Inducing Schemes with Finite Weighted Complexity

TL;DR

This paper develops a thermodynamic formalism for certain dynamical systems with inducing schemes under finite weighted complexity, showing liftability of measures with large pressure when a compatible partition exists.

Contribution

It introduces a thermodynamic formalism for systems with inducing schemes under finite weighted complexity and establishes liftability of measures with large pressure.

Findings

Thermodynamic formalism established for systems with inducing schemes.

Liftability of ergodic measures with large pressure under certain conditions.

Conditions under which the formalism applies are specified.

Abstract

In this paper, we consider a Borel measurable map of a compact metric space which admits an inducing scheme. Under the finite weighted complexity condition, we establish a thermodynamic formalism for a parameter family of potentials $\varphi+t\psi$ in an interval containing $t=0$. Furthermore, if there is a generating partition compatible to the inducing scheme, we show that all ergodic invariant measures with sufficiently large pressure are liftable.

Equations

$$\underset{\mu \in \mathcal{M}(f,X)}{\sup }{P}_{\mu }(\phi )$$

$$P_{\mu }(\phi ):=h_{\mu }(f)+\int \phi d\mu .$$

$${\mathcal{M}}^W(f,X):=\left\{\mu \in \mathcal{M}(f,X):\mu (W)>0\right\}.$$

$$h(f):=\underset{\mu \in \mathcal{M}(f,X)}{\sup }{h}_{\mu }(f)<\infty ,$$

$$f^{\tau (a)}(J_a)\subset W\text{and}\bigcup _{a\in S}{f}^{\tau (a)}(J_a)=W.$$

$$F|J_a=f^{\tau (a)}|J_a\text{for any}a\in S,$$

$$\check{S}:=\left\{\underline{a}\in S^{\mathbb{Z}}:x_n(\underline{a})\in J_{a_n}\text{for all}n\in \mathbb{Z}\right\}.$$

$$gcd\left\{\tau (a):a\in S\right\}=1.$$

$$S_n:=\{a\in S:\tau (a)=n\}.$$

$$\#S_n<\infty \text{for all}n\ge 1,\text{and}\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log \#S_n<\infty .$$

$$Y:=\bigcup _{k=0}^{\infty }{f}^k(W)=\left\{f^n(x):x\in W,0\le n<\tau (x)\right\}.$$

$$Q_{\nu }:={\int }_W\tau d\nu .$$

$$\mu (E):=\frac{1}{Q_{\nu }}\sum _{a\in S}\sum _{k=0}^{\tau (a)-1}\nu (f^{-k}E\cap J_a)$$

$${\mathcal{M}}_L(f,Y):=\left\{\mu \in \mathcal{M}(f,Y):\text{there is}\nu \in \mathcal{M}(F,W)\text{with}\mathcal{L}(\nu )=\mu \right\}.$$

$$P_L(\phi ):=\underset{\mu \in {\mathcal{M}}_L(f,Y)}{\sup }{P}_{\mu }(\phi ),$$

$$P^W(\phi ):=\sup \left\{P_{\mu }(\phi ):\mu \in {\mathcal{M}}^W(f,X)\text{with}\int |\phi |d\mu <\infty \right\},$$

$$\pi (\underline{a}):=x_0(\underline{a})=\bigcap _{n=-\infty }^{\infty }{F}^{-n}\overline{J_{a_n}}$$

$$\overline{\phi }(x):=\sum _{k=0}^{\tau (x)-1}\phi (f^k(x))$$

$$V_n(\Phi ):=\underset{[\underline{c}]\in {\mathcal{C}}_{-n+1}^{n-1}}{\sup }\underset{\underline{a},{\underline{a}}^{\prime }\in [\underline{c}]}{\sup }\left\{\left|\Phi (\underline{a})-\Phi ({\underline{a}}^{\prime })\right|\right\}\le H{r}^n,$$

$$[\underline{c}]=[c_n\dots c_m]:=\left\{\underline{a}\in S^{\mathbb{Z}}:a_k=c_k,k=n,\dots ,m\right\}.$$

$$\sum _{a\in S}\underset{x\in J_a}{\sup }\exp \overline{\phi }(x)<\infty .$$

$${\phi }^{+}:=\overline{\phi -P_L(\phi )}=\overline{\phi }-P_L(\phi )\tau .$$

$$\sum _{a\in S}\tau (a)\underset{x\in J_a}{\sup }\exp ({\phi }^{+}(x)+\epsilon \tau (x))<\infty .$$

$$K^{-1}\le \frac{{\nu }_{{\phi }^{+}}(J_{\underline{c}})}{\exp \left({\sum }_{k=0}^{n-1}{\phi }^{+}(F^k(x))\right)}\le K.$$

$$K^{-1}\le \frac{{\nu }_{{\phi }^{+}}(J_a)}{\exp \left({\phi }^{+}(x)\right)}\le K.$$

$${\nu }_{{\phi }^{+}}(\{x\in W:\tau (x)\ge n\})\le C{\theta }^n.$$

$$\left|\int h_1\cdot h_2\circ f^{n}d\mu -\int h_{1}d\mu \int h_{2}d\mu \right|\le K{\Theta }^n,$$

$$\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}\left(h\circ f^k-\int hd\mu \right)⟶N(0,{\sigma }^2)\text{in law},$$

$${\rho }_a(t):=\underset{x\in J_a}{\sup }\exp \left(\overline{\phi +t\psi }(x)\right),$$

$$u_n(t):=U_n\left(\overline{\phi +t\psi }\right)=\sum _{a\in S_n}{\rho }_a(t).$$

Full text

Keywords: Inducing schemes, Weighted Complexity, Equilibrium measures.

1. 1 Introduction

2. 2 Inducing Schemes

3. 2.1 Weighted Complexity

4. 2.2 Pressure and Equilibrium Measures

5. 2.3 Regular and Nice Potentials

6. 2.4 Thermodynamics of Inducing Schemes

7. 3 Thermodyanmic Formalism

8. 3.1 Auxiliary functions

9. 3.2 Proof of Theorem A

10. 4 Liftability Problem

11. 4.1 Compatible Partition and Upper bound for $P_{\mu}(\varphi)$

12. 4.2 Proof of Theorem B

13. 4.3 Proof of Corollary C

1 Introduction

The main goal in the thermodynamic formalism is to study the equilibrium measures of a dynamical system $f:X\to X$ for a potential function $\varphi:X\to{\mathbb{R}}$, i.e., the measures for which the supremum

[TABLE]

is attained, where $\mathcal{M}(f,X)$ is the class of all $f$-invariant ergodic Borel probability measure on $X$, and $P_{\mu}(\varphi)$ is the free energy given by

[TABLE]

Here $h_{\mu}(f)$ denotes the Kolmogorov-Sinai entropy of the system $(f,\mu)$.

The classical works of Sinai, Ruelle and Bowen (see e.g. [22, 21, 2]) demonstrated the existence, uniqueness and ergodic properties of equilibrium measures for uniformly hyperbolic systems. Great efforts have later been made for systems beyond uniform hyperbolicity, using various extension/inducing techniques (see e.g. [13, 5, 14, 15, 6, 3, 28, 29, 7, 8, 9, 4, 10, 11, 12]). Using principle results obtained by Sarig (see e.g. [23, 24, 25]) on the thermodynamic formalism for the countable Markov shifts, Pesin, Senti and Zhang [17, 18, 19, 20] developed a version of the inducing scheme method, which is applicable to some multimodal interval maps, the Young’s diffeomorphisms, the Hénon family and the Katok map.

In this paper, we consider a Borel measurable map $f:X\to X$ of a compact metric space, possibly with discontinuities and singularities. We assume that the map $f$ admits an inducing scheme $\{\mathcal{S},\tau\}$ satisfying Conditions (I1)-(I4) (see Definition 1), which is either of hyperbolic type or of expanding type. The inducing scheme $\{\mathcal{S},\tau\}$ might not have finite complexity (see (2.5)), in particular, there may be infinitely many blocks with the same inducing time. In such situation, we could only expect some finite weighted complexity condition (see Definition 2) for particular weight functions.

In the context of continuous maps admitting inducing schemes, Pesin, Senti and Zhang [17, 18, 19, 20] have established a thermodynamic formalism with respect to a class of nice potential functions, i.e., functions satisfying the verifiable conditions (P1)-(P4) in Section 4 of [20] (see Definition 3 for the precise statements). In this paper, we adapt their results to a measurable map $f:X\to X$ with discontinuities and singularities, which is assumed to have finite variational entropy (see (2.1)). Once an inducing scheme $\{\mathcal{S},\tau\}$ is chosen, we denote $Y$ the forward invariant subset associated with $\{\mathcal{S},\tau\}$ (see (2.6)). The potential functions under consideration is a one parameter family $\{\varphi+t\psi\}_{t\in{\mathbb{R}}}$, where $\varphi$ is a strongly nice potential and $\psi$ is a regular potential (see Definition 3). Under Condition (H) below, which assumes the finite weighted complexity (see Definition 2) at two parameters $\underline{t}_{0}<0<\overline{t}_{0}$, we are able to establish a thermodynamic formalism for the potentials $\varphi+t\psi$ in an interval containing $t=0$. Applying a result by Shahidi and Zelerowicz [26], we also obtain the analyticity of the variational lifted pressure (see (2.9)) in this interval.

The precise statements of our first main result is as follows.

Theorem A**.**

Let $\{\mathcal{S},\tau\}$ be an inducing scheme. Assume that $\varphi$ is a strongly nice potential and $\psi$ is a regular potential. Furthermore, we assume that

1. (H)

there are $\underline{t}_{0}<0<\overline{t}_{0}$ such that the inducing scheme $\{\mathcal{S},\tau\}$ has finite $\overline{\varphi+t\psi}$-complexity (see Definition 2) at $t=\underline{t}_{0}$ and $t=\overline{t}_{0}$, where $\overline{\varphi+t\psi}$ is the induced potential of $\varphi+t\psi$ (see (2.11)).

Then there exist $\underline{t}<0<\overline{t}$ such that the following hold:

1. (1)

for every $t\in(\underline{t},\overline{t})$, there exists a unique equilibrium measure $\mu_{t}$ for the potential $\varphi+t\psi$ in the class $\mathcal{M}_{L}(f,Y)$ of all liftable measures (see (2.8)); 2. (2)

if the inducing scheme $\{\mathcal{S},\tau\}$ satisfies the aperiodic condition (2.3), then the measure $\mu_{t}$ is mixing and in fact Bernoulli. Moreover, $\mu_{t}$ has exponential decay of correlations and satisfies the Central Limit Theorem (with respect to a class of functions which contains all bounded Hölder continuous functions on $Y$). 3. (3)

the function $\displaystyle t\mapsto P_{L}(\varphi+t\psi)$ is finite and real analytic in $(\underline{t},\overline{t})$, where $P_{L}(\cdot)$ is the variational liftable pressure given by (2.9).

We remark that $\underline{t}$ and $\overline{t}$ are chosen by the explicit formula in (3.11).

Our second result deals with the liftability problem over the inducing schemes, that is, we would like to determine whether an ergodic invariant measure $\mu\in\mathcal{M}(f,X)$ is liftable, i.e., $\mu\in\mathcal{M}_{L}(f,Y)$. For continuous maps admitting inducing schemes, Pesin, Senti and Zhang [19, 20] provide some conditions under which all ergodic invariant measures of sufficiently large entropy are liftable. A crucial assumption therein is the finite complexity for non-first-return inducing times (see Condition (L2) in Theorem 5.1 of [20]). For systems with nasty discontinuities and singularities, one could only impose certain finite weighted complexity condition like Condition (L2) below. Together with Condition (L1) below, which asserts the existence of a finite generating partition compatible to the inducing scheme (see Definition 4), we are able to show that all ergodic invariant measures with sufficiently large free energy are liftable.

Our second main result is stated as follows.

Theorem B**.**

Let $\{\mathcal{S},\tau\}$ be an inducing scheme. We assume that

1. (L1)

the map $f:X\to X$ has a generating partition $\mathcal{P}$, which is compatible to the inducing scheme $\{\mathcal{S},\tau\}$ (see Definition 4); 2. (L2)

the inducing scheme $\{\mathcal{S},\tau\}$ has finite $\overline{\varphi}$-complexity (see Definition 2), for some potential function $\varphi:X\to{\mathbb{R}}$.

Then any ergodic measure $\mu\in\mathcal{M}(f,X)$ with $\mu(W)>0$ and $\mathcal{K}(\overline{\varphi})<P_{\mu}(\varphi)<\infty$ is liftable, where $\mathcal{K}(\overline{\varphi})$ and $P_{\mu}(\varphi)$ are given by Definition 2 and (1.1) respectively.

An immediate corollary of Theorem A and Theorem B is the following.

Corollary C**.**

Let $\{\mathcal{S},\tau\}$ be an inducing scheme with inducing domain $W$. Assume that $\varphi$ is a strongly nice potential and $\psi$ is a regular potential. If Conditions (H) and (L1) hold, then the measure $\mu_{t}$ obtained in Theorem A is in fact the unique equilibrium measure for the potential $\varphi+t\psi$ in the class

[TABLE]

Therefore, the function $\displaystyle t\mapsto P^{W}(\varphi+t\psi)$ is finite and real analytic in $(\underline{t},\overline{t})$, where $P^{W}(\cdot)$ is the variational pressure given by (2.10).

2 Inducing Schemes

To work on the thermodynamics for continuous maps of compact metric spaces, Pesin, Senti and Zhang developed an inducing scheme method in [17, 18, 19, 20]. Our goal is to adapt this general framework to systems with discontinuities and singularities, for which the finite complexity condition (see (2.5)) usually fails. To this end, we first introduce the basic notations as follows.

Let $f:X\to X$ be a measurable map of a compact metric space, which is allowed to have discontinuities. If further $X$ is a smooth manifold, there may be singularity points at which $f$ is continuous but $Df$ does not exist. We assume that $f$ has finite variational entropy, that is,

[TABLE]

where $\mathcal{M}(f,X)$ is the set of all $f$-invariant ergodic Borel probability measures on $X$ and $h_{\mu}(f)$ is the Kolmogorov-Sinai entropy of the system $(f,\mu)$.

The definition of inducing schemes is given as follows.

Definition 1**.**

Given a countable collection of disjoint Borel subsets $\mathcal{S}=\{J_{a}\}_{a\in S}$ and a positive integer value function $\tau:S\to{\mathbb{N}}$, we say that $f$ admits an inducing scheme $\{\mathcal{S},\tau\}$ of hyperbolic type, with the inducing domain $W:=\bigcup_{a\in S}J_{a}$ and the inducing time $\tau:X\to{\mathbb{N}}$ defined by $\displaystyle\tau(x):=\begin{cases}\tau(a),&x\in J_{a},\\ 0,&x\not\in W,\end{cases}$ provided the following conditions (I1)-(I4) hold:

1. (I1)

For any $a\in S$, we have

[TABLE]

Moreover, let $F:W\to W$ be the induced map defined by

[TABLE]

and assume that $F|J_{a}$ can be extended to a homeomorphism of a neighborhood of the closure $\overline{J_{a}}$. 2. (I2)

For every bi-infinite sequence $\underline{a}=(a_{n})_{n\in{\mathbb{Z}}}\in S^{\mathbb{Z}}$, there exists a unique sequence $\underline{x}=(x_{n})_{n\in{\mathbb{Z}}}$ with coordinates $x_{n}=x_{n}(\underline{a})$ such that

• (a)

$\displaystyle x_{n}\in\overline{J_{a_{n}}}$ and $\displaystyle f^{\tau(J_{a_{n}})}(x_{n})=x_{n+1}$;

• (b)

if $\displaystyle x_{n}(\underline{a})=x_{n}(\underline{b})$ for all $n\leq 0$, then $\underline{a}=\underline{b}$.

Denote by $\sigma:S^{\mathbb{Z}}\to S^{\mathbb{Z}}$ the full left shift and let

[TABLE]

1. (I3)

The set $S^{\mathbb{Z}}\backslash\check{S}$ supports no ergodic $\sigma$-invariant measure which gives positive mass to any open subset. 2. (I4)

The induced map $F$ has at least one periodic orbit in $W$.

Remark 2.1**.**

We say $f$ admits an inducing scheme $\{\mathcal{S},\tau\}$ of expanding type if Conditions (I1)-(I4) hold, with the following changes:

• (1)

the second equation of (2.2) in (I1) is replaced by $f^{\tau(a)}(J_{a})=W$;

• (2)

the two-sided infinite sequence in $S^{\mathbb{Z}}$ is replaced by one-sided infinite sequence in $S^{{\mathbb{N}}_{0}}$, and Condition (I2)(b) is removed.

In this paper, we shall only state and prove results for inducing schemes of hyperbolic type, as all the results for the expanding type can be obtained in a similar fashion, thereby we omit the term “of hyperbolic type” when we mention inducing schemes.

We say that an inducing scheme $\{\mathcal{S},\tau\}$ satisfies the aperiodicity condition if

[TABLE]

In what follows, we shall fix an inducing scheme $\{\mathcal{S},\tau\}$ and explain the definitions of weighted complexity, liftable measures and regular/nice potentials, and then we state a main theorem in [18, 20] concerning on the thermodynamics of the inducing scheme $\{\mathcal{S},\tau\}$.

2.1 Weighted Complexity

For any $n\in{\mathbb{N}}$, we set

[TABLE]

We say that the inducing scheme $\{\mathcal{S},\tau\}$ has finite complexity if

[TABLE]

Due to the presence of discontinuities and singularities, the finite complexity condition (2.5) usually fails. In fact, when $X$ is a smooth manifold and $Df$ blows up near the singularity points, it is very likely that $\#S_{n}=\infty$ for infinitely many $n$’s, or every $\#S_{n}$ is finite but grows super-exponentially fast.

Nevertheless, when we deal with some specific class of potentials, it would be useful to introduce the following concept.

Definition 2**.**

Given a weight function $\omega:W\to{\mathbb{R}}$, the inducing scheme $\{\mathcal{S},\tau\}$ is said to have finite $\omega$-complexity if

• (a)

$\displaystyle U_{n}(\omega):=\sum_{a\in S_{n}}\sup_{x\in J_{a}}\exp{(\omega(x))}<\infty$ for all $n\geq 1$;

• (b)

$\displaystyle\mathcal{K}(\omega):=\limsup_{n\to\infty}\frac{1}{n}\log U_{n}(\omega)<\infty.$

We shall call $\mathcal{K}(\omega)$ the complexity function of $\omega$.

It is clear that the finite complexity condition (2.5) is a special case of the finite weighted complexity condition with the weight function $\omega\equiv 0$.

2.2 Pressure and Equilibrium Measures

The forward invariant hull of the inducing domain $W$ is defined by

[TABLE]

Denote the set of $f$-invariant ergodic Borel probability measures on $X$ (or $Y$) by $\mathcal{M}(f,X)$ (or $\mathcal{M}(f,Y)$), and denote the set of $F$-invariant ergodic Borel probability measures on $W$ by $\mathcal{M}(F,W)$. For any $\nu\in\mathcal{M}(F,W)$, let

[TABLE]

If $Q_{\nu}<\infty$, then $\nu$ is liftable to a measure $\mu=\mathcal{L}(\nu)\in\mathcal{M}(f,Y)$ given by

[TABLE]

for any Borel subset $E\subset Y$. We denote the class of liftable measures by

[TABLE]

The variational liftable pressure of a function $\varphi:X\to{\mathbb{R}}$ is defined by

[TABLE]

where $P_{\mu}(\varphi)$ is the pressure of $\varphi$ with respect to $\mu$ given by (1.1). Condition (I4) implies that $\mathcal{M}_{L}(f,Y)\neq\emptyset$ and $P_{L}(\varphi)>-\infty$. Note that it is possible that $P_{L}(\varphi)=+\infty$. Also, $P_{L}(\varphi-q)=P_{L}(\varphi)-q$ for any $q\in{\mathbb{R}}$.

A measure $\mu\in\mathcal{M}_{L}(f,Y)$ is called an equilibrium measure of $\varphi$ (in the class of $\mathcal{M}_{L}(f,Y)$ of liftable measures) if it attains the supremum of (2.9). Note that the potentials $\varphi-q$ and $\varphi$ share the same equilibrium measures for any $q\in{\mathbb{R}}$.

We remark that the $f$-invariant hull $Y$, the measure class $\mathcal{M}_{L}(f,Y)$ and the variational liftable pressure $P_{L}(\varphi)$ depend on the choice of the inducing schemes $\{\mathcal{S},\tau\}$. A more independent quantity is the variational pressure given by

[TABLE]

where the class $\mathcal{M}^{W}(f,X)$ of measures is given by (1.2). Notice that the only requirement on $\mathcal{M}^{W}(f,X)$ is to give positive mass on the inducing domain $W$.

2.3 Regular and Nice Potentials

Condition (I2) allows one to define the coding map $\pi:S^{\mathbb{Z}}\to\widehat{W}:=\bigcup_{a\in S}\overline{J_{a}}$ by

[TABLE]

for every bi-infinite sequence $\underline{a}=(a_{n})_{n\in{\mathbb{Z}}}\in S^{\mathbb{Z}}$. Note that $\pi$ is a semi-conjugacy from $(S^{\mathbb{Z}},\sigma)$ to $(\widehat{W},F)$, i.e., $\pi\circ\sigma=F\circ\pi$, and it is one-to-one on $\check{S}$.

Let $\varphi:X\to{\mathbb{R}}$ be a potential function. In below, we list some verifiable conditions that were introduced in [18, 20].

1. (P1)

the induced potential $\overline{\varphi}:W\to{\mathbb{R}}$ given by

[TABLE]

can be extended by continuity to a function on $\overline{J_{a}}$ for any $a\in S$; 2. (P2)

the function $\Phi=\overline{\varphi}\circ\pi$ is a locally Hölder continuous function on $S^{\mathbb{Z}}$, i.e., there are $H>0$ and $0<r<1$ such that for all $n\geq 1$,

[TABLE]

where $\mathcal{C}_{n}^{m}$, with integers $n\leq m$, is the collection of cylinders of the form

[TABLE] 3. (P3)

the following summability condition holds:

[TABLE]

By Theorem 4.5 in [18], if $\varphi$ satisfies (P1)-(P3), then $P_{L}(\varphi)<\infty$. We then define the normalized induced potential $\varphi^{+}:W\to{\mathbb{R}}$ by

[TABLE]

1. (P4)

there exists $\varepsilon>0$ such that

[TABLE]

Again by Theorem 4.5 in [18], if $\varphi$ satisfies (P1)-(P4), the normalized induced potential $\varphi^{+}$ has zero topological pressure, and there is a unique $F$-invariant equilibrium measure $\nu_{\varphi^{+}}$ for $\varphi^{+}$ among all the measures in $\mathcal{M}(F,W)$. Moreover, $\nu_{\varphi^{+}}$ has the Gibbs property, i.e., there exists a constant $K>0$ such that for any $n\geq 1$, any cylinder $[\underline{c}]\in\mathcal{C}_{0}^{n-1}$ and any $x\in J_{\underline{c}}:=\pi[\underline{c}]$, we have

[TABLE]

In particular, for any $a\in S$ and any $x\in J_{a}$, we have

[TABLE]

(P5) We say that $\nu_{\varphi^{+}}$ has exponential tail, if there are $C>0$ and $\theta\in(0,1)$ such that for any $n\geq 1$,

[TABLE]

We now introduce the following definitions for potential functions.

Definition 3**.**

We say that a function $\varphi:X\to{\mathbb{R}}$ is

• (1)

regular if $\varphi$ satisfies Conditions (P1)(P2);

• (2)

nice if $\varphi$ satisfies Conditions (P1)-(P4);

• (3)

strongly nice if $\varphi$ satisfies Conditions (P1)-(P5).

We remark that the above properties depend on the choice of the inducing scheme. It is possible that a function is regular with respect to one inducing scheme but not regular with respect to some others. We also stress that the induced potential $\overline{\varphi}$ may be unbounded even if $\varphi$ is a regular potential.

2.4 Thermodynamics of Inducing Schemes

Let us describe some statistical properties of $\mu\in\mathcal{M}(f,X)$. We say that $\mu$ has exponential decay of correlations with respect to a class $\mathcal{H}$ of functions if there exists $\Theta\in(0,1)$ such that for any $h_{1},h_{2}\in\mathcal{H}$ and any $n\geq 1$,

[TABLE]

for some $K=K(h_{1},h_{2})>0$. We say that $\mu$ satisfies the Central Limit Theorem with respect to the class $\mathcal{H}$ if for any $h\in\mathcal{H}$, which is not cohomologous to a constant, there exists $\sigma>0$ such that

[TABLE]

where $N(0,\sigma^{2})$ denotes the normal distribution.

We now state some main results by Pesin, Senti and Zhang [18, 20] on the thermodynamic formalism of inducing schemes. We also include the result on the Bernoulli property established by Shahidi and Zelerowicz [26]. Although we are dealing with maps with discontinuities and singularities in this paper, the proof of the following theorem can still be carried out by the same arguments.

Theorem 2.2**.**

Let $\{S,\tau\}$ be an inducing scheme, and assume that the potential function $\varphi$ is nice. Then

1. (1)

there exists a unique equilibrium measure $\mu_{\varphi}$ for $\varphi$ among all liftable measures in $\mathcal{M}_{L}(f,Y)$; 2. (2)

if the inducing scheme $\{\mathcal{S},\tau\}$ satisfies the aperiodic condition (2.3), then the meausre $\mu_{\varphi}$ is mixing and in fact Bernoulli. Furthermore, if $\nu_{\varphi^{+}}=\mathcal{L}^{-1}(\mu_{\varphi})$ has exponential tail, then $\mu_{\varphi}$ has exponential decay of correlations and satisfies the Central Limit Theorem (with respect to a class of functions which contains all bounded Hölder continuous functions on $Y$).

Recall that $P_{L}(\cdot)$ is the variational lifted pressure given by (2.9). The following theorem was established by Shahidi and Zelerowicz [26], which provides a criterion for the analyticity of the variational lifted pressure.

Theorem 2.3**.**

Let $\varphi_{1}$ and $\varphi_{2}$ be two potential functions on $X$. Assume there exists $\varepsilon_{0}>0$ such that $\varphi_{1}+s\varphi_{2}$ is nice for all $|s|<\varepsilon_{0}$. Then for some $0<\varepsilon<\varepsilon_{0}$, the function $s\mapsto P_{L}(\phi_{1}+s\phi_{2})$ is real analytic in $(-\varepsilon,\varepsilon)$.

3 Thermodyanmic Formalism

3.1 Auxiliary functions

Let $\varphi$ be a strongly nice potential satisfying Conditions (P1)-(P5) and let $\psi$ be a regular potential satisfying Conditions (P1)(P2).

We need the following auxiliary functions. For any $t\in(-\infty,\infty)$, we set

[TABLE]

where $\overline{\varphi+t\psi}$ is the induced potentials of $\varphi+t\psi$. Recall that $S_{n}$ is the set defined by (2.4) and $U_{n}(\cdot)$ is given by Definition 2. For any $n\geq 1$, we set

[TABLE]

We allow $u_{n}(t)=\infty$ if the above sum diverges, and we set $u_{n}(t)=0$ if $S_{n}$ is empty. Hence $u_{n}$ is a function from $(-\infty,\infty)$ to $[0,\infty]$.

By convention, we set $\log 0=-\infty$ and $\log\infty=\infty$. Recall that $\mathcal{K}(\cdot)$ is the complexity function given by Definition 2. We define a function $\kappa:(-\infty,\infty)\to[-\infty,\infty]$ by

[TABLE]

Lemma 3.1**.**

The following statements hold:

• (1)

The function $\kappa(t)$ is convex on $(-\infty,\infty)$;

• (2)

If Condition (H) holds, then

• (a)

$u_{n}(t)<\infty$* for any $t\in[\underline{t}_{0},\overline{t}_{0}]$ and any $n\geq 1$;*

• (b)

either $\kappa\equiv-\infty$ or $\kappa$ is a real-valued continuous function on $(\underline{t}_{0},\overline{t}_{0})$;

• (3)

$\kappa(0)<p:=P_{L}(\varphi)<\infty$.

Proof.

(1) For any $n\in{\mathbb{N}}$, we set

[TABLE]

for all $t\in(-\infty,\infty)$, where

[TABLE]

Since the potential $\psi$ satisfies Condition (P2), there is $H>0$ such that for any $a\in S$, we have

[TABLE]

and hence for any $t\in{\mathbb{R}}$,

[TABLE]

For any $t_{1},t_{2}\in(-\infty,\infty)$ and $\alpha\in(0,1)$, by the Hölder inequality,

[TABLE]

and hence

[TABLE]

Taking $\limsup\limits_{n\to\infty}\frac{1}{n}\log$ on both sides of (3.7), we have that

[TABLE]

Hence the function $\kappa$ is convex on $(-\infty,\infty)$.

(2) It immediately follows from Condition (H) that

[TABLE]

For any $n\geq 1$ and $t\in(\underline{t}_{0},\overline{t}_{0})$, we can write $t=\alpha\underline{t}_{0}+(1-\alpha)\overline{t}_{0}$ for some $\alpha\in(0,1)$, thus $u_{n}(t)<\infty$ by (3.7).

Furthermore, by the convexity of $\kappa$, if $\kappa\not\equiv-\infty$, then we must have $\kappa(t)>-\infty$ for all $t\in(-\infty,\infty)$. Since $\kappa(\underline{t}_{0})<\infty$ and $\kappa(\overline{t}_{0})<\infty$, by convexity of $\kappa$, we get $\kappa(t)<\infty$ for any $t\in[\underline{t}_{0},\overline{t}_{0}]$. In other words, $\kappa$ is a real-valued convex function on $[\underline{t}_{0},\overline{t}_{0}]$, and hence $\kappa$ is continuous in the interior $(\underline{t}_{0},\overline{t}_{0})$.

(3) Recall that $\varphi$ is a strongly nice potential, that is, $\varphi$ satisfies Conditions (P1)-(P5). Conditions (P1)-(P3) imply that $p:=P_{L}(\varphi)<\infty$. Together with Condition (P4), we have that $\varphi^{+}=\overline{\varphi}-p\tau$ has a unique $F$-invariant equilibrium measure $\nu_{\varphi^{+}}$. By (2.12), there is $K>0$ such that for any $a\in J$,

[TABLE]

where $\rho_{a}$ is given by (3.4). Condition (P5) further says that $\nu_{\varphi^{+}}$ has exponential tail, i.e., there are $C>0$ and $\theta\in(0,1)$ such that

[TABLE]

Hence

[TABLE]

The proof of this lemma is complete. ∎

3.2 Proof of Theorem A

Let $\varphi$ be a strongly nice potential satisfying Conditions (P1)-(P5) and let $\psi$ be a regular potential satisfying Conditions (P1)(P2).

We shall consider the potentials of the form $\varphi+t\psi-q_{t}$ for all $t\in{\mathbb{R}}$, where $q_{t}$ is a constant with regard to $x$ (see the precise formula in (3.10)). Note that the equilibrium measures of $\varphi+t\psi-q_{t}$ are the same as those of $\varphi+t\psi$ in the liftable class $\mathcal{M}_{L}(f,Y)$.

Proposition 3.2**.**

For any $t\in{\mathbb{R}}$ and any $q\in{\mathbb{R}}$, the potential $\varphi+t\psi-q$ satisfies Conditions (P1) and (P2).

Proof.

Note that $\displaystyle\overline{\varphi+t\psi-q}=\overline{\varphi}+t\overline{\psi}-q\tau$, and $\overline{\varphi}$ and $\overline{\psi}$ both satisfy Conditions (P1) and (P2). Since $\tau$ is constant on each $J_{a}$, it is continuous on $\overline{J_{a}}$ and $\tau\circ\pi$ is automatically locally Hölder continuous. Therefore, the potential $\varphi+t\psi-q$ satisfies Conditions (P1) and (P2). ∎

In the rest of this section, we show that

Proposition 3.3**.**

Under Condition (H), there exist $\underline{t}<0<\overline{t}$ such that for all $t\in(\underline{t},\overline{t})$, there exists $q_{t}\in{\mathbb{R}}$ so that the potential $\varphi+t\psi-q_{t}$ satisfies Conditions (P3)-(P5).

From now on, we shall assume that Condition (H) holds. To prove Proposition 3.3, we need the following preparations. Set

[TABLE]

where $\mu_{\varphi}$ is the equilibrium measure for the strongly nice potential $\varphi$, which is ensured by Theorem 2.2.

Lemma 3.4**.**

The number $\lambda$ given in (3.9) is well defined, and $\displaystyle-\infty<\lambda<\infty$.

Proof.

Since $\mu_{\varphi}=\mathcal{L}(\nu_{\varphi^{+}})\in\mathcal{M}_{L}(f,Y)$, we have $1\leq Q_{\nu_{\varphi^{+}}}<\infty$, where $Q_{\nu_{\varphi^{+}}}$ is given by (2.7). By Kac’s formula (see e.g. Theorem 2.3 in [18]), if $\int\overline{\psi}d\nu_{\varphi^{+}}$ is finite, then

[TABLE]

It then suffices to show that $\displaystyle-\infty<\int\overline{\psi}d\nu_{\varphi^{+}}<\infty$. By (3.5), there is $H>0$ such that

[TABLE]

Recall the definitions of $\rho_{a}$ and $\xi_{a}$ in (3.4), and also by (3.8), we have

[TABLE]

Thus it boils down to show that

[TABLE]

Recall that $u_{n}(t)$ and $v_{n}(t)$ are defined by (3.1) and (3.3) respectively, and $\kappa(t)$ is the auxiliary function defined in (3.2). By Condition (H) and Lemma 3.1, for any $p_{1}\in(\kappa(0),p)$, there exist $\underline{t}_{1}\in(\underline{t}_{0},0)$ and $\overline{t}_{1}\in(0,\overline{t}_{0})$ such that $\kappa(\underline{t}_{1})<p_{1}$ and $\kappa(\overline{t}_{1})<p_{1}$. Hence there exists $D_{1}>0$ such that $\displaystyle u_{n}(\underline{t}_{1})\leq D_{1}e^{p_{1}n}$ and $\displaystyle u_{n}(\overline{t}_{1})\leq D_{1}e^{p_{1}n}$. By the inequality $\displaystyle\log\xi_{a}\leq\frac{\xi_{a}^{\overline{t}_{1}}}{\overline{t}_{1}}$ and (3.6), we get

[TABLE]

Similarly, using the inequality $\displaystyle\frac{\xi_{a}^{\underline{t}_{1}}}{\underline{t}_{1}}\leq\log\xi_{a}$ and that $\underline{t}<0$, we get

[TABLE]

Hence we have shown that $\displaystyle-\infty<\int\overline{\psi}d\nu_{\varphi^{+}}<\infty$. The proof is complete. ∎

We further set

[TABLE]

where $P_{L}(\cdot)$ is the variational liftable pressure given by (2.9), which is greater than $-\infty$ but may be $+\infty$. Since $\varphi$ is strongly regular, we have $p=p_{0}=P_{L}(\varphi)$ is a finite number. Also, let $\lambda$ be the number given by (3.9). We further define a constant (with regard to $x$) by

[TABLE]

which can be viewed as a linear function in terms of $t$.

Lemma 3.5**.**

For any $t\in{\mathbb{R}}$, we have $\displaystyle p_{t}\geq q_{t}.$

Proof.

Since $\mu_{\varphi}$ is the unique equilibrium measure for $\varphi$ in the class $\mathcal{M}_{L}(f,Y)$ of liftable measures, we have

[TABLE]

Hence

[TABLE]

∎

Recall that the auxiliary function $\kappa:(-\infty,\infty)\to[-\infty,\infty]$ that we have defined in (3.2).

Lemma 3.6**.**

There are $\underline{t}<0<\overline{t}$ such that $\displaystyle\kappa(t)<q_{t}$ for all $t\in(\underline{t},\overline{t})$.

Proof.

If $\kappa\equiv-\infty$, then we simply take $\underline{t}=\underline{t}_{0}$ and $\overline{t}=\overline{t}_{0}$.

Otherwise, by Lemma 3.1, $\kappa$ is a real-valued convex continuous function on the interval $(\underline{t}_{0},\overline{t}_{0})$. Take $\displaystyle\kappa_{1}(t)=\kappa(t)-q_{t}.$ Since $q_{t}$ is linear in $t$, the function $\kappa_{1}$ is a continuous convex function on $(\underline{t}_{0},\overline{t}_{0})$, and $\kappa_{1}(0)=\kappa(0)-p<0$. Then this lemma holds if we set

[TABLE]

It is clear that $\underline{t}<0<\overline{t}$ by the continuity of $\kappa_{1}$. ∎

Now we are ready to prove Proposition 3.3.

Proof of Proposition 3.3.

Let $\underline{t}<0<\overline{t}$ be given by Lemma 3.6. Recall that $q_{t}$ is given by (3.10). For any $t\in(\underline{t},\overline{t})$, by Lemma 3.6, we can choose $\varepsilon_{t}$ such that $0<2\varepsilon_{t}<q_{t}-\kappa(t)$. By Lemma 3.1, there is $D_{t}>0$ such that

[TABLE]

We now verify Conditions (P3)-(P5) for the potential $\varphi+t\psi-q_{t}$.

(i) Note that $\overline{\varphi+t\psi-q_{t}}=\overline{\varphi+t\psi}-q_{t}\tau$, by (3.12), we get

[TABLE]

Thus Condition (P3) holds.

(ii) Since now Conditions (P1)-(P3) hold for the potential $\varphi+t\psi-q_{t}$, we have

[TABLE]

Also note that

[TABLE]

We take $\varepsilon=\varepsilon_{t}/2$, then by (3.12) and Lemma 3.5, Condition (P4) holds since

[TABLE]

(iii) Since now the potential $\varphi+t\psi-q_{t}$ satisfies Conditions (P1)-(P4), by Theorem 4.5 in [18], the normalized induced potential $\displaystyle(\varphi+t\psi-q_{t})^{+}=\overline{\varphi+t\psi}-p_{t}\tau$ has a unique $F$-invariant equilibrium measure $\nu_{t}:=\nu_{(\varphi+t\psi-q_{t})^{+}}$ in $\mathcal{M}(F,W)$. Moreover, $\nu_{t}$ has the Gibbs property, and in particular, there is $K_{t}>0$ such that for any $a\in J_{a}$,

[TABLE]

Hence by (3.12) and Lemma 3.5, Condition (P5) holds since

[TABLE]

for some constant $C_{t}>0$. This completes the proof of Lemma 3.3. ∎

Now we are ready to prove Theorem A.

Proof of Theorem A.

Proposition 3.2 and Proposition 3.3 show that Conditions (P1)-(P5) hold for the potential $\varphi+t\psi-q_{t}$ for all $t\in(\underline{t},\overline{t})$. In other words, $\varphi+t\psi-q_{t}$ is a strongly nice potential.

It follows from Theorem 2.2 that $\varphi+t\psi-q_{t}$ has a unique equilibrium measure $\mu_{t}:=\mu_{\varphi+t\psi-q_{t}}$ in the class $\mathcal{M}_{L}(f,Y)$; moreover, if $\{\mathcal{S},\tau\}$ satisfies the aperiodic condition (2.3), then $\mu_{t}$ is mixing and in fact Bernoulli, combining with Condition (P5), $\mu_{t}$ has exponential decay of correlations and satisfies the Central Limit Theorem. Therefore, Statement (1)(2) of Theorem A follows from the fact that $\varphi+t\psi$ and $\varphi+t\psi-q_{t}$ admit the same equilibrium measures.

We now show Statement (3) of Theorem A. Since the potential $\varphi+t\psi-q_{t}$ satisfies Conditions (P1)-(P3) for all $t\in(\underline{t},\overline{t})$, then by Theorem 4.5 in [18], we have $P_{L}(\varphi+t\psi-q_{t})<\infty$. Moreover, since

[TABLE]

is nice for all $t\in(\underline{t},\overline{t})$ and sufficiently small $|s|$, it follows from Theorem 2.3 that the function $t\mapsto P_{L}(\varphi+t\psi-q_{t})$ is real analytic in $(\underline{t},\overline{t})$. Therefore, the function

[TABLE]

is also finite and real analytic in $(\underline{t},\overline{t})$. ∎

4 Liftability Problem

4.1 Compatible Partition and Upper bound for $P_{\mu}(\varphi)$

In this subsection, we first explain Condition (L1) in Theorem B.

Definition 4**.**

We say that a measurable partition $\mathcal{P}$ of $X$ is compatible to the inducing scheme $\{\mathcal{S},\tau\}$ if the following property holds: for any $a\in S$ and any $0\leq i<\tau(a)$, the set $f^{i}(J_{a})$ is contained in an element of $\mathcal{P}$.

The compatibility condition implies that the iterates of any block $J_{a}$ would not be cut by $\partial\mathcal{P}$ into two or more pieces before it returns to the base $W$. Such condition is similar to Condition (C) in [19].

Let $\mathcal{P}$ be the partition given by Condition (L1), that is, $\mathcal{P}$ is a generating partition which is compatible to the inducing scheme $\{\mathcal{S},\tau\}$. Note that $\mathcal{P}$ is generating means the smallest $\sigma$-algebra containing $\bigcup_{n\geq 0}\mathcal{P}_{n}$ (or $\bigcup_{n\geq 0}\widetilde{\mathcal{P}}_{n}$ if $f$ is invertible) is the Borel $\sigma$-algebra on $X$, where $\mathcal{P}_{n}:=\bigvee_{k=0}^{n}f^{-k}\mathcal{P}$ (or $\widetilde{\mathcal{P}}_{n}:=\bigvee_{k=-n}^{n}f^{-k}\mathcal{P}$).

Let $\varphi:X\to{\mathbb{R}}$ be the potential given by Condition (L2), that is, the inducing scheme $\{\mathcal{S},\tau\}$ has finite $\overline{\varphi}$-complexity, and thus the complexity function $\mathcal{K}(\overline{\varphi})$ is finite (see Definition 2).

Theorem B claims that under Conditions (L1)(L2), all ergodic measures $\mu\in\mathcal{M}(f,X)$ are liftable, i.e., $\mu\in\mathcal{M}_{L}(f,Y)$, provided that

• •

$\mu$ gives positive weight to the base, i.e., $\mu(W)>0$;

• •

$\mu$ has sufficiently large but finite pressure, i.e., $\mathcal{K}(\overline{\varphi})<P_{\mu}(\varphi)<\infty$.

Recall that $P_{\mu}(\varphi)$ is the pressure of $\varphi$ with respect to $\mu$ (see (1.1)). As we assume that $f$ has finite variational entropy (see (2.1)), the condition that $-\infty<P_{\mu}(\varphi)<\infty$ is equivalent to that $\displaystyle\int|\varphi|d\mu<\infty$. Hence it suffices to consider ergodic measures $\mu\in\mathcal{M}(f,X)$ such that $\varphi$ is $\mu$-integrable.

To effectively estimate the pressure $P_{\mu}(\varphi)$ from above, we introduce the following Caratheodory-Pesin type quantity. Given an integer $m\geq 1$, let $\mathcal{G}_{m}$ be the collection of all elements in $\cup_{n\geq m}\mathcal{P}_{n}$ whose depth are marked, that is,

[TABLE]

Given a Borel subset $Z\subset X$ and a real number $\alpha\in{\mathbb{R}}$, we let

[TABLE]

where $S_{n}\varphi:=\sum_{k=0}^{n-1}\varphi\circ f^{k}$ and the infimum is taken over all sub-collection $\mathcal{G}$ of $\mathcal{G}_{m}$ which covers $Z$, that is, $\displaystyle Z\subset\bigcup_{(A,n)\in\mathcal{G}}A$. Since $M(Z,\varphi,\alpha,m)$ is non-decreasing in terms of $m$, we then define

[TABLE]

Moreover, we define the pressure of $\varphi$ on $Z$ by

[TABLE]

We remark that our definition of $P_{Z}(\varphi)$ is slightly different from the standard definition given in Section 11 of [16], as the collection $\mathcal{G}$ is not taken as an open cover but a sub-collection related to the generating partition $\mathcal{P}$. Moreover, since the map $f$ and the potential $\varphi$ that we consider are not assumed to be continuous, we may not have the inverse variational principle, i.e., $\displaystyle P_{\mu}(\varphi)=\inf\{P_{Z}(\varphi):\mu(Z)>0\}$. Nevertheless, the following lemma is enough for our purpose, which provides an upper bound for $P_{\mu}(\varphi)$.

Lemma 4.1**.**

For any ergodic measure $\mu\in\mathcal{M}(f,X)$ with $\displaystyle\int|\varphi|d\mu<\infty$, and for any Borel subset $Z\subset X$ with $\mu(Z)>0$, we have

[TABLE]

Proof.

For any $\varepsilon>0$, we set $\displaystyle\alpha_{\varepsilon}=P_{\mu}(\varphi)-2\varepsilon$. Recall that $\mathcal{P}$ is a generating partition, and hence $h_{\mu}(f)=h_{\mu}(f,\mathcal{P})$.

Since $\mu$ is ergodic, by the Birkhoff’s ergodic theorem and the Shannon-McMillan-Breiman theorem, there exists $m_{0}\in{\mathbb{N}}$ and a Borel subset $X_{0}\subset X$ with $\mu(X\backslash X_{0})<\frac{1}{2}\mu(Z)$, such that for all $n\geq m_{0}$ and $x\in X_{0}$,

[TABLE]

where $\mathcal{P}_{n}(x)$ denotes the element of $\mathcal{P}_{n}$ containing $x$.

We take $Z_{0}=Z\cap X_{0}$, then $\mu(Z_{0})>0$. Given any $m\geq m_{0}$, let $\mathcal{G}$ be a sub-collection of $\mathcal{G}_{m}$ which covers $Z_{0}$. For any $(A,n)\in\mathcal{G}$ with $A\cap Z_{0}\neq\emptyset$, we have $A=\mathcal{P}_{n}(x)$ for any $x\in A\cap Z_{0}$, and hence

[TABLE]

Therefore,

[TABLE]

which implies that $P_{Z}(\varphi)\geq\alpha_{\varepsilon}=P_{\mu}(\varphi)-2\varepsilon$. Thus $P_{Z}(\varphi)\geq P_{\mu}(\varphi)$ since $\varepsilon$ is arbitrarily chosen. ∎

4.2 Proof of Theorem B

Section 5 in [20] provides a necessary condition for an ergodic measure to be non-liftable. We briefly describe it below.

Define a tower associated with the inducing scheme $\{\mathcal{S},\tau\}$ by

[TABLE]

and define $\widehat{f}:\widehat{Y}\to\widehat{Y}$ by

[TABLE]

For any $N\geq 1$, we set

[TABLE]

For any $x\in W$ and any $n\in{\mathbb{N}}$, we define the frequency of the $\widehat{f}$-orbit of the point $(x,0)$ falling into $\widehat{E}_{N}$ during the first $n$ iterates by

[TABLE]

Following the ideas from Keller [15] and Zweimüller [30], Pesin, Senti and Zhang obtained the following property for non-liftable measures (Lemma 5.4 in [20]).

Lemma 4.2**.**

Let $\mu\in\mathcal{M}(f,X)$ be such that $\mu(W)>0$. If $\mu$ is non-liftable, then there exists an increasing sequence $\{n_{k}\}_{k\geq 1}$ of positive integers such that for any $\varepsilon>0$, there exists $Z\subset W$ satisfying that $\mu(Z)\geq(1-\varepsilon)\mu(W)$ and

[TABLE]

In fact, [20] considered an frequency $A_{n}^{N}$ falling into a larger set containing $\widehat{E}_{N}$, and thus Lemma 4.2 here is a corollary of Lemma 5.4 in [20].

Later in the proof of Theorem B, we shall use the following combinatoric inequality (see e.g. Section I.5 of [27]): for any $n\geq m\geq 0$, we have

[TABLE]

where

[TABLE]

Note that $h(t)$ is increasing for $t\in[0,\frac{1}{2}]$ and $\displaystyle\max_{t\in[0,1]}h(t)=\log 2<1$.

We are now ready to prove Theorem B.

Proof of Theorem B.

Let $\mu\in\mathcal{M}(f,X)$ be an ergodic measure with $\mu(W)>0$ and $\mathcal{K}(\overline{\varphi})<P_{\mu}(\varphi)<\infty$. Fix some $\mathcal{K}\in\left(\mathcal{K}(\overline{\varphi}),P_{\mu}(\varphi)\right)$. By Definition 2, we can pick $D>1$ and $n_{0}\in{\mathbb{N}}$ such that

[TABLE]

We fix a sufficiently small $\delta\in(0,\frac{1}{10})$ such that

[TABLE]

where $h(\cdot)$ is the function given by (4.3).

Now suppose that $\mu$ is non-liftable. We shall prove Theorem B by contradiction.

Fix $\displaystyle N=\lceil 1/\delta\rceil.$ By Lemma 4.2, there exists a Borel subset $Z\subset W$ with $\mu(Z)>0$ and an increasing sequence $\{n_{k}\}_{k\geq 1}$ of positive integers (by choosing a subsequence of the original sequence in Lemma 4.2 if necessary) such that

[TABLE]

Without loss of generality, we may assume that for all $k\geq 1$,

[TABLE]

We would like to estimate the quantity $M\left(Z,\varphi,\alpha,n_{k}\right)$ introduced in (4.1). To this end, we construct a particular covering of $Z$ as follows.

An $n_{k}$-code via the inducing scheme $\{\mathcal{S},\tau\}$ is the $s$-tuple of the form

[TABLE]

such that $m_{s-1}<n_{k}\leq m_{s}$, where $m_{0}=0$ and $m_{j}=\sum_{i=0}^{j-1}\tau(a_{i})$ for $1\leq j\leq s$. For convenience, we denote $\displaystyle n(\mathbf{a})=m_{s}$ and $\displaystyle{\boldsymbol{\tau}}(\mathbf{a})=(\tau(a_{0}),\dots,\tau(a_{s-1}))$.

Condition (L1) assumes that the partition $\mathcal{P}$ is compatible to the inducing scheme, which implies that for any $a\in S$, the block $J_{a}$ is contained in an element of $\mathcal{P}_{\tau(a)}$. Hence an $n_{k}$-code $\mathbf{a}$ defines a Borel subset

[TABLE]

which is contained in an element of $\mathcal{P}_{n(\mathbf{a})}$.

Subject to the Borel subset $Z$ and the sequence $\{n_{k}\}_{k\geq 1}$ that are obtained from Lemma 4.2, we make the following notions:

• •

let $\Gamma(Z,n_{k})$ be the collection of all $n_{k}$-codes $\mathbf{a}$ such that $A_{\mathbf{a}}\cap Z\neq\emptyset$.

• •

we say that a string ${\boldsymbol{\tau}}=(\tau_{0},\dots,\tau_{s-1})$ is $(Z,n_{k})$-admissible if there is $\mathbf{a}\in\Gamma(Z,n_{k})$ such that ${\boldsymbol{\tau}}(\mathbf{a})={\boldsymbol{\tau}}$. Let $\mathcal{A}(Z,n_{k})$ be the collection of $(Z,n_{k})$-admissible strings.

• •

given a $(Z,n_{k})$-admissible string ${\boldsymbol{\tau}}$, we let $\Gamma(Z,{\boldsymbol{\tau}})$ be the collection of all $n_{k}$-codes $\mathbf{a}\in\Gamma(Z,n_{k})$ such that ${\boldsymbol{\tau}}(\mathbf{a})={\boldsymbol{\tau}}$. It is clear that $\Gamma(Z,n_{k})=\bigcup_{{\boldsymbol{\tau}}\in\mathcal{A}(Z,n_{k})}\Gamma(Z,{\boldsymbol{\tau}}).$

• •

given a $(Z,n_{k})$-admissible string ${\boldsymbol{\tau}}=(\tau_{0},\dots,\tau_{s-1})$, we denote $n({\boldsymbol{\tau}})=\sum_{i=0}^{s-1}\tau_{i}$. We claim that

[TABLE]

Indeed, let $n^{\prime}:=\sum_{0\leq i<s-1:\ \tau_{i}\leq N}\tau_{i}$, then (4.6) implies that $\displaystyle n^{\prime}/n_{k}<\delta/2,$ and hence

[TABLE]

It is clear that $Z\subset\bigcup_{\mathbf{a}\in\Gamma(Z,n_{k})}A_{\mathbf{a}}=\bigcup_{{\boldsymbol{\tau}}\in\mathcal{A}(Z,n_{k})}\bigcup_{\mathbf{a}\in\Gamma(Z,{\boldsymbol{\tau}})}A_{\mathbf{a}}$, where each $A_{\mathbf{a}}$ lies inside an element of $\mathcal{P}_{n(\mathbf{a})}$. Hence for any $\alpha\in{\mathbb{R}}$, we have

[TABLE]

Note that for any ${\boldsymbol{\tau}}=(\tau_{0},\dots,\tau_{s-1})\in\mathcal{A}(Z,n_{k})$ and any $\mathbf{a}=(a_{0},\dots,a_{s-1})\in\Gamma(Z,{\boldsymbol{\tau}})$, we set $m_{0}=0$ and $m_{j}=\sum_{i=0}^{j-1}\tau_{i}$ for $1\leq j\leq s$, then we have

[TABLE]

Recall that $U_{n}(\cdot)$ given by Definition 2, then by (4.4) and (4.8), we have

[TABLE]

Therefore,

[TABLE]

where $\gamma_{n}:=\#\{{\boldsymbol{\tau}}\in\mathcal{A}(Z,n_{k}):\ n({\boldsymbol{\tau}})=n\}$. Note that $\gamma_{n}$ is no more than the number of ways to rewrite $n$ as

[TABLE]

such that each $\tau_{i}\geq 1$ and $\sum_{\tau_{i}\leq N}\ \tau_{i}<\delta n$. Let $s_{1}$ be the number of $i$’s such that $\tau_{i}\leq N$, and then $s_{2}=s-s_{1}$ is the number of $i$’s such that $\tau_{i}>N$. Then we have $\displaystyle 0\leq s_{1}<\delta n$ and $\displaystyle 1\leq s_{2}\leq n/N<\delta n$. Therefore, by the combinatoric inequality (4.2) and the monotonicity of the function $h(t)$ given by (4.3) for $t\in[0,\frac{1}{2}]$, as well as the choices of $\delta$ and $n_{k}$ given by (4.5) and (4.7) respectively, we get for any $n\geq n_{k}$,

[TABLE]

Finally, let $\mathcal{K}_{1}$ given by (4.5) and take some $\alpha\in(\mathcal{K}_{1},P_{\mu}(\varphi))$, then we obtain

[TABLE]

which implies that $M(Z,\varphi,\alpha)=\lim_{k\to\infty}M(Z,\varphi,\alpha,n_{k})=0$ and thus $P_{Z}(\varphi)\leq\alpha<P_{\mu}(\varphi)$. However, by Lemma 4.1, we must have $P_{\mu}(\varphi)\leq P_{Z}(\varphi)$, which is a contradiction. Therefore, the measure $\mu$ has to be liftable. The proof of Theorem B is complete. ∎

4.3 Proof of Corollary C

We now proceed the proof of Corollary C.

Proof of Corollary C.

Let $\{\mathcal{S},\tau\}$ be an inducing scheme with inducing domain $W$. Let $\varphi$ be a strongly nice potential satisfying Conditions (P1)-(P5) and let $\psi$ be a regular potential satisfying Conditions (P1)(P2). Recall that $\mathcal{M}_{L}(f,Y)$, $\mathcal{M}^{W}(f,X)$, $P_{L}(\cdot)$ and $P^{W}(\cdot)$ are defined by (2.8) (1.2), (2.9) and (2.10) respectively.

If Condition (H) holds, then by Statement (1) of Theorem A, we let $\mu_{t}$ be the unique equilibrium measure for the potential $\varphi+t\psi$ in the class $\mathcal{M}_{L}(f,Y)$, then $\displaystyle P_{\mu_{t}}(\varphi+t\psi)=P_{L}(\varphi+t\psi)$ is finite for all $t\in(\underline{t},\overline{t})$. It follows that $\displaystyle\int|\varphi+t\psi|d\mu_{t}<\infty$, and hence $P_{L}(\varphi+t\psi)\leq P^{W}(\varphi+t\psi)$ for all $t\in(\underline{t},\overline{t})$.

If now Condition (L1) holds, Lemma 3.1, Lemma 3.5 and Lemma 3.6 imply that the inducing scheme $\{\mathcal{S},\tau\}$ has finite $\overline{\varphi+t\psi}$-complexity, and

[TABLE]

For any ergodic measure $\mu_{*}\in\mathcal{M}^{W}(f,X)$ such that

[TABLE]

we have

[TABLE]

Then by Theorem B, the measure $\mu_{*}$ is liftable and hence $\mu_{*}=\mu_{t}$. In other words, $\mu_{t}$ is in fact the unique equilibrium measure for the potential $\varphi+t\psi$ in the class $\mathcal{M}^{W}(f,X)$. Moreover, by Statement (3) of Theorem A, the function $\displaystyle t\mapsto P^{W}(\varphi+t\psi)=P_{L}(\varphi+t\psi)$ is finite and real analytic in $(\underline{t},\overline{t})$. ∎

Acknowledgements

J. Chen is partially supported by the National Key Research and Development Program of China (No.2022YFA1005802), the NSFC Grant 12001392 and NSF of Jiangsu BK20200850. F. Wang is supported by NSFC grant 11871045 and the State Scholarship Fund from China Scholarship Council (CSC). H.-K. Zhang is partially supported by the Simons Foundation Collaboration Grants for Mathematicians (706383).