, y)^{θ}.
ii) For x, y ∈ Q_{k}, w = w_{0}w_{1} ⋯ w_{n−1} with wx, wy ∈ Q_{m},
|S_{n}φ(wx) − S_{n}φ(wy)| + JD_{m}^{α}d(wx, wy)^{θ} ≤ JD_{k}^{α}d(x, y)^{θ},
and if w_{n−1} ≠ 0, then for x, y ∈ Σ_{A}^{+},
|S_{n}φ(wx) − S_{n}φ(wy)| + JD_{m}^{α}d(wx, wy)^{θ} ≤ JK_{0}^{α}d(x, y)^{θ}.

Proof. i) First we assume k ≥ K_{0} and x, y ∈ Q_{k}. It is easy to check by the definition that d(0x, 0y) ≤ k^{γ+1}(k + 1)^{−(γ+1)}d(x, y). By Assumption A(II),

|φ(0x) − φ(0y)| + J(k + 1)^{α}d(0x, 0y)^{θ}
≤C_{φ}(k + 1)^{α−1}d(0x, 0y)^{θ} + J(k + 1)^{α}d(0x, 0y)^{θ}
≤J · (C_{φ} \frac{k^{θ(γ+1)−α}}{J(k + 1)^{θ(γ+1)−α+1}} + \frac{k^{θ(γ+1)−α}}{(k + 1)^{θ(γ+1)−α}} )k^{α}d(x, y)^{θ}.

If J_{φ} is large enough and J ≥ J_{φ}, then the quantity in the parentheses is less than 1.

If k < K_{0} or s ≠ 0, then we take J_{φ} > 0 such that (C_{φ}K_{0}^{α} + JK_{0}^{α})κ^{θ} ≤ JK_{0}^{α} whenever J ≥ J_{φ}. Therefore we can get the results in a similar way.

ii) It can be obtained from i) by induction. □

Recall that ĝ is defined in (1.7).

Corollary 3.2. Let J ≥ J_{φ} and x = 0x̄, y = 0ȳ ∈ O_{2}. If g(sȳ) ≤ g(sx̄)e^{JK_{0}^{α}d(sx̄,sȳ)^{θ}} ∀s ≠ 0, then ĝ(y) ≤ ĝ(x)e^{JK_{0}^{α}d(x,y)^{θ}}.

Proof. It follows from the above proposition and the fact that

\frac{ĝ(y)}{ĝ(x)} ≤ max_{s≠0}\{\frac{e^{φ(sȳ)}g(sȳ)}{e^{φ(sx̄)}g(sx̄)}\} ≤ max_{s≠0}\{e^{φ(sȳ)−φ(sx̄)}e^{JK_{0}^{α}d(sx̄,sȳ)^{θ}}\}.

By Proposition 3.1.i), the right side is less than or equal to e^{JK_{0}^{α}d(x,y)^{θ}}. □

Proof of Theorem A. By Lemma 3.3 there exist λ^{∗} ≥ e^{φ(¯0)} = 1, and ν ∈ M(Σ_{A}^{+}), which is positive on nonempty open sets, such that L_{φ}^{∗}ν = λ^{∗}ν.

Fix J ≥ J_{φ} and take a constant J^{∗} > J_{φ} large enough, which can be determined in the proof of Lemma 3.5. Let B = C^{0}(Σ_{A}^{+}\{¯0}) but with the norm

∥g∥ = sup_{x∈Σ_{A}^{+}\{¯0}}{e^{−J^{∗}k(x)}g(x)},  (3.1)

where k(x) = k if x ∈ P_{k}. It is easy to check that B is a Banach space. Lemma 3.4 below implies that L_{φ} : B → B is continuous. Take

H = \{g ∈ \overline{G}_{J} : g(¯1) = 1\}.

H is not empty since it contains a constant function g(x) = 1. Clearly, H is a convex set. By Lemma 3.5, H is compact.

Define an operator L̄ : B → B by L̄g = L_{φ}g/(L_{φ}g)(¯1). L̄ is continuous because L_{φ} is continuous. By Lemma 3.6, L̄H ⊂ H. By the Schauder-Tychonoff Fixed Point Theorem (see e.g. [DS]), L̄ has a fixed point h ∈ H. So we have h ∈ \overline{G}_{J}, ∀J ≥ J_{φ}, and L_{φ}h = ((L_{φ}h)(¯1))h. Denote λ = (L_{φ}h)(¯1).

By Lemma 3.7, λ = λ^{∗}. Hence, we also have L_{φ}^{∗}ν = λν.

To prove that μ is σ-invariant, we can check directly that L_{φ}(h·(g∘σ)) = g·(L_{φ}h); then μ(g ∘ σ) = ν(h · (g ∘ σ)) = ν(λ^{−1}L_{φ}(h · [g ∘ σ])) = ν((λ^{−1}L_{φ}h) · g) = ν(h · g) = μ(g). (See e.g. [B] for more details.)

The part λ > 1 if −1 < β′ ≤ 0 is proved in Lemma 4.5.

The last part of the theorem, concerning conditions under which μ is finite or infinite, follows from Corollary A.2 and the fact that Σ_{A}^{+} = ⋃_{i=0}^{∞} P_{i}. □

Lemma 3.3. There is a real number λ^{∗} ≥ e^{φ(¯0)}, and a measure ν ∈ M(Σ_{A}^{+}), which is positive on nonempty open sets, such that L_{φ}^{∗}ν = λ^{∗}ν.

Proof. The map μ → L_{φ}^{∗}μ /(L_{φ}^{∗}μ)(1) is a continuous map from M(Σ_{A}^{+}) to itself. Since M(Σ_{A}^{+}) is compact in weak^{∗} topology, by the Schauder-Tychonoff Fixed Point Theorem the map has a fixed point ν. So if we take λ^{∗} = (L_{φ}^{∗}μ)(1), then L_{φ}^{∗}ν = λ^{∗}ν.

To prove νU > 0 for any open set U, it is enough to prove that for any word u, νR_{u} > 0. Since σ is topologically mixing, σ^{n}R_{u} = Σ_{A}^{+} for some n > 0. Hence for any x ∈ Σ_{A}^{+}, there is an n-word v such that vx ∈ R_{u}. We have

L_{φ}^{n}χ_{R_{u}}(x) = ∑_{w∈ξ_{n}} e^{S_{n}φ(wx)}χ_{R_{u}}(wx) ≥ e^{S_{n}φ(vx)}χ_{R_{u}}(vx) ≥ e^{−n∥φ∥} > 0,

where ξ_{n} is defined in (1.1). So

νR_{u} = \frac{1}{(λ^{∗})^{n}} (L_{φ}^{∗})^{n}ν(χ_{R_{u}}) ≥ \frac{1}{(λ^{∗})^{n}} ν(L_{φ}^{n}χ_{R_{u}}) ≥ \frac{ν(e^{−n∥φ∥})}{(λ^{∗})^{n}} = \frac{1}{(λ^{∗}e^{∥φ∥})^{n}} > 0.

Now we prove λ^{∗} ≥ e^{φ(¯0)}. Suppose λ^{∗} < e^{φ(¯0)}. Since φ is a continuous function, there is k > 0 such that (λ^{∗})^{−1}e^{φ(x)} > 1 ∀x ∈ O_{k}. Note that if y ∈ P_{i+1}, then σy ∈ P_{i}. We get that for i ≥ k,

νP_{i+1} = ν((λ^{∗})^{−1}L_{φ}χ_{P_{i+1}}) = ∫ \frac{1}{λ^{∗}} ∑_{σy=x} e^{φ(y)}χ_{P_{i+1}}(y)dν(x)

= ∫ \frac{1}{λ^{∗}} e^{φ(0x)}χ_{P_{i}}(x)dν(x) > ∫ χ_{P_{i}}(x)dν(x) = νP_{i}.  (3.2)

Hence, we have νP_{i} > νP_{k} ∀i > k, contradicting finiteness of ν. □

Lemma 3.4. L_{φ} is a bounded linear operator.

Proof. Take g(x) = e^{J^{∗}k(x)}, where k(x) = k if x ∈ P_{k}. Clearly g is the maximal element in the unit ball with respect to the norm in (3.1). Since L_{φ} is a positive operator, we only need to prove that e^{−J^{∗}k(x)}L_{φ}g(x) is bounded.

Note that k(0x) = k(x) + 1 and k(sx) = 0 if s ≠ 0. So

e^{−J^{∗}k(x)}L_{φ}g(x) = e^{−J^{∗}k(x)}(e^{φ(0x)}e^{J^{∗}k(0x)} +∑_{s≠0} e^{φ(sx)}e^{J^{∗}k(sx)}) ≤ e^{∥φ∥+J^{∗}} +r^{∗}e^{∥φ∥},

where r^{∗} is the number of different symbols used in Σ_{A}^{+}. □

Lemma 3.5. The set H is compact.

Proof. For any g ∈ H ⊂ \overline{G}_{J}, we have

g(y) ≤ g(x)e^{JD_{k}^{α}d(x,y)^{θ}} ≤ g(x)e^{Jk^{−θ(1+γ)+α}}

if x ∈ P_{k}, y ∈ P′_{k} and k > K_{1}. So if x ∈ P_{k}, then we can choose x^{(i)} ∈ P_{k−i}, i = 1, ⋯, k, and x^{(k)} = ¯1. Denote x = x^{(0)}. By the above inequality we get

g(x) ≤ g(¯1) ∏_{i=0}^{k−1} g(x^{(i)})/g(x^{(i+1)}) ≤ e^{∑_{i=K_{0}}^{k−1} J(k−i)^{−θ(1+γ)+α}} ≤ e^{J^{∗}k^{1−θ(1+γ)+α}} (3.3)

for some J^{∗} > 0 independent of g and x^{(i)}. Since α ≤ θ(1 + γ), k^{1−θ(1+γ)+α} < k. So e^{−J^{∗}k(x)}g(x) → 0 as x → ¯0, and the convergence is uniform for all g ∈ H.

Since H ⊂ \overline{G}_{J}, it is easy to see that H is uniformly bounded and equicontinuous outside O_{k} for any large k. With the above arguments we know that H is uniformly bounded and equicontinuous.

Clearly, H is closed in B. We get the result. □

Lemma 3.6. L̄H ⊂ H.

Proof. Take g ∈ H. We prove L̄g ∈ H.

Take x ∈ P_{k}, y ∈ P′_{k}. Then

\frac{L̄g(y)}{L̄g(x)} = \frac{∑_{sy∈Σ_{A}^{+}} e^{φ(sy)}g(sy)}{∑_{sx∈Σ_{A}^{+}} e^{φ(sx)}g(sx)} ≤ max_{sx,sy∈Σ_{A}^{+}}\{\frac{e^{φ(sy)}g(sy)}{e^{φ(sx)}g(sx)}\}.

Note that \frac{g(sy)}{g(sx)} ≤ e^{JK_{0}^{α}d(sx,sy)^{θ}} if s ≠ 0 and \frac{g(0y)}{g(0x)} ≤ e^{JD_{k+1}^{α}d(0x,0y)^{θ}}. By Propo-

sition 3.1, the right side of the above inequality is bounded by e^{JD_{k}^{α}d(x,y)^{θ}}. So L̄g ∈ \overline{G}_{J}.

It is clear that L̄_{φ}g > 0 and (L̄_{φ}g)(¯1) = 1. So L̄g ∈ H. □

Lemma 3.7. λ = λ^{∗}.

Proof. Define h^{(n)} by

h^{(n)}(x) = \{ \begin{matrix}h(x) & if x ∈ Q_{n}; \\ 0 & otherwise. \end{matrix}

We have L_{φ}h^{(n)} = λh^{(n−1)} + ĥχ_{O_{n}}, where ĥ(x) is defined in (1.7). So

\begin{matrix}ν(h^{(n)}) = & (λ^{∗})^{−1}L_{φ}^{∗}(ν)(h^{(n)}) = (λ^{∗})^{−1}ν(L_{φ}h^{(n)}) \\ = & (λ^{∗})^{−1}λν(h^{(n−1)}) + (λ^{∗})^{−1}ν(ĥχ_{O_{n}}). \end{matrix}

Since ν(h^{(n)}) = ν(h^{(n−1)}) + ν(hχ_{P_{n}}), we have

(1 − (λ^{∗})^{−1}λ)ν(h^{(n−1)}) = −ν(hχ_{P_{n}}) + (λ^{∗})^{−1}ν(ĥχ_{O_{n}}).  (3.4)

Note that ν(ĥχ_{O_{n}}) → 0 as n → ∞ since h is bounded on Q_{0} and νO_{n} → 0. Now we prove ν(hχ_{P_{n}}) → 0. This implies λ = λ^{∗}, because ν(h^{(n−1)}) increases with n.

Since ν(hχ_{P_{n}}) = μP_{n} and μP_{n} decreases, the sequence {ν(hχ_{P_{n}})} decreases with n. So if it does not converge to 0, then it is bounded away from 0. Hence by (3.4), ν(hχ_{P_{n}}) is roughly proportional to ν(h^{(n−1)}). Since ν(h^{(n)}) = ν(h^{(n−1)})+ν(hχ_{P_{n}}), it implies that ν(h^{(n)}) and ν(hχ_{P_{n}}) increase exponentially fast. So restricted to P_{n}, h(x) increases exponentially fast. It contradicts (3.3) which says that functions in \overline{G}_{J} increase subexponentially. □

4. The density function: Proof of Corollary A.1

Lemma 4.1. For x ≠ ¯0, lim_{n→∞} h(0^{n}x) = \frac{ĥ(¯0)}{λ − 1} if λ > 1 and lim_{n→∞} h(0^{n}x) = ∞ if λ = 1. Proof. Since L_{φ}h = λh, we have e^{φ(0^{n+1}x)}h(0^{n+1}x) + ĥ(0^{n+1}x) = λh(0^{n}x), where ĥ is defined in (1.7). Hence

h(0^{n+1}x) = e^{−φ(0^{n+1}x)}(λh(0^{n}x) − ĥ(0^{n+1}x)),  (4.1)

or

\frac{h(0^{n+1}x)}{h(0^{n}x)} = e^{−φ(0^{n+1}x)}(λ − \frac{ĥ(0^{n+1}x)}{h(0^{n}x)}).

Since h ∈ Ḡ_{J}, h(0^{n+1}x)/h(0^{n}x) ≤ e^{J_{φ}d(0^{n+1}x,0^{n}x)} → 1 as n → ∞. We also have e^{−φ(0^{n+1}x)} → e^{−φ(¯0)} = 1 and ĥ(0^{n+1}x) → ĥ(¯0). The result of the lemma follows. □

Recall that h^{∗} is defined in the statement of Theorem A. Lemma 4.2. Let x ≠ ¯0.

h(x) = λ^{−n}e^{S_{n}φ(0^{n}x)}h(0^{n}x) + ∑_{j=1}^{n} ∑_{s≠0} λ^{−j}e^{S_{j}φ(s0^{j−1}x)}h(s0^{j−1}x),  (4.2)

and if λ = 1, then h^{∗}(x) = lim_{n→∞} e^{S_{n}φ(0^{n}x)}h(0^{n}x). Proof. Note that for any x, h(x) = λ^{−1}e^{φ(0x)}h(0x) + ∑_{s≠0} λ^{−1}e^{φ(sx)}h(sx). The equality holds by induction.

Let λ = 1. The sum in (4.2) increases with n. So we know that e^{S_{n}φ(0^{n}x)}h(0^{n}x) decreases with n and therefore has a limit as n → ∞. Then we use (1.5). □ Lemma 4.3. i) λ > 1 if and only if h^{∗}(x) < 0 for any x ≠ ¯0.

ii) If λ = 1, then either h^{∗}(x) = 0 for all x ≠ ¯0, or h^{∗}(x) > 0 for all x ≠ ¯0. Proof. i) “⇒” If λ > 1, then lim_{n→∞} λ^{−n}e^{S_{n}φ(0^{n}x)}h(0^{n}x) = 0 because h(0^{n}x) increases at most subexponentially, and e^{S_{n}φ(0^{n}x)} decreases by Assumption A(III). So by (4.2), for any x ≠ ¯0, h^{∗}(x) < 0.

ii) Let λ = 1. By Proposition 3.1 and the fact h ∈ \overline{G}_{J}, for any x ∈ P_{k}, y ∈ P′_{k}, e^{S_{n}φ(0^{n}y)}h(0^{n}y)/e^{S_{n}φ(0^{n}x)}h(0^{n}x) ≤ e^{JK_{0}^{α}d(x,y)^{θ}} ≤ e^{JK_{0}^{α}}. So we know that the limit lim_{n→∞} e^{S_{n}φ(0^{n}x)}h(0^{n}x) is either 0 or bounded away from 0 on any P_{k}, and therefore on any Q_{k}. Since h^{∗}(x) is equal to the limit, the result follows.

i) “⇐” If h^{∗}(x) < 0 for some x ≠ ¯0, then by part ii), λ ≠ 1. So λ > 1. □

Lemma 4.4. For any k > 0, there is B_{φ} = B_{φ,k} > 0 with lim_{k→∞} B_{φ,k} = 1 such that

for all n > 0, x ∈ P_{k}, e^{S_{n}φ(0^{n}x)} ≤ B_{φ}(\frac{k}{k + n})^{β+1}.

Suppose Assumption A (III′) also holds. Then for any k > 0, there is B′_{φ} =

B′_{φ,k} > 0 with lim_{k→∞} B′_{φ,k} = 1 such that for all n > 0, x ∈ P_{k}, e^{S_{n}φ(0^{n}x)} ≥

B′_{φ}(\frac{k}{k + n})^{β′+1}.

Moreover, if β = β′, then the limit B_{φ}^{∗} = lim_{n→∞} n^{β+1}e^{S_{n}φ(0^{n}x)} exists.

Proof. By Assumption A (III), there exists A′_{φ}, A_{φ} > 0 such that for all k > 0, if x ∈ P_{k}, then

e^{φ(x)} ≤ 1 − \frac{β}{k + 1} + \frac{A′_{φ}}{(k + 1)^{1+δ}} ≤ (1 − \frac{1}{k + 1})^{β+1}(1 + \frac{A_{φ}}{(k + 1)^{1+δ}} ).

Taking product, we get

e^{S_{n}φ(0^{n}x)} ≤ (\frac{k}{k + n})^{β+1} ∏_{i=k}^{k+n−1}(1+ \frac{A_{φ}}{(i + 1)^{1+δ}} ) ≤ (\frac{k}{k + n})^{β+1} ∏_{i=k}^{∞}(1+ \frac{A_{φ}}{(i + 1)^{1+δ}} ).

We let B_{φ,k} be the product, which is convergent. Clearly, lim_{k→∞} B_{φ,k} = 1.

The results corresponding to Assumption A (III′) can be obtained in a similar way.

Let β = β′. We know that the sequence {n^{β+1}e^{S_{n}φ(0^{n}x)}} is bounded. Note that e^{S_{n}φ(0^{n}x)} = e^{S_{n−k}φ(0^{n}x)}e^{S_{k}φ(0^{k}x)} is bounded between B′_{φ,k}(k/n)^{β+1}e^{S_{k}φ(0^{k}x)} and B_{φ,k}(k/n)^{β+1}e^{S_{k}φ(0^{k}x)}. We have

B′_{φ,k}k^{β+1}e^{S_{k}φ(0^{k}x)} ≤ n^{β+1}e^{S_{n}φ(0^{n}x)} ≤ B_{φ,k}k^{β+1}e^{S_{k}φ(0^{k}x)}

for any n ≥ k. Also, lim_{k→∞} B′_{φ,k} = lim_{k→∞} B_{φ,k} = 1. So {n^{β+1}e^{S_{n}φ(0^{n}x)}} is a Cauchy

sequence, and therefore is convergent. □

Lemma 4.5. For any k ≥ 0, there exist B_{ν} = B_{ν,k} > 0 and C_{ν} = C_{ν,k} > 0 such that the limits lim_{k→∞} B_{ν,k} and lim_{k→∞} C_{ν,k} exist, and for all n ≥ k,

νP_{n} ≤ λ^{−n}B_{ν}n^{−(β+1)}, νO_{n} ≤ λ^{−n}C_{ν}n^{−β}.

Suppose Assumption A (III′) also holds. Then the above inequalities hold if we replace β, B_{ν}, C_{ν} and “ ≤” by β′, B′_{ν}, C′_{ν} and “ ≥”, respectively. Hence, if β′ ≤ 0, then λ > 1.

Moreover, if β = β′, then B_{ν,k} and B′_{ν,k} can be chosen in such a way that

lim_{k→∞} B_{ν,k} = lim_{k→∞} B′_{ν,k} = lim_{n→∞} λ^{n}n^{β+1}νP_{n} = B_{ν}^{∗}.  (4.3)

Proof. By a similar method for (3.2), then by Proposition 3.1 and Lemma 4.4, for all large k, we have

νP_{n} = \frac{1}{λ^{n−k}} ∫_{P_{k}} e^{S_{n−k}φ(0^{n−k}x)}dν(x) ≤ B_{φ,k} \frac{1}{λ^{n−k}} (\frac{k}{n})^{β+1}νP_{k}.  (4.4)

So λ^{n}n^{−(β+1)}νP_{n} is bounded. We can take B_{ν,k} = B_{φ,k} max_{n≥k}{λ^{n}n^{−(β+1)}νP_{n}}; then

we get the upper bound estimates for νP_{n}.

The estimate for νO_{n} follows from the fact that O_{n} = ⋃_{i≥n} P_{i}.

If Assumption A (III′) also holds, the lower bound estimates can be made similarly. So if β′ ∈ (−1, 0], then λ > 1 since ν is a probability measure.

Let β = β′. The sequence {λ^{n}n^{−(β+1)}νP_{n}} is bounded. By (4.4) we have

B′_{ν,k}λ^{k}k^{−(β+1)}νP_{k} ≤ λ^{n}n^{−(β+1)}νP_{n} ≤ B_{ν,k}λ^{k}k^{−(β+1)}νP_{k}

for any n ≥ k. Also, lim_{k→∞} B′_{φ,k} = lim_{k→∞} B_{φ,k} = 1. So {λ^{n}n^{−(β+1)}νP_{n}} is a Cauchy

sequence, and therefore the limits in (4.3) exist. □

The next lemma is for the proof of uniqueness of equilibriums in Theorem D and phase transition in Corollary D.1.

Lemma 4.6. i) If h^{∗}(¯1) > 0, then there exists a continuous function φ^{+} > φ on Σ_{A}^{+}\{¯0} such that for any φ^{+} ≥ φ′ ≥ φ satisfying Assumptions (I)-(III), h′^{∗}(¯1) > 0, where h′ is the density function given in Theorem A for φ′.

If Assumption A(III′) also holds with β′ = β, then φ^{+} can be taken in such a way that φ^{+}(x) − φ(x) > a/k for some a > 0 as x ∈ P_{k} and for all large k.

ii) If there exists φ′ ≥ φ with φ′ ≠ φ that satisfies Assumptions (I)-(III) such that h′^{∗}(¯1) ≥ 0, then h^{∗}(¯1) ≥ 0.

Proof. i) From the proof of Lemma 4.3 we know that if h^{∗}(¯1) > 0, then h^{∗}(x) ≥ c > 0 for some c as x ∈ P_{0}. By the definition of h^{∗} given in (1.5) and L̃φ̃ given in (1.4) we know that h^{∗}(¯1) > 0 implies h(x) > L̃φ̃h(x) + c for any x ∈ P_{0}. So by continuity we can find φ^{+} > φ such that for any φ^{+} ≥ φ′ ≥ φ, h(x) > L̃_{}_{φ̃′}h(x).

Take φ that satisfies Assumptions (I)-(III), and φ^{+} ≥ φ′ ≥ φ. Note that L̃_{}_{φ̃′} maps the set of continuous functions on P_{0} to itself. So by the similar method as in the proof of Lemma 3.3, we get that there is a conformal measure ν′ on P_{0} and a constant λ′ such that L̃_{φ̃′}^{∗}ν′ = λ′ν′. Hence,

λ′ν′(h) = (L̃_{φ̃′}^{∗}ν′)(h) = ν′(L̃_{}_{φ̃′}h) < ν′(h).

That is, λ′ < 1.

Let h′ be the density function obtained in Theorem A for φ′. We claim h′^{∗}(¯1) > 1. In fact, if not, then by Lemma 4.3 we have h′(x) ≤ L̃_{}_{φ̃′}h′(x) for all x ∈ P_{0}. Then the same argument gives λ′ν′(h′) ≥ ν′(h). It implies λ′ ≥ 1, a contradiction.

Now we assume that Assumption A(III′) also holds and β = β′. By Lemma 4.3, h′^{∗}(¯1) > 1 implies λ = 1, and then by Lemma 4.5 we have β = β′ > 0. Take φ_{1}^{+} ≥ φ such that φ_{1}^{+} > φ on Q_{K_{1}}, where K_{1} is given in Assumption (III), and such that

∑_{j=1}^{∞} ∑_{s≠0} e^{S_{j}φ_{1}^{+}(s0^{j−1}x)}h(s0^{j−1}x) − ∑_{j=1}^{∞} ∑_{s≠0} e^{S_{j}φ(s0^{j−1}x)}h(s0^{j−1}x) ≤ c/2.

This is possible; for example, we can let φ_{1}^{+} = φ on O_{K_{1}+1}. By Lemma 4.4 we know that there exists B > 0 such that e^{S_{j}φ_{1}^{+}(s0^{j−1}x)} ≤ Bj^{−(1+β)} for all j > 0 and x ∈ P_{0}. So if we take φ_{a}(x) = φ_{1}^{+}(x) + a/k for all x ∈ P_{k} and k ≥ K_{1}, and φ_{a}(x) = φ_{1}^{+}(x) for all x ∈ Q_{K_{1}}, then

∑_{j=K_{1}}^{∞} ∑_{s≠0} e^{S_{j}φ_{a}(s0^{j−1}x)}h(s0^{j−1}x) → ∑_{j=K_{1}}^{∞} ∑_{s≠0} e^{S_{j}φ_{1}^{+}(s0^{j−1}x)}h(s0^{j−1}x),

as a → 0. Hence we can take a > 0 small enough such that the difference is less than c/2. Therefore φ^{+} = φ_{a} is the function we need.

ii) It can be proved in a similar way. □

The next two lemmas are for the case h^{∗}(¯1) = 0.

Lemma 4.7. Suppose h^{∗}(¯1) = 0. Then there exists H > 0 such that for all k > 0, h(x) ≤ Hk if x ∈ P_{k}.

Proof. Since h^{∗}(¯1) = 0, we have that for any x ≠ ¯0,

h(x) = ∑_{j=1}^{∞} ∑_{s≠0} e^{S_{j}φ(s0^{j−1}x)}h(s0^{j−1}x).

Since h is bounded on P_{0}, we denote by H_{0} the upper bound. Also, e^{S_{j}φ(s0^{j−1}x)} = e^{φ(s0^{j−1}x)}e^{S_{j−1}φ(0^{j−1}x)}. So if x ∈ P_{k}, then by Lemma 4.4, we get

h(x) ≤ r^{∗}e^{∥φ∥}H_{0} ∑_{j=1}^{∞} e^{S_{j−1}φ(0^{j−1}x)} ≤ r^{∗}e^{∥φ∥}H_{0}B_{φ} ∑_{j=1}^{∞}(\frac{k}{k + j − 1})^{β+1} ≤ Hk

for some H > 0. □

Lemma 4.8. Suppose h^{∗}(¯1) = 0. Then there is C_{h} > 0 such that for all x ∈ P_{0}, for all large n,

\frac{β}{n}h(0^{n}x) ≤ ĥ(¯0)(1 + \frac{C_{h}}{n^{δ}} ).  (4.5)

If φ also satisfies Assumption A(III′) and h^{∗}(¯1) = 0, then there is C′_{h} > 0 such that for all x ∈ P_{0}, for all large n,

\frac{β′}{n} h(0^{n}x) ≥ ĥ(¯0)(1 − \frac{C′_{h}}{n^{δ}} ).  (4.6)

Proof. We only prove (4.5). The inequality (4.6) can be proved similarly.

By Lemma 4.3, we have λ = 1 and therefore by Theorem A, β > 0. Recall that we assume δ ≤ min{1, γθ} after Assumption A(III′) is stated. Take C_{h} > 0 such that for all large n,

\frac{βn − 2 − β}{2(n + 1)^{2}} · \frac{C_{h}}{(n + 1)^{δ}} − \frac{(1 + β)^{2}}{(n + 1)^{2}} − \frac{n + β + 2}{(n + 1)^{2}} · \frac{2βJ_{0}C_{γ}}{n^{γθ}} − \frac{C_{δ}(n + ϵn − β)}{(n + 1)^{2+δ}} > 0,

where J_{0} = J_{φ}K_{0}^{α}.

First we claim that if there is x ∈ P_{0} such that for ϵ ≥ C_{h}n^{−δ},

\frac{β}{n}h(0^{n}x) ≥ ĥ(¯0)(1 + ϵ)  (4.7)

holds for some large n, then

\frac{β}{n + 1}h(0^{n+1}x) ≥ ĥ(¯0)(1 + ϵ + \frac{β}{2(n + 1)}ϵ).  (4.8)

Now we prove the claim. By Lemma 3.3 and Assumption A(III),

e^{φ(¯0)−φ(0^{n+1}x)} ≥ 1 + \frac{β + 1}{n + 1} − \frac{C_{δ}}{(n + 1)^{1+δ}} .  (4.9)

Note that h ∈ \overline{G}_{J_{φ}}. By Corollary 3.2 and (1.2), if n is large enough, then

ĥ(0^{n+1}x) ≤ ĥ(¯0)e^{J_{0}d(0^{n}x,¯0)^{θ}} ≤ ĥ(¯0)e^{J_{0}C_{γ}n^{−γθ}} ≤ ĥ(¯0)(1 + 2J_{0}C_{γ}n^{−γθ}). (4.10)

So by (4.7) and (4.10), if n is large enough, then

\frac{1}{ĥ(¯0)}(λh(0^{n}x) − ĥ(0^{n+1}x)) ≥\frac{n}{β}(1 + ϵ) − (1 + \frac{2J_{0}C_{γ}}{n^{γθ}} )
=\frac{n − β}{β} + \frac{n}{β}ϵ − \frac{2J_{0}C_{γ}}{n^{γθ}} ≥ \frac{n − 2β}{β} . (4.11)

By (4.1) and then by (4.9) and (4.11), we have

\frac{β}{n + 1} \frac{h(0^{n+1}x)}{ĥ(¯0)} ≥ \frac{β}{n + 1} · e^{−φ(0^{n+1}x)} · \frac{1}{ĥ(¯0)}(h(0^{n}x) − ĥ(0^{n+1}x))

≥\frac{β}{n + 1} · \frac{n + β + 2}{n + 1} · [\frac{n − β}{β} + \frac{n}{β}ϵ − \frac{2J_{0}C_{γ}}{n^{γθ}} ] − \frac{β}{n + 1} · \frac{C_{δ}}{(n + 1)^{1+δ}} · \frac{n + ϵn − 2β}{β}
=\frac{(n + β + 2)(n − β)}{(n + 1)^{2}} + \frac{(n + β + 2)n}{(n + 1)^{2}}ϵ − \frac{n + β + 2}{(n + 1)^{2}} · \frac{2βJ_{0}C_{γ}}{n^{γθ}} − \frac{C_{δ}(n + ϵn − 2β)}{(n + 1)^{2+δ}}.

Note that

\frac{(n + β + 2)(n − β)}{(n + 1)^{2}} = 1 − \frac{(1 + β)^{2}}{(n + 1)^{2}} ,

\frac{(n + β + 2)n}{(n + 1)^{2}}ϵ = ϵ + \frac{β}{2(n + 1)}ϵ + \frac{βn − 2 − β}{2(n + 1)^{2}} ϵ.

Also note that ϵ ≥ C_{h}n^{−δ}. By the choice of C_{h}, we get (4.8). The claim is true.

Using this claim we can get the result of the lemma. Otherwise we have an ϵ > C_{h}n_{0}^{−δ} such that (4.7) holds for some large n_{0}. Then using the claim repeatedly, we get

\frac{β}{n_{0} + k} h(0^{n_{0}+k}x) ≥ ĥ(¯0)(1 + ϵ + ϵ ∑_{i=n_{0}}^{n_{0}+k−1} \frac{β}{2(i + 1)}) ∀k ≥ 0.

Since the summation goes to infinity as k → ∞, it contradicts the fact given by Lemma 4.7 that h(0^{n}x) ≤ Hn for all n > 0. □

Proof of Corollary A.1. The results for the case h^{∗}(¯1) < 0 and h^{∗}(¯1) = 0 are given in Lemma 4.1 and Lemma 4.8, respectively.

For the case h^{∗}(¯1) > 0, by Lemma 4.2, lim_{n→∞} e^{S_{n}φ(0^{n}x)}h(0^{n}x) = h^{∗}(x), and by

Lemma 4.4, n^{−(β+1)}e^{−S_{n}φ(0^{n}x)} ≥ B_{φ,0}^{−1}. So the first part of the case follows. The rest can be proved similarly.

5. The function ψ and measure μ: Proof of Corollaries A.2 and A.3

Lemma 5.1. Let w be an n-word, n > 0. Then for any integrable function g,

∫ e^{S_{n}ψ(wx)}g(wx)dμ(x) = ∫_{R_{w}} g(x)dμ(x).

Hence, if we take g = χ_{R_{wu}} for any word u, then ∫_{R_{u}} e^{S_{n}ψ(wx)}dμ(x) = μR_{wu}.

Proof. Define g^{∗}(x) = g(x) if x ∈ R_{w} and g^{∗}(x) = 0 otherwise. Then we have e^{S_{n}ψ(wx)}g(wx) = L_{ψ}^{n}g^{∗}(x) for any x. So we get

∫ e^{S_{n}ψ(wx)}g(wx)dμ(x) = μ(L_{ψ}^{n}g^{∗}) = μ(g^{∗}) = ∫_{R_{w}} g(x)dμ(x).

By using χ_{R_{wu}}(wx) = χ_{R_{u}}(x), we can get the second part of the lemma. □

Lemma 5.2. e^{ψ(0x)} = \frac{e^{φ(0x)}h(0x)}{λh(x)} = 1 − \frac{1}{λh(x)}ĥ(0x). In particular, e^{ψ(¯0)} = \frac{1}{λ}.

Proof. Since L_{ψ}h(x) = λh(x), we have e^{φ(0x)}h(0x) + ĥ(0x) = λh(x). So by the definition of ψ,

e^{ψ(0x)} = \frac{e^{φ(0x)}h(0x)}{λh(x)} = 1 − \frac{1}{λh(x)}ĥ(0x).  (5.1)

For the case x = ¯0, the result follows from the definition of ψ and (1.8). □

Lemma 5.3. There exists B_{δ} > 0 such that for all large n, the following hold:

i) if h^{∗}(¯1) < 0, then e^{ψ(0x)} ≤ \frac{1}{λ} · (1 + \frac{B_{δ}}{(n + 1)^{min\{1,α−θ(1+γ)\}}} ) ∀x ∈ P_{n};

ii) if h^{∗}(¯1) = 0, then e^{ψ(0x)} ≤ 1 − \frac{β}{n} + \frac{B_{δ}}{n^{1+δ}} ∀x ∈ P_{n};

iii) if h^{∗}(¯1) > 0, then e^{ψ(0x)} ≥ 1 − \frac{B_{δ}}{n^{1+β}} ∀x ∈ P_{n}.

Suppose Assumption A (III′) also holds. Then there exists B′_{δ} > 0 such that for all large n, the above inequalities hold if we interchange “ ≤” and “ ≥”, and then replace B_{ψ} and β by −B′_{ψ} and β′ in i) and ii), and by B′_{ψ} and β′ in iii).

Proof. i) Note that by Assumption A (III), φ(0x) ≤ −\frac{β + 1}{n + 2} + \frac{C_{δ}}{(n + 2)^{1+δ}}. Also

since h ∈ \overline{G}_{J_{φ}}, h(0x)/h(x) ≤ e^{J_{φ}n^{α}d(x,0x)^{θ}} ≤ e^{J_{φ}(n+1)^{α−θ(1+γ)}}. So if h^{∗}(¯1) < 0, the result can be obtained from the first equality of (5.1).

ii) By Lemma 4.8 and a similar method for (4.10), we have that for all large n,

\frac{1}{λh(x)} · ĥ(0x) ≥ \frac{β}{nĥ(¯0)}(1 + \frac{C_{h}}{n^{δ}} )^{−1} · ĥ(¯0)(1 − \frac{2C_{γ}J_{φ}K_{0}^{α}}{n^{γθ}} ) ≥ \frac{β}{n} − \frac{B_{δ}}{n^{1+δ}}

for some large B_{δ}. Now the result follows from the second equality of (5.1).

iii) Since h^{∗}(¯1) > 0, by Lemma 4.2, h(x) is of the same order as e^{−S_{n}φ(x)}. By Lemma 4.3, if x = 0^{n}y ∈ P_{n}, y ∈ P_{0}, then e^{−S_{n}φ(x)} ≥ B_{φ}^{−1}(n + 1)^{β+1}. Now we use the second equality of (5.1).

The other direction of the inequalities can be estimated in a similar way. □

Lemma 5.4. For any k > 0, there is C_{ψ} = C_{ψ,k} > 0 and C′_{ψ} = C′_{ψ,k} > 0 with lim_{k→∞} C_{ψ,k} = lim_{k→∞} C′_{ψ,k} = 1 such that for all n > 0, x ∈ P_{k}, the following holds:

i) if h^{∗}(¯1) < 0, then e^{S_{n}ψ(0^{n}x)} ≤ C_{ψ}λ^{−n}(\frac{k}{k + n})^{β+1};

ii) if h^{∗}(¯1) = 0, then e^{S_{n}ψ(0^{n}x)} ≤ C_{ψ}(\frac{k}{k + n})^{β};

iii) if h^{∗}(¯1) > 0, then C′_{ψ}h^{∗}(x) ≤ e^{S_{n}ψ(0^{n}x)} ≤ C_{ψ}h^{∗}(x).

Suppose Assumption A (III′) also holds. Then i) and ii) are true if we replace “ ≤”, β and C_{ψ} by “ ≥”, β′ and C′_{ψ}, respectively.

Proof. i) Use Lemma 4.4 and the fact that both h(0^{n}x), h(x) → λ^{−1} as k → ∞.

ii) It can be proved by using Lemma 5.3 and the same methods as for the proof of Lemma 4.4.

iii) Since x ∈ P_{k}, we know that x = 0^{k}y for some y ∈ P_{0}. By the definition of ψ, we have

e^{S_{n}ψ(0^{n}x)} = e^{S_{n}φ(0^{n}x)} \frac{h(0^{n}x)}{h(x)} = \frac{e^{S_{n+k}φ(0^{n+k}y)}h(0^{n+k}y)}{e^{S_{k}φ(0^{k}y)}h(0^{k}y)}.

Then we take C_{ψ,k} = max_{y∈P_{0}} sup_{n≥0}\{\frac{e^{S_{n+k}φ(0^{n+k}y)}h(0^{n+k}y)}{e^{S_{k}φ(0^{k}y)}h(0^{k}y)}\} and C′_{ψ,k} = C_{ψ,k}^{−1}. By

Lemma 4.2, C_{ψ,k}, C′_{ψ,k} → 1 as k → ∞.

The other direction of the inequalities for i) and ii) can be obtained similarly. □

Proof of Corollary A.2. Note that by the definition of μ, μP_{k} = ν(hχ_{P_{k}}) ≤ νP_{k} max{h(x) : x ∈ P_{k}}. Also note that O_{n} = ⋃_{i≥n} P_{i}. Hence, part i) follows from Corollary A.1 and Lemma 4.5 with B_{μ,k} = B_{ν,k} max{h(x) : x ∈ P_{k}}, and part ii) follows from Lemma 4.8 and 4.5 with B_{μ,k} = B_{ν,k}β^{−1}ĥ(¯0)(1 + C_{h}n^{−δ}).

By Lemma 5.1, μP_{n+k} = ∫_{P_{k}} e^{S_{n}ψ(0^{n}x)}dμ(x). So we use part iii) of Lemma 5.4 to get μP_{n+k} ≥ C′_{ψ}μP_{k} min{h^{∗}(x) : x ∈ P_{k}} > 0. It means that {μP_{n}} are bounded away from 0. Since this is a decreasing sequence, we get part iii).

If Assumption A (III′) also holds, the lower bounds can be estimated similarly. If β′ = β, the existence of limits follows from (4.3) and the choice of these

constants. □

Proof of Corollary A.3. (I). It follows from continuity of φ on Σ_{A}^{+} and h on Σ_{A}^{+}\{¯0}. (II). Take an n-word w = w_{0}w_{1} ⋯ w_{n−1}, and suppose wx, wy ∈ P_{m}. We have

S_{n}ψ(wy) − S_{n}ψ(wx) = S_{n}φ(wy) − S_{n}φ(wx) + log \frac{h(wy)}{h(wx)} + log \frac{h(x)}{h(y)}.

Since h ∈ H_{J_{φ}}, log \frac{h(wy)}{h(wx)} ≤ J_{φ}D_{m}^{α}d(wx, wy)^{θ} and log \frac{h(x)}{h(y)} ≤ J_{φ}D_{k}^{α}d(x, y)^{θ}. By

Proposition 3.1 ii), we get the first part of (II) with J_{ψ} = 2J_{φ}.

For the second part, we write

(S_{n}ψ(wy) + log h(y))−(S_{n}ψ(wx) + log h(x))
=S_{n}φ(wy) − S_{n}φ(wx) + log h(wy) − log h(wx)

and then use the second part of Proposition 3.1 ii).

(III) & (III′). The results follow from Lemma 5.3. □

6. Exactness: Proof of Theorem B

Recall that L_{ψ} is defined in Section 1, after Corollary D.2 is stated.

Lemma 6.1. i) L_{ψ}c = c for any constant function c.

ii) μ(L_{ψ}g) = μ(g) for any integrable function g.

iii) μ(|L_{ψ}g|) ≤ μ(|g|) for any function g in L^{1}(Σ_{A}^{+}, μ). Further, if g is continuous, and there exists x ∈ Σ_{A}^{+}, and n-words u and v such that g(ux) > 0 and g(vx) < 0, then μ(|L_{ψ}^{n}g|) < μ(|g|).

Proof. i) Since L_{φ}h = λh, we get L_{ψ}c = \frac{1}{λh}L_{φ}(ch) = \frac{c}{λh}L_{φ}h = c.

ii) This is because μ(L_{ψ}g) = ν(h · \frac{1}{λh}L_{φ}(hg)) = ν(\frac{1}{λ}L_{φ}(hg)) = ν(hg) = μ(g).

iii) It is easy to check by using part i) that for any x, |L_{ψ}g(x)| ≤ L_{ψ}|g(x)|. Hence, by part ii), μ(|L_{ψ}g|) ≤ μ(L_{ψ}|g|) = μ(|g|).

For the second part, we have

|L_{ψ}^{l}g(x)| = | ∑_{w∈ξ_{l}} e^{ψ(wx)}g(wx)| < ∑_{w∈ξ_{l}} e^{ψ(wx)}|g(wx)| = (L_{ψ}^{l}|g|)(x)

for x given in the lemma. So we have μ(|L_{ψ}^{l}g|) < μ(L_{ψ}^{l}|g|) = μ(|g|). □

For any x, we denote

ḡ(0x) = \frac{∑_{s≠0} e^{ψ(sx)}g(sx)}{∑_{s≠0} e^{ψ(sx)}} = \frac{∑_{s≠0} e^{φ(sx)}h(sx̃)g(sx)}{∑_{s≠0} e^{φ(sx)}h(sx)}.  (6.1)

That is, ḡ(0x) is the average of g(sx), s ≠ 0, with weights e^{φ(sx)}. Since ∑_{s≠0} e^{ψ(sx)} =

1 − e^{ψ(0x)}, we have

L_{ψ}g(x) = e^{ψ(0x)}g(0x) + (1 − e^{ψ(0x)})ḡ(0x).  (6.2)

Lemma 6.2. For any x ∈ Σ_{A}^{+},

L_{ψ}^{n}g(x) = g(0^{n}x)e^{S_{n}ψ(0^{n}x)} + ∑_{j=1}^{n}(\overline{L_{ψ}^{n−j}g})(0^{j}x)(1 − e^{ψ(0^{j}x)})e^{S_{j−1}ψ(0^{j−1}x)}.

Proof. It can be obtained by (6.2) and induction. □

Lemma 6.3. For any continuous function g on Σ_{A}^{+} with μ(|g|) < ∞ and μ(g) = 0,

lim_{n→∞} L_{ψ}^{n}g(x) = 0 ∀x ∈ Σ_{A}^{+} \ {¯0},

and the convergence is in L^{1}(μ) and uniform on Q_{k} for any k ≥ 0. Also,

lim_{n→∞} L_{ψ}^{n}g(¯0) = \{ \begin{matrix}0 & if λ > 1; \\ g(¯0) & if λ = 1. \end{matrix}

Proof. Let x ∈ P_{i}, y ∈ P′_{i}. By Corollary A.3(II), for any n ≥ 0 we have

|L_{ψ}^{n}g(x) − L_{ψ}^{n}g(y)|
≤ ∑_{w∈ξ_{n}} e^{S_{n}ψ(wx)}|g(wx) − g(wy)| + ∑_{w∈ξ_{n}} g(wy)e^{S_{n}ψ(wy)}|e^{S_{n}ψ(wx)−S_{n}ψ(wy)} − 1|
≤var_{n}(g) ∑_{w∈ξ_{n}} e^{S_{n}ψ(wx)} + (e^{J_{ψ}D_{i}^{α}d(x,y)^{θ}} − 1) ∑_{w∈ξ_{n}} g(wy)e^{S_{n}ψ(wy)}
≤var_{n}(g) + (e^{J_{ψ}D_{i}^{α}d(x,y)^{θ}} − 1)|L_{ψ}^{n}g(y)|,

where var_{n}(g) is defined in (1.3). Note that e^{J_{ψ}D_{i}^{α}d(x,y)^{θ}} −1 → 0 as d(x, y) → 0, and the convergence is uniform for all x, y ∈ Q_{k} whenever k > 0 is fixed. Since L_{ψ}g(x) is the average value of the g(sx) with weight e^{ψ(sx)}, {L_{ψ}^{n}g : n ≥ 0} is uniformly bounded. The above arguments says that restricted to Q_{k}, {L_{ψ}^{n}g : n ≥ 0} is equicontinuous. So the closure of {L_{ψ}^{n}g : n ≥ 0} is compact. Therefore, there is a subsequence {n_{i}} and a continuous function g^{(k)} on Q_{k} such that L_{ψ}^{n_{i}}g → g^{(k)}. Since Q_{k} ⊂ Q_{k+1} and ⋃_{k≥0} Q_{k} = Σ_{A}^{+}\{¯0}, by applying the diagonalization method, we know that there is a subsequence {n_{i}} and a continuous function g^{∗} on Σ_{A}^{+} \ {¯0} such that L_{ψ}^{n_{i}}g(x) → g^{∗}(x) for all x ∈ Σ_{A}^{+} \ {¯0}.

Now we prove that g^{∗} = 0 on Σ_{A}^{+} \ {¯0}. It is enough to show μ(|g^{∗}|) = 0. By Lemma 6.1, {μ(|L_{ψ}^{n}g|)} is decreasing. So c = lim_{n→∞} μ(|L_{ψ}^{n}g|) exists. By taking

subsequences, we obtain that c = μ(|L_{ψ}^{j}g^{∗}|) for all limit points g^{∗} of {L_{ψ}^{n}g} and for all j ≥ 0. If c ≠ 0, then by continuity of g^{∗}, we can find x ∈ Σ_{A}^{+}\{¯0}, and l-words u and v such that g^{∗}(ux) > 0 and g^{∗}(vx) < 0. By Lemma 6.1, we have μ(|L_{ψ}^{l}g^{∗}|) < μ(|g^{∗}|), a contradiction. So we get c = 0. Hence, L_{ψ}^{n_{i}}g(x) → 0 in L^{1}(μ). Since this is true for any subsequence, we get L_{ψ}^{n}g(x) → 0 in L^{1}(μ). The arguments in the previous paragraph imply that the convergence is uniform on Q_{k} for any k.

For the case x = ¯0 and λ > 1, we first note that as j → ∞, L_{ψ}^{j}g(s¯0) → 0 for

any s ≠ 0 by the above argumentsand therefore (\overline{L_{ψ}^{j}g})(¯0) → 0 by (6.1). Also, by Lemma 5.2, e^{ψ(¯0)} = λ^{−1} < 0 and therefore e^{S_{n}ψ(0^{n}¯0)} = λ^{−n} → 0 as n → ∞. So by Lemma 6.2, L_{ψ}^{n}g(¯0) → 0 as n → ∞.

It remains to show that lim_{n→∞}(L_{ψ}^{n}g)(¯0) = g(¯0) if λ = 1. In fact, λ = 1 implies that

e^{ψ(¯0)} = 1 by Lemma 5.2, and hence e^{ψ(s¯0)} = 0 for any s ≠ 0. So (L_{ψ}g)(¯0) = g(¯0). We get the result. □

Proposition 6.4. (a) If μ is a probability measure, then for any continuous function g on Σ_{A}^{+},

lim_{n→∞} L_{ψ}^{n}g(x) = \{ \begin{matrix}μ(g) & if λ > 1 or x ≠ ¯0; \\ g(¯0) & if λ = 1 and x = ¯0. \end{matrix}

(b) If μ is an infinite measure, then for any continuous function g on Σ_{A}^{+} with with μ(|g|) < ∞ and μ(g) = 0,

lim_{n→∞} L_{ψ}^{n}g(x) = 0 ∀x ∈ Σ_{A}^{+}.

The convergence in both cases is in L^{1}(μ) and uniform on Q_{k} for any k ≥ 0.

Proof. Note that L_{ψ}^{n}(g(x) − μ(g)) = L_{ψ}^{n}g(x) − μ(g). So part (a) can be obtained from Lemma 6.3 by applying the function g − μ(g).

For part (b), by Lemma 6.3 we only need to show that g(¯0) = 0. In fact, if g(¯0) > 0, then by continuity g(x) > g(¯0)/2 on O_{k} for some k > 0. Since μ(O_{k}) = ∞, we have μ(gχ_{O_{k}}) = ∞. It implies that g is not integrable, a contradiction. □

Proof of Theorem B. By a theorem of Lin ([Li]; see also [A], Theorem 1.3.3), for a nonsingular system (σ, μ), it is exact if and only if ∥L_{ψ}^{n}g∥_{1} → 0 for any g ∈ L^{1}(μ) with μ(g) = 0, where ∥ · ∥_{1} denotes the L^{1} norm.

Take g ∈ L^{1}(μ) with μ(g) = 0. We need show that for any ϵ > 0, there is N > 0 such that for any n > N, ∥L_{ψ}^{n}g∥_{1} ≤ ϵ.

Since g ∈ L^{1}(μ), we can take k > 0 such that μ(|gχ_{O_{k+1}}|) ≤ ϵ/6. Then take a function g′_{ϵ} such that g′_{ϵ}|_{O_{k+1}} = 0, g′_{ϵ}|_{Q_{k}} continuous, and μ(|g′_{ϵ} − gχ_{Q_{k}}|) ≤ ϵ/6. By triangle inequality we have μ(|g′_{ϵ} − g|) ≤ μ(|g′_{ϵ} − gχ_{Q_{k}}|) + μ(|gχ_{O_{k+1}}|) ≤ ϵ/3. So we can take a continuous function g_{ϵ} such that g_{ϵ}(¯0) = 0, μ(|g_{ϵ} − g′_{ϵ}|) ≤ ϵ/3 and μ(g_{ϵ}) = 0. Now we have μ(|g − g_{ϵ}|) ≤ 2ϵ/3. By Lemma 6.1, for any n > 0, μ(|L_{ψ}^{n}g − L_{ψ}^{n}g_{ϵ}|) ≤ μ(L_{ψ}^{n}|g − g_{ϵ}|) ≤ 2ϵ/3.

By Proposition 6.4, L_{ψ}^{n}g_{ϵ} → 0 in L^{1}(μ) as n → ∞. So there exists N > 0 such that for all n > N, μ(|L_{ψ}^{n}g_{ϵ}|) ≤ ϵ/3. Hence we get

μ(|L_{ψ}^{n}g|) ≤ μ(|L_{ψ}^{n}g − L_{ψ}^{n}g_{ϵ}|) + μ(|L_{ψ}^{n}g_{ϵ}|) ≤ ϵ.

This is what we need. □

7. Gibbs properties: Proof of Theorem C

Lemma 7.1. There is an increasing sequence of positive numbers {C̄_{n}} such that

lim_{n→∞} \frac{1}{n} log C̄_{n} = 0 and e^{S_{n}φ(0^{n}x)} ≥ C̄_{n}^{−1} for any x ∈ Σ_{A}^{+}.

Proof. Recall that var_{i}(φ) is defined in (1.3). Take C̄_{n} = e^{∑_{i=0}^{n−1} var_{i}(φ)}. Since φ is

continuous, lim_{n→∞} var_{n}(φ) → 0. Hence lim_{n→∞} \frac{1}{n} log C̄_{n} = lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} var_{i}(φ) → 0.

Since 0^{i}x ∈ O_{i} and φ(¯0) = 0, φ(0^{i}x) ≥ −var_{i}(φ). So we get

e^{S_{n}φ(0^{n}x)} ≥ e^{− ∑_{i=0}^{n−1} var_{i}(φ)} = C̄_{n}^{−1}

for any x ∈ Σ_{A}^{+}. □

Lemma 7.2. There is a sequence of positive numbers \{C_{n}\} with lim_{n→∞} \frac{1}{n} log C_{n} = 0 such that for any x ∈ P_{k}, n-word w with μR_{w} < ∞,

C_{n}^{−1} ≤ \frac{μR_{w}}{exp\{−n log λ + S_{n}φ(wx)\}} ≤ C_{n}.

Proof. By Lemma 5.1 and the definition of ψ, for any n-word w = uv, where u is an n_{0}-word, n_{0} ≤ n, we have

μR_{w} = ∫_{R_{v}} e^{S_{n_{0}} ψ(wz)}dμ(z) = \frac{1}{λ^{n_{0}}} ∫_{R_{v}} e^{S_{n_{0}} φ(wz)}h(wz)dν(z).  (7.1)

First, we consider the case w = 0^{n}. If μO_{n} < ∞, then by Corollary A.2 and Lemma 4.3, μO_{n} ≤ λ^{−n}C_{μ,0}n^{−β} ≤ λ^{−n}C_{μ,0} if h^{∗}(¯1) < 0, i.e. λ > 1, and μO_{n} ≤ 1 if h^{∗}(¯1) = 0, i.e. λ = 1. We may assume C_{μ,0} ≥ 1 and then get

μR_{w} = μO_{n} ≤ λ^{−n}C_{μ,0} ≤ λ^{−n}C_{μ,0}C̄_{n}e^{S_{n}φ(0^{n}x)}.  (7.2)

On the other hand, by Lemma 4.4, e^{S_{n}φ(0^{n}x)} ≤ B_{φ} for any x. So applying (7.1) with n_{0} = n and Lemma 7.1, we get

μR_{w} = μO_{n} ≥ λ^{−n}C̄_{n}^{−1}h_{−} ≥ λ^{−n}C̄_{n}^{−1}h_{−}B_{φ}^{−1}e^{S_{n}φ(0^{n}x)},

where h_{−} = min{h(x) : x ∈ Σ_{A}^{+}}. By Lemma 4.1, h_{−} > 0. So we can take

C_{n} = C̄_{n} max{C_{μ,0}, h_{−}^{−1}B_{φ}}.  (7.3)

For the case w ≠ 0^{n}, we may assume w = u0^{n_{1}}, where u is an n_{0}-word whose last symbol is not equal to 0, and n = n_{0} + n_{1}. By Proposition 3.1 and (1.2), for any z,

e^{S_{n_{0}} φ(wz)}h(wz) ≤ e^{J_{φ}K_{0}^{α}d(0^{n_{1}} x,0^{n_{1}} z)^{θ}}e^{S_{n_{0}} φ(wx)}h(wx) ≤ e^{J_{φ}C_{γ}K_{0}^{α}}e^{S_{n_{0}} φ(wx)}h(wx).

Hence, using (7.1) with v = 0^{n_{1}}, we get

μR_{w} ≤ λ^{−n_{0}}e^{J_{φ}C_{γ}K_{0}^{α}}e^{S_{n_{0}} φ(wx)}h(wx)νO_{n_{1}}.

Similarly to (7.2) we get νO_{n_{1}} ≤ λ^{−n_{1}}C_{ν,0}C̄_{n}e^{S_{n_{1}} φ(0^{n_{1}} x)}. Note that

S_{n}φ(wx) = S_{n_{0}}φ(wx)S_{n_{1}}φ(0^{n_{1}}x).

By (3.3), h(wx) ≤ e^{J^{∗}n_{0}^{1−θ(1+γ)+α}} ≤ e^{J^{∗}n^{1−θ(1+γ)+α}}. So we have

μR_{w} ≤ λ^{−n}e^{J_{φ}C_{γ}K_{0}^{α}}C_{ν,0}e^{J^{∗}n^{1−θ(1+γ)+α}} C̄_{n}e^{S_{n}φ(wx)}.  (7.4)

Similarly, we have

μR_{w} ≥ λ^{−n_{0}}e^{−J_{φ}C_{γ}K_{0}^{α}}e^{S_{n_{0}} φ(wx)}h(wx)νO_{n_{1}}

and νO_{n_{1}} ≥ λ^{−n_{1}} C̄_{n}^{−1}B_{φ}^{−1}e^{S_{n}φ(0^{n}x)}. So

μR_{w} ≥ λ^{−n}e^{−J_{φ}C_{γ}K_{0}^{α}} C̄_{n}^{−1}h_{−}B_{φ}^{−1}e^{S_{n}φ(wx)}.  (7.5)

By (7.4) and (7.5), in this case we can take

C_{n} = C̄_{n}e^{J_{φ}C_{γ}K_{0}^{α}} max{C_{ν,0}e^{J^{∗}n^{1−θ(1+γ)+α}}, h_{−}^{−1}B_{φ}}.  (7.6)

Now we take C_{n} as the larger one in (7.3) and (7.6). Clearly {C_{n}} is subexponential and the inequalities of the lemma are satisfied. □

Lemma 7.3. There exists a constant A_{φ} > 0 such that for any n-word w = w_{0}w_{1} ⋯ w_{n−1} with w_{n−1} ≠ 0,

A_{φ}^{−1}h(wx) ≤ \frac{μR_{w}}{exp\{−n log λ + S_{n}φ(wx)\}} ≤ A_{φ}h(wx).

Proof. We may assume that d(x, z)^{θ} ≤ C_{γ} for any x, z ∈ Σ_{A}^{+}. Since h ∈ \overline{G}_{J_{φ}}, by Proposition 3.1 ii), we have e^{−J_{φ}C_{γ}K_{0}^{α}} ≤ e^{S_{n}φ(wz)}h(wz)/e^{S_{n}φ(wx)}h(wx) ≤ e^{J_{φ}C_{γ}K_{0}^{α}}. So by (7.1)

μR_{w} ≤ \frac{1}{λ^{n}} ∫ e^{J_{φ}C_{γ}K_{0}^{α}}e^{S_{n}φ(wx)}h(wx)dν(z) ≤ \frac{1}{λ^{n}} e^{J_{φ}C_{γ}K_{0}^{α}}e^{S_{n}φ(wx)}h(wx).

On the other hand, if wx ∈ Σ_{A}^{+}, then

μR_{w} ≥ \frac{1}{λ^{n}} ∫ e^{−J_{φ}C_{γ}K_{0}^{α}}e^{S_{n}φ(wx)}h(wx)dν(z) ≥ \frac{1}{λ^{n}} e^{−J_{φ}C_{γ}K_{0}^{α}}e^{S_{n}φ(wx)}h(wx).

So we can take A_{φ} ≥ e^{J_{φ}C_{γ}K_{0}^{α}}. □

Note that by [W2] Theorem 9.6, the topological pressure of σ for φ is given by

P (σ, φ) = lim sup_{n→∞} \frac{1}{n} log ∑_{w∈ξ_{n}} inf_{wx∈R_{w}} e^{S_{n}φ(wx)}.  (7.7)

Lemma 7.4. A probability invariant measure μ is an equilibrium state for a continuous function φ whenever it is a weak Gibbs measure for φ, and the constant P in the definition of weak Gibbs measure is equal to the topological pressure P (σ, φ) for φ.

Proof. Since μ is a weak Gibbs measure, by (1.9) we have

C_{n}^{−1}e^{nP} μR_{x_{0}x_{1}···x_{n−1}} ≤ e^{S_{n}φ(x)} ≤ C_{n}e^{nP} μR_{x_{0}x_{1}···x_{n−1}}.  (7.8)

Hence,

P − \frac{1}{n} log C_{n} ≤ \frac{1}{n}S_{n}φ(x) − \frac{1}{n} log μ(R_{x_{0}x_{1}···x_{n−1}}) ≤ P + \frac{1}{n} log C_{n}.

Note that R_{x_{0}x_{1}···x_{n−1}} is the element of ξ_{n} containing x. Let n → ∞, by the Birkhoff Ergodic Theorem, the Shannon-McMillan-Breiman Theorem, and the fact (1/n) log C_{n} → 0, we have

P = ∫ φdμ + h_{μ}(σ).

We show that P is equal to the topological pressure of φ. We replace x by wx in (7.8), where w is an n word, and then use the fact ∑_{w∈ξ_{n}} μ(R_{w}) = 1 to get

C_{n}^{−1}e^{nP} ≤ ∑_{R_{w}∈ξ_{n}} inf_{wx∈R_{w}} e^{S_{n}φ(wx)} ≤ C_{n}e^{nP} .

By (7.7),

P (σ, φ) = lim sup_{n→∞} \frac{1}{n} log ∑_{w∈ξ_{n}} inf_{wx∈R_{w}} e^{S_{n}φ(wx)} = P.  □

The result in the lemma is also obtained by M. Yuri (see e.g. [Yu1]) in a slightly different setting.

Lemma 7.5. Any probability or σ-finite measure μ′ that satisfies (1.10) with a function p′(x, n) satisfying a) and b) in Theorem C coincides with μ up to a constant coefficient.

Proof. Assume μ′ satisfies (1.10) with constant P′ and function p′(x, n). Similar arguments as in the proof for Lemma 7.4 show that P′ = P (σ, φ) = P. Fix k > 0. Denote B_{k} = sup_{x∈Q_{k}∩σ^{−n+1}Q_{0}} p(x, n). Let w = w_{0} ⋯ w_{n−1} be an

n-word, n ≥ k, with R_{w} ⊂ Q_{k}.

If w_{n−1} ≠ 0, then by part b), p′(wx, n) ≤ B′_{k} for some B′_{k} > 0. So by (1.10),

μ′R_{w} ≤ B′_{k}e^{−nP +S_{n}φ(wx)} ≤ B′_{k}B_{k}μR_{w}.  (7.9)

If w_{n−1} = 0, then we can always find a sequence of words {u^{(i)}}_{i=1}^{∞} whose last

symbols are nonzero such that R_{w} = {w¯0} ∪ ⋃_{i=1}^{∞} R_{wu^{(i)}}. Since μ is an invariant

measure, μ{w¯0} = 0 for any w ≠ ¯0 because ¯0 ∈ σ^{−n}¯0. By (7.9) we know that

μ′R_{w} = ∑_{i=1}^{∞} μ′R_{wu^{(i)}} ≤ B′_{k}B_{k} ∑_{i=1}^{∞} μR_{wu^{(i)}} = B′_{k}B_{k}μR_{w}.

Since this is true for all cylinders in Q_{k}, by taking a limit we know that μ′(E) ≤ B′_{k}B_{k}^{−1}μ(E) for all Borel set E ⊂ Q_{k}. It implies that μ′ is absolutely continuous with respect to μ on Q_{k} and therefore on Σ_{A}^{+}. By the Radon-Nykodym Theorem we know that dμ′/dμ exists. Since both μ′ and μ are σ-invariant, dμ′/dμ is a σinvariant function. Since μ is ergodic, dμ′/dμ is equal to a constant c μ-almost everywhere. Clearly, c > 0. So we get μ′ = cμ. □

Proof of Theorem C. By Lemma 7.2 with wx replaced by x, μ satisfies (1.9) and therefore is a weak Gibbs measure.

Take p(x, n) = min{C_{n}, A_{φ} max{h(x), h(x)^{−1}}} if x_{n−1} ≠ ¯0 and p(x, n) = C_{n} otherwise. Clearly (1.10) holds by Lemma 7.2 and 7.3. Also by Lemma 7.2, p(x, n) satisfies a). Since h(x) is bounded on Q_{k} for each k, p(x, n) also satisfies b).

The uniqueness follows from Lemma 7.5. □

Proof of Corollary C.1. By Lemma 7.2 and 7.3, we know that P in (1.9) and (1.10) is equal to log λ. By Lemma 7.4, it is equal to P (σ, φ). □

8. Equilibriums: Proof of Theorem D

Since by Theorem C and Lemma 7.4 we know that μ is an equilibrium for φ if μ is a probability measure, the main work in this section is to deal with the case that μ is an infinite measure, in particular, the case h^{∗}(¯1) = 0.

Lemma 8.1. Let {a_{n}} be a decreasing sequence of positive numbers with

∑_{n=1}^{∞} \frac{a_{n}}{n} < ∞.  (8.1)

Then a_{n} log n → 0 as n → ∞.

Proof. First we know that there is a subsequence n_{i} such that

a_{n_{i}} log(n_{i} + 1) → 0 as i → ∞, (8.2)

because if otherwise, there would be M > 0 and ϵ > 0 such that a_{n} > ϵ/ log(n + 1) for all n ≥ M, and therefore the series in (8.1) would diverge.

Denote

S_{n} = ∑_{i=1}^{n} a_{i} log(1 + \frac{1}{i}), T_{n} = ∑_{i=2}^{n}(a_{i−1} − a_{i}) log i.

Since log(1 + t) < t for any t > 0, by (8.1) S = lim_{n→∞} S_{n} exists. Note that

S_{n} = ∑_{i=1}^{n} a_{i}(log(i+1)−log i) = a_{n} log(n+1)+∑_{i=2}^{n}(a_{i−1}−a_{i}) log i = a_{n} log(n+1)+T_{n}.

So T_{n} is bounded and therefore T = lim_{n→∞} T_{n} exists. By (8.2) we have S = T. Hence a_{n} log(n + 1) → 0. This implies the result. □

Lemma 8.2. If h^{∗}(¯1) ≤ 0, then for any σ-invariant measure ρ with ∑_{i=1}^{∞} i^{−1}ρP_{i} <

∞, for any n > 0,

∫_{σ^{−1}O_{n}\O_{n+1}} log h(σx)dρ(x) < ∞.

Suppose Assumption A (III′) also holds; then the condition h^{∗}(¯1) ≤ 0 can be removed.

Proof. Since h^{∗}(¯1) ≤ 0, by Corollary A.1, h(σx) ≤ Cn for some C > 0 if x ∈ P_{n}. So

∫_{σ^{−1}P_{i}\P_{i+1}} log h(σx)dρ(x) ≤ log(Ci)ρ(σ^{−1}P_{i} \ P_{i+1}) = log(Ci)(ρP_{i} − ρP_{i+1}),

and therefore for any k ≥ n,

∑_{i=n}^{k} ∫_{σ^{−1}P_{i}\P_{i+1}} log h(σx)dρ(x) ≤ ∑_{i=n}^{k} log(Ci)(ρP_{i} − ρP_{i+1})

=ρP_{n} log(Cn) + ∑_{i=n}^{k} ρP_{i+1}[log(C(i + 1)) − log(Ci)] − ρP_{k+1} log(C(k + 1))

≤ρP_{n} log(Cn) + ∑_{i=n}^{k} i^{−1}ρP_{i+1} − ρP_{k+1} log(C(k + 1)).

Since ∑_{i=n}^{∞} i^{−1}ρP_{i+1} is convergent and ρP_{k+1} log(C(k + 1)) ≥ 0 for all k ≥ n, the

sum in the left side of the inequality is bounded. Note that σ^{−1}O_{n} \ O_{n+1} is the pairwise disjoint union of the sets σ^{−1}P_{i} \ P_{i+1}, i ≥ n; we get the result.

If Assumption A (III′) also holds, then by Corollary A.1, we have h(x) ≤ Cn^{β′+1} and log h(x) ≤ (β + 1) log C′n for some C, C′ > 0 if x ∈ P_{n}. Then the same arguments can be applied. □

Lemma 8.3. If h^{∗}(¯1) ≤ 0, then for any σ-invariant measure ρ with ∑_{i=1}^{∞} i^{−1}ρP_{i} <

∞,

∫ (log h(x) − log h(σx))dρ(x) = 0.

Suppose Assumption A (III′) also holds; then the condition h^{∗}(¯1) ≤ 0 can be removed.

Proof. It is obvious if ρ(log h) < ∞ because ρ is an invariant measure. This is the case if h^{∗}(¯1) < 0. So we assume h^{∗}(¯1) = 0.

Since σ^{−1}Q_{n−1} = Q_{n} \ σ^{−1}O_{n} = P_{n} ∪ (Q_{n−1} \ σ^{−1}O_{n}) and ρ is an invariant measure, we have

∫_{P_{n}} log h(σx)dρ(x) + ∫_{Q_{n−1}\σ^{−1}O_{n}} log h(σx)dρ(x) = ∫_{Q_{n−1}} log h(x)dρ(x).

Note that Q_{n−1} can be partitioned into {Q_{n−1} \ σ^{−1}O_{n}, σ^{−1}O_{n} \ O_{n+1}}. So

∫_{Q_{n−1}} log \frac{h(x)}{h(σx)}dρ(x) = ∫_{Q_{n−1}} log h(x)dρ(x) − ∫_{Q_{n−1}} log h(σx)dρ(x)
= ∫_{P_{n}} log h(σx)dρ(x) − ∫_{σ^{−1}O_{n}\O_{n+1}} log h(σx)dρ(x).

Since Σ_{A}^{+} can be partitioned into {Q_{n−1}, O_{n}}, we only need to prove

∫_{O_{n}} log \frac{h(x)}{h(σx)}dρ(x) = ∫_{σ^{−1}O_{n}\O_{n+1}} log h(σx)dρ(x) − ∫_{P_{n}} log h(σx)dρ(x). (8.3)

Since σ^{−1}P_{i} = P_{i+1} ∪ (σ^{−1}P_{i} \ P_{i+1}),

∫_{P_{i+1}} log h(σx)dρ(x) + ∫_{σ^{−1}P_{i}\P_{i+1}} log h(σx)dρ(x) = ∫_{P_{i}} log h(x)dρ(x).

Therefore if we denote O_{n,k} = ⋃_{i=n}^{n+k} P_{i} for k > n, then

∫O_{n,k} log \frac{h(x)}{h(σx)}dρ(x) = ∫_{P_{n+k}} log h(x)dρ(x)

+ ∑_{i=n}^{n+k−1}[∫_{P_{i}} log h(x)dρ(x) − ∫_{P_{i+1}} log h(σx)dρ(x)] − ∫_{P_{n}} log h(σx)dρ(x)

= ∫_{P_{n+k}} log h(x)dρ(x) + ∑_{i=n}^{n+k−1} ∫_{σ^{−1}P_{i}\P_{i+1}} log h(σx)dρ(x) − ∫_{P_{n}} log h(σx)dρ(x).

Let k → ∞; we get that this equality implies (8.3). To see this, we first note that by Lemma 8.2 the integral

∑_{i=n}^{∞} ∫_{σ^{−1}P_{i}\P_{i+1}} log h(σx)dρ(x) = ∫_{σ^{−1}O_{n}\O_{n+1}} log h(σx)dρ(x)

converges. Also, by Corollary A.1, we know that there is C > 0 such that for x ∈ P_{i}, either h(σx) ≤ Ci if h^{∗}(¯1) = 0, or h(σx) ≤ Ci^{β′+1} if Assumption (III′) holds. Using Lemma 8.1 with a_{i} = ρP_{i} we have that if k → ∞, then ∫_{P_{n+k}} log h(x)dρ(x) ≤

ρP_{n+k} log C(n + k)^{max{1,β′+1}} → 0. This completes the proof of the lemma. □

Lemma 8.4. For any sequence \{a_{n}\} with a_{n} ≥ \frac{β + 1}{n + 1} − \frac{C_{δ}}{(n + 1)^{1+δ}} and a_{n} → 0,

where δ, C_{δ} and β are given in Assumption A (III) with β > 0,

∑_{k=n}^{∞} a_{k}ke^{− ∑_{i=n}^{k} a_{i}} < ∞  ∀n > 0.  (8.4)

Proof. By adding the first n−1 terms and multiplying by e^{− ∑_{i=1}^{n−1} a_{i}}, we know that if the result is true for some n > 0, then it is true for n = 1 and therefore for any n > 0.

Let b_{n} = \frac{β + 1}{n + 1} − \frac{C_{δ}}{(n + 1)^{1+δ}}.

By the same arguments as in the proof of Lemma 4.4, we know that there exist B(n) ≥ B′(n) > 0 with lim_{n→∞} B(n) = lim_{n→∞} B′(n) = 1 such that for all k ≥ n,

B′(n)(\frac{n}{k + 1})^{β+1} ≤ e^{− ∑_{i=n}^{k} b_{i}} ≤ B(n)(\frac{n}{k + 1})^{β+1}.

Hence, we have

∑_{k=n}^{∞} b_{k}ke^{− ∑_{i=n}^{k} b_{i}} ≤ ∑_{k=n}^{∞} \frac{β + 1}{k + 1}k · B(n)(\frac{n}{k + 1})^{β+1} < ∞.  (8.5)

Take ϵ > 0 small such that \frac{β + 1 − ϵ}{β} > 1. Then take N > 0 such that for any n ≥

N, b_{n} ≥ \frac{β + 1 − ϵ}{n} and B′(n)\frac{β + 1 − ϵ}{β}(\frac{n}{n + 1})^{β} > 1. Note that ∑_{k=n}^{∞} \frac{1}{(k + 1)^{β+1}} >

∫_{n+1}^{∞} \frac{1}{t^{β+1}} dt = \frac{1}{β(n + 1)^{β}}. We have that for any n ≥ N,

∑_{k=n}^{∞} b_{k}ke^{− ∑_{i=n}^{k} b_{i}} ≥ ∑_{k=n}^{∞} \frac{β + 1 − ϵ}{k}k · B′(n)(\frac{n}{k + 1})^{β+1}

≥(β + 1 − ϵ)B′(n)\frac{n^{β+1}}{β(n + 1)^{β}} ≥ \frac{β + 1 − ϵ}{β}B′(n)\frac{n^{β}}{(n + 1)^{β}} · n > n. (8.6)

Let

c_{k}^{(j)} = \{ \begin{matrix}a_{k} & if k < j; \\ b_{k} & if k ≥ j. \end{matrix}

Then let

S(j) = ∑_{k=n}^{∞} c_{k}^{(j)}ke^{− ∑_{i=n}^{k} c_{k}^{(j)}}.

S(n) is the left side of (8.5) and S(∞) is the left side of (8.4). So by (8.5) we only need to prove that S(j) decreases with j.

Let T_{j}(t) be the function obtained from S(j) by replacing c_{j}^{(j)} by t. So we have T_{j}(b_{j}) = S(j) and T_{j}(a_{j}) = S(j + 1). Since a_{j} ≥ b_{j}, we only need to prove that for any t ≥ b_{k}, T′_{j}(t) < 0. In fact,

T′_{j}(t) = je^{−t−∑_{i=n}^{j−1} c_{i}^{(j)}} − tje^{−t−∑_{i=n}^{j−1} c_{i}^{(j)}} − ∑_{k=j+1}^{∞} c_{k}^{(j)}ke^{−(t−c_{j}^{(j)})−∑_{i=n}^{k} c_{i}^{(j)}}.

Hence by definition of c_{k}^{(j)} and (8.6), we get

e^{t+∑_{i=n}^{j−1} a_{i}}T′_{j}(t) = j − tj − ∑_{k=j+1}^{∞} b_{k}ke^{− ∑_{i=j+1}^{k} b_{i}} < j − tj − (j + 1) < 0.

It implies T′_{j}(t) < 0. □

Lemma 8.5. If h^{∗}(¯1) ≤ 0, then μ(|φ|) < ∞.

Proof. If μ is finite, then the result is trivial. So we only need to consider the case that μ is infinite. In this case, h^{∗}(¯1) = 0 and λ = 1.

By Assumption A(III), φ ≤ 0 on O_{K_{1}}. Since μQ_{n} ≤ ∞ for any n ≥ K_{1} and φ is continuous, μ(|φ|) < ∞ if and only if μ(−φχ_{O_{n}}) < ∞.

Denote a_{i} = max{|φ(y)| : y ∈ P_{i}}.

The same arguments as for the proof of Proposition 3.1 give that for any k ≥ n,

|− ∑_{i=n}^{k} a_{i} − S_{k−n+1}φ(0^{k−n+1}x)| ≤ J_{φ}(diam P_{n})^{θ} ≤ C_{γ}J_{φ}K_{0}^{α} ∀x ∈ P_{n}.

Since h^{∗}(¯1) = 0, by Corollary A.1, we can take n large enough such that h(x) ≤ 2kĥ(¯0)/β for any x ∈ P_{k} and k ≥ n. Hence by (4.5) and the definition of ψ, we get

\begin{matrix}μP_{k} = & ∫_{P_{n}} e^{S_{k−n+1}φ(0^{k−n+1}x)} \frac{h(0^{k−n+1}x)}{h(x)}dμ(x) \\ ≤ & e^{− ∑_{i=n}^{k} a_{i}+C_{γ}J_{φ}K_{0}^{α}} · \frac{2kĥ(¯0)}{β min\{h(x) : x ∈ P_{n}\}} · μP_{n} ≤ Cke^{− ∑_{i=n}^{k} a_{i}}μP_{n}, \end{matrix}

for some C independent of k. So we have

∫_{O_{n}}(−φ)dμ = ∑_{k=n}^{∞} ∫_{P_{k}}(−φ)dμ ≤ C μP_{n} ∑_{k=n}^{∞} a_{k}ke^{− ∑_{i=n}^{k} a_{i}}.

By Lemma 8.4, the series on the right side converges if n is large enough. This is what we need. □

Recall that the definition of entropy for an infinite measure is given in (1.11).

Lemma 8.6 (Rohilin’s formula). If μ is an infinite measure, then

h_{μ}(σ) = − ∫ ψdμ < ∞.

Moreover, if h^{∗}(¯1) = 0, then

h_{μ}(σ) = − ∫ φdμ.

Proof. Note that β ≤ 0 implies λ > 1 by Theorem A, and therefore h^{∗}(¯1) < 0 by Lemma 4.3. So we have that h^{∗}(¯1) = 0 implies β > 0 and λ = 1. By Corollary A.2,

μP_{i} is of the order i^{−β}, and therefore ∑_{i=1}^{∞} i^{−1}μP_{i} < ∞. Hence we can use Lemma 8.3,

the definition of ψ, and the fact that λ = 1 to get ∫ φdμ = ∫ ψdμ. By Lemma 8.5, the integrals are finite. So we only need to prove the first equality.

The other possibility for μ being an infinite measure is h^{∗}(¯1) > 1. In this case the integrals are finite as well because by Lemma 5.3, |ψ| = −ψ is at most of the order i^{−(β+1)} with β > 0 on P_{i}.

Now we prove h_{μ}(σ) = − ∫ ψdμ.

Take Γ = P_{0} in (1.11). Denote by σ̃ and ψ̃ the first return map of σ with respect to P_{0} and the corresponding potential, respectively, that is, σ̃x = σ^{n(x)}x and ψ̃(x) = S_{n(x)}ψ(x), where n(x) is the smallest positive integer such that s^{n(x)}x ∈ P_{0}. Denote by μ̃ the conditional measure of μ restricted to P_{0}. Then we define the Perron-Frobenius Operator L̃ψ̃ as in (1.4). Using these facts and Lemma 5.1 we can get

∫_{P_{0}} L̃ψ̃ψ̃(x)dμ(x) = ∫_{Σ_{A}^{+}} ψ(x)dμ(x).

It means that L̃ψ̃ψ̃ is integrable with respect to μ and therefore with respect to μ̃. Hence we can get that

∫_{P_{0}} ψ̃(x)dμ̃(x) = ∫_{P_{0}} L̃ψ̃ψ̃(x)dμ̃(x) = \frac{1}{μP_{0}} ∫_{P_{0}} L̃ψ̃ψ̃(x)dμ(x).  (8.7)

Now we calculate hμ̃(σ̃). Take a partition ξ̃ of P_{0} into

{R_{s0^{n}} \ R_{s0^{n+1}} : s ≠ 0, n = 0, 1, ⋯ }.

Clearly, ξ̃^{−} = ⋁_{i=0}^{∞} σ̃^{−i}ξ̃ is a partition into single points. So ξ̃ is a generator and

hμ̃(σ̃) = hμ̃(σ̃, ξ̃) = Hμ̃(ξ̃|σ̃^{−1}ξ̃^{−}) = ∫_{P_{0}} Iμ̃(ξ̃|σ̃^{−1}ξ̃^{−})dμ̃,  (8.8)

where Iμ̃(ξ̃|σ̃^{−1}ξ̃^{−}) is the conditional information of ξ̃ given σ̃^{−1}ξ̃^{−} (see [R] or [Ke]). Since the smallest σ-algebra containing ξ̃^{−} is the Borel algebra B over P_{0}, we have

Iμ̃(ξ̃|σ̃^{−1}ξ̃^{−}) = − log Eμ̃(χ_{ξ̃(·)}|σ̃^{−1}B),  (8.9)

where Eμ̃(χ_{ξ̃(·)}|σ̃^{−1}B) is the conditional expectation of the characteristic function χ_{ξ̃(·)} and ξ̃(x) is the element of ξ̃ that contains x. Note that L̃ψ̃ is the dual operator of the operator T̃ defined by T̃g(x) = g(σ̃x), and μ̃ is a σ̃-invariant measure. So for any g ∈ L^{1}(μ̃) and any Borel set E ⊂ P_{0}, we have

\begin{aligned}& ∫_{σ̃^{−1}E} gdμ̃ = ∫ χ_{E}(σ̃x)g(x)dμ̃(x) = ∫ χ_{E}(x)L̃ψ̃g(x)dμ̃(x) \\ = & ∫_{E} L̃ψ̃g(x)dμ̃(x) = ∫_{σ̃^{−1}E} L̃ψ̃g(σ̃x)dμ̃(x) = ∫_{σ̃^{−1}E} ∑_{σ̃y=σ̃x} e^{ψ̃(y)}g(y)dμ̃(x). \end{aligned}

Since L̃ψ̃g ∘ σ̃ is σ̃^{−1}B-measurable, we know Eμ̃(g|σ̃^{−1}B)(x) = ∑_{σ̃y=σ̃x} e^{ψ̃(y)}g(y).

Hence,

Eμ̃(χ_{ξ̃(x)}|σ̃^{−1}B)(x) = ∑_{σ̃y=σ̃x} e^{ψ̃(y)}χ_{ξ̃(x)}(y) = e^{ψ̃(x)}.  (8.10)

By (8.8)-(8.10),

hμ̃(σ̃) = − ∫_{P_{0}} ψ̃(x)dμ̃(x).

By (1.11) and (8.7), we get what we need. □

Lemma 8.7. If h^{∗}(¯1) > 0, then − ∫ φdμ = ∞ and − ∫ ψdμ < ∞.

Proof. By Assumption A(III) we know that −φ is bounded below by a function of order n^{−1} over P_{n}. Also by Corollary A.2, μP_{n} decreases to a nonzero constant. So we have − ∫ φdμ = ∞.

By Lemma 5.3 iii), −ψ is of order k^{−(β+1)} on P_{k}. So it is easy to see that − ∫_{O_{n}} ψdμ < ∞ for any n > 0.

Note that ψ(x) → −∞ as x → s¯0 for any s ≠ 0. So we need to estimate ∫_{σ^{−1}O_{n−1}\O_{n}} ψdμ as well. Let R_{s,k} = {sx : x ∈ P_{k−1}}; then σ^{−1}O_{n−1} \ O_{n} =

⋃_{k=n}^{∞} ⋃_{s≠0} R_{s,k}. By a similar method as in the proof of Lemma 5.1, we have

∫_{R_{s,k}} ψ(x)dμ(x) = ∫_{P_{k−1}} e^{ψ(sx)}ψ(sx)dμ(x).

So

∫_{σ^{−1}O_{n−1}\O_{n}} ψ(x)dμ(x) = ∑_{s≠0} ∑_{k=n}^{∞} ∫_{P_{k−1}} e^{ψ(sx)}ψ(sx)dμ(x).

Note that φ(sx) and h(sx) are bounded. By the definition of ψ and Corollary A.2 iii),

−e^{ψ(sx)}ψ(sx) = \frac{e^{φ(sx)}h(sx)}{h(x)}(−φ(sx) − log h(sx) + log h(x)) ≤ \frac{C_{1}}{k^{1+β}} (C_{2} + log k^{1+β})

for some C_{1},C_{2} > 0 independent of k. Since β > 0, the series ∑_{k=n}^{∞} k^{−(1+β)} log k^{1+β} converges. So we get − ∫_{σ^{−1}O_{n−1}\O_{n}} ψdμ < ∞. □

Proof of Theorem D. For the case h^{∗}(¯1) ≤ 0, if μ is a probability measure, then it is an equilibrium by Theorem C and Lemma 7.4. If it is an infinite measure, then by Theorem A we have h^{∗}(¯1) = 0, and therefore by Corollary A.2 we have ∑_{i=1}^{∞} i^{−1}μP_{i} < ∞. Hence by Lemma 8.6 we have Rohlin’s formula. Since by Corollary C.1 and Lemma 4.3, P (φ) = log λ = 0, we get (1.12). On the other hand, if h^{∗}(¯1) > 0, then by Lemmas 8.6 and 8.7 we know that Rohilin’s formula and therefore (1.12) do not hold for μ.

Consider the case h^{∗}(¯1) ≥ 0. We know by Lemma 4.3 that h^{∗}(¯1) ≥ 0 if and only if P (φ) = 0, and obviously this is true if and only if δ¯0 satisfies (1.12).

Now we prove uniqueness. First we consider the case h^{∗}(¯1) ≤ 0.

Suppose ρ is a probability ergodic measure. Since the topological entropy of σ is finite and φ is continuous, we have h_{ρ}(σ) < ∞ and P (φ) < ∞. So ρ(φ) < ∞. By Assumption (III) we have φ(x) < 0 and |φ(x)| > c/k for some c > 0 if x ∈ P_{k}. Hence we have ∑_{i=1}^{∞} i^{−1}ρP_{i} < ∞. Then we can apply Lemma 8.3 to get ∫ ψdρ = ∫ φdρ − log λ. Also P (ψ) = P (φ) − log λ. Hence, P (φ) = h_{ρ}(σ) + ∫ φdρ if and only if P (ψ) = h_{ρ}(σ) + ∫ ψdρ. That is, ψ and φ have the same equilibriums.

By the same arguments used in the proof of Theorem 10 in [W1], ρ is an equilibrium of ψ if and only if L_{ψ}^{∗}ρ = ρ, where L_{ψ}^{∗} is the dual operator of L_{ψ}. Note that by Proposition 6.4, for any continuous function g,

L_{ψ}^{n}g(x) → \{ \begin{matrix}μ(g) & if λ > 1 or x ≠ ¯0; \\ g(¯0) & if λ = 1 and x = ¯0. \end{matrix}

So if ρ is an ergodic probability measure and equilibrium, then

ρ(g) = (L_{ψ}^{∗}^{n}ρ)(g) = ρ(L_{ψ}^{n}g) → \{ \begin{matrix}ρ(μ(g)) = μ(g) & if λ > 1 or ρ(\{¯0\}) = 0; \\ ρ(g(¯0)) = g(¯0) & if λ = 1 and ρ(\{¯0\}) = 1. \end{matrix}

Hence, if P (φ) > 0, then ρ = μ, and if P (φ) = 0, then either ρ = μ or ρ = δ¯0.

Now we consider the case h^{∗}(¯1) > 0. If a probability measure ρ is an equilibrium for φ, then by Lemma 4.6 i), there is φ′ ≥ φ satisfying Assumptions (I)-(III) with φ′(x) > φ(x) for some x ≠ ¯0 such that the corresponding h′^{∗}(¯1) > 0. Hence by Lemma 4.3, we have λ′ = 1 and therefore P (φ′) = 0. Now it follows that

P (φ′) = 0 = P (φ) = h_{ρ}(σ) + ∫ φdρ < h_{ρ}(σ) + ∫ φ′dρ.

It contradicts the variational principle. So there is no probability equilibrium ρ with ρ(Σ_{A}^{+} \ {¯0}) > 0.

Lastly, we assume that Assumption (III′) also holds. If ρ is an infinite measure such that ρ(φ) < ∞, then the same argument as above gives ∑_{i=1}^{∞} i^{−1}ρP_{i} < ∞. Hence we can use Lemma 8.3 to get ρ(φ) = ρ(ψ). Then the same arguments as in the case h^{∗}(¯1) ≤ 0 gives that if ρ satisfies (1.12), then ρ = μ. □

Proof of Corollary D.1. Note that if φ satisfies Assumptions A(I)-(III), then so is tφ for all t > 0. Since φ(x) ≤ 0 for all x and t > 0, P (tφ) decreases with t. It is easy to see that if t is large enough, then P (tφ) = 0.

Let t_{0} = min{t : P (tφ) = 0}. Clearly t_{0} > 0 since P (0) > 0. So if t < t_{0}, then P (tφ) > 0 and by Theorem D, μ is the unique equilibrium. We get part i).

For t = t_{0}, by Lemma 4.6 i) we know that the corresponding h_{t_{0}φ}^{∗}(¯1) = 0. In fact, if it is less than 0, then we can find t > t_{0} such that h_{tφ}^{∗}(¯1) < 0, and by Lemma 4.3, we have λ_{tφ} = 0 and therefore P (tφ) = 0, contradicting the choice of t_{0}. So by Theorem D, we know that both μ and δ¯0 satisfy (1.12). Whether μ is an equilibrium depends whether μ is finite. By Theorem A, this depends on whether t_{0}(β + 1) − 1 > 1. So part ii) is true.

For part iii), we know by Lemma 4.6 ii) that h_{tφ}^{∗}(¯1) > 0. By Theorem D we know that δ¯0 is an equilibrium for tφ, and μ_{tφ} is an infinite measure and does not satisfy (1.12). □

Proof of Corollary D.2. By Lemma 6.3, any σ-invariant weak Gibbs measure ρ for φ is an equilibrium for φ. Since μ, and probably δ¯0, are the only equilibriums for φ, and δ¯0 is not a weak Gibbs measure, we must have ρ = μ. □

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Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

   E-mail address: hu@math.msu.edu