# Inducing Schemes with Finite Weighted Complexity

Jianyu Chen School of Mathematical Sciences & Center for Dynamical Systems and
Differential Equations, Soochow University, Suzhou, Jiangsu 215006, P.R.China.
Email<EMAIL_ADDRESS>Fang Wang School of Mathematical Sciences, Capital
Normal University, Beijing, 100048, P.R.China. Email<EMAIL_ADDRESS>Hong-Kun Zhang Department of Mathematics and Statistics, University of
Massachusetts, Amherst MA 01003, USA. Email<EMAIL_ADDRESS>
###### 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.

Keywords: Inducing schemes, Weighted Complexity, Equilibrium measures.

###### Contents

  1. 1 Introduction
  2. 2 Inducing Schemes
    1. 2.1 Weighted Complexity
    2. 2.2 Pressure and Equilibrium Measures
    3. 2.3 Regular and Nice Potentials
    4. 2.4 Thermodynamics of Inducing Schemes
  3. 3 Thermodyanmic Formalism
    1. 3.1 Auxiliary functions
    2. 3.2 Proof of Theorem A
  4. 4 Liftability Problem
    1. 4.1 Compatible Partition and Upper bound for $P_{\mu}(\varphi)$
    2. 4.2 Proof of Theorem B
    3. 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

$\sup_{\mu\in\mathcal{M}(f,X)}P_{\mu}(\varphi)$

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

$P_{\mu}(\varphi):=h_{\mu}(f)+\int\varphi d\mu.$ (1.1)

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

$\mathcal{M}^{W}(f,X):=\left\\{\mu\in\mathcal{M}(f,X):\mu(W)>0\right\\}.$
(1.2)

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,

$h(f):=\sup_{\mu\in\mathcal{M}(f,X)}h_{\mu}(f)<\infty,$ (2.1)

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

$f^{\tau(a)}(J_{a})\subset W\ \ \text{and}\ \ \bigcup_{a\in
S}f^{\tau(a)}(J_{a})=W.$ (2.2)

Moreover, let $F:W\to W$ be the _induced map_ defined by

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

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

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

  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

$\displaystyle gcd\left\\{\tau(a):a\in S\right\\}=1.$ (2.3)

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

$S_{n}:=\\{a\in S:\ \tau(a)=n\\}.$ (2.4)

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

$\\#S_{n}<\infty\ \text{for all}\ n\geq 1,\ \text{and}\
\limsup_{n\to\infty}\frac{1}{n}\log\\#S_{n}<\infty.$ (2.5)

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

$Y:=\bigcup_{k=0}^{\infty}f^{k}(W)=\left\\{f^{n}(x):\ x\in W,\ 0\leq
n<\tau(x)\right\\}.$ (2.6)

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

$Q_{\nu}:=\int_{W}\tau\ d\nu.$ (2.7)

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

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

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

$\mathcal{M}_{L}(f,Y):=\left\\{\mu\in\mathcal{M}(f,Y):\ \text{there is}\
\nu\in\mathcal{M}(F,W)\ \text{with}\ \mathcal{L}(\nu)=\mu\right\\}.$ (2.8)

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

$P_{L}(\varphi):=\sup_{\mu\in\mathcal{M}_{L}(f,Y)}P_{\mu}(\varphi),$ (2.9)

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

$P^{W}(\varphi):=\sup\left\\{P_{\mu}(\varphi):\ \mu\in\mathcal{M}^{W}(f,X)\
\text{with}\ \int|\varphi|d\mu<\infty\right\\},$ (2.10)

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

$\pi(\underline{a}):=x_{0}(\underline{a})=\bigcap_{n=-\infty}^{\infty}F^{-n}\overline{J_{a_{n}}}$

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

$\overline{\varphi}(x):=\sum^{\tau(x)-1}_{k=0}\varphi(f^{k}(x))$ (2.11)

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 $n\geq 1$,

$V_{n}(\Phi):=\sup_{[\underline{c}]\in\mathcal{C}_{-n+1}^{n-1}}\
\sup_{\underline{a},\underline{a}^{\prime}\in[\underline{c}]}\left\\{\left|\Phi(\underline{a})-\Phi(\underline{a}^{\prime})\right|\right\\}\leq
Hr^{n},$

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

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

  3. (P3)

the following summability condition holds:

$\displaystyle\sum_{a\in S}\sup_{x\in J_{a}}\exp\overline{\varphi}(x)<\infty.$

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

$\varphi^{+}:=\overline{\varphi-
P_{L}(\varphi)}=\overline{\varphi}-P_{L}(\varphi)\tau.$

  1. (P4)

there exists $\varepsilon>0$ such that

$\displaystyle\sum_{a\in S}\tau(a)\sup_{x\in
J_{a}}\exp(\varphi^{+}(x)+\varepsilon\tau(x))<\infty.$

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

$K^{-1}\leq\dfrac{\nu_{\varphi^{+}}(J_{\underline{c}})}{\exp\left(\sum_{k=0}^{n-1}\varphi^{+}(F^{k}(x))\right)}\leq
K.$

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

$K^{-1}\leq\dfrac{\nu_{\varphi^{+}}(J_{a})}{\exp\left(\varphi^{+}(x)\right)}\leq
K.$ (2.12)

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

$\nu_{\varphi^{+}}(\\{x\in W:\ \tau(x)\geq n\\})\leq C\theta^{n}.$

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 $n\geq 1$,

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

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

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

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

$\rho_{a}(t):=\sup_{x\in J_{a}}\exp\left(\overline{\varphi+t\psi}(x)\right),$

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

$u_{n}(t):=U_{n}\left(\overline{\varphi+t\psi}\right)=\sum_{a\in
S_{n}}\rho_{a}(t).$ (3.1)

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

$\kappa(t):=\mathcal{K}\left(\overline{\varphi+t\psi}\right)=\limsup_{n\to\infty}\frac{1}{n}\log
u_{n}(t).$ (3.2)

######  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

$v_{n}(t)=\sum_{a\in S_{n}}\rho_{a}\xi_{a}^{t}$ (3.3)

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

$\rho_{a}:=\rho_{a}(0)=\sup_{x\in J_{a}}\exp\overline{\varphi}(x)\ \
\text{and}\ \ \xi_{a}:=\sup_{x\in J_{a}}\exp\overline{\psi}(x).$ (3.4)

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

$\displaystyle\sup_{x\in J_{a}}\overline{\psi}(x)-\inf_{x\in
J_{a}}\overline{\psi}(x)\leq H,$ (3.5)

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

$v_{n}(t)e^{-H|t|}\leq u_{n}(t)\leq v_{n}(t)e^{H|t|}.$ (3.6)

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

$v_{n}(\alpha t_{1}+(1-\alpha)t_{2})=\sum_{a\in
S_{n}}\left[\rho_{a}\xi_{a}^{t_{1}}\right]^{\alpha}\left[\rho_{a}\xi_{a}^{t_{2}}\right]^{1-\alpha}\leq
v_{n}(t_{1})^{\alpha}v_{n}(t_{2})^{1-\alpha},$

and hence

$u_{n}(\alpha t_{1}+(1-\alpha)t_{2})\leq
u_{n}(t_{1})^{\alpha}u_{n}(t_{2})^{1-\alpha}e^{2H\left[\alpha|t_{1}|+(1-\alpha)|t_{2}|\right]}.$
(3.7)

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

$\kappa(\alpha
t_{1}+(1-\alpha)t_{2})\leq\alpha\kappa(t_{1})+(1-\alpha)\kappa(t_{2}).$

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

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

$\begin{split}&u_{n}(\underline{t}_{0})<\infty\ \text{for all}\ n\geq 1,\ \
\text{and}\ \ \kappa(\underline{t}_{0})<\infty,\\\
&u_{n}(\overline{t}_{0})<\infty\ \text{for all}\ n\geq 1,\ \ \text{and}\ \
\kappa(\overline{t}_{0})<\infty.\end{split}$

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 $a\in
J$,

$K^{-1}\leq\dfrac{\nu_{\varphi^{+}}(J_{a})}{\rho_{a}e^{-p\tau(a)}}\leq K,$
(3.8)

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

$u_{n}(0)=\sum_{a\in S_{n}}\rho_{a}\leq Ke^{pn}\nu_{\varphi^{+}}\left(\\{x\in
W:\tau(x)=n\\}\right)\leq CKe^{pn}\theta^{n}.$

Hence

$\kappa(0)=\limsup\limits_{n\to\infty}\frac{1}{n}\log u_{n}(0)\leq
p+\log\theta<p.$

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

$\lambda:=\int\psi d\mu_{\varphi},$ (3.9)

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

$-\infty<\int\overline{\psi}d\nu_{\varphi^{+}}=Q_{\nu_{\varphi^{+}}}\int\psi
d\mu_{\varphi}<\infty.$

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

$\sum\limits_{a\in S}\sup_{x\in
J_{a}}\overline{\psi}(x)\cdot\nu_{\varphi^{+}}(J_{a})-H\leq\int\overline{\psi}d\nu_{\varphi^{+}}\leq\sum\limits_{a\in
S}\sup_{x\in J_{a}}\overline{\psi}(x)\cdot\nu_{\varphi^{+}}(J_{a}).$

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

$K^{-1}\leq\dfrac{\sum\limits_{a\in S}\sup_{x\in
J_{a}}\overline{\psi}(x)\cdot\nu_{\varphi^{+}}(J_{a})}{\sum\limits_{a\in
S}\rho_{a}e^{-p\tau(a)}\log\xi_{a}}\leq K.$

Thus it boils down to show that

$-\infty<\sum\limits_{a\in
S}\rho_{a}e^{-p\tau(a)}\log\xi_{a}=\sum_{n=1}^{\infty}e^{-pn}\sum_{a\in
S_{n}}\rho_{a}\log\xi_{a}<\infty.$

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

$\sum_{n=1}e^{-pn}\sum_{a\in
S_{n}}\rho_{a}\log\xi_{a}\leq\frac{1}{\overline{t}_{1}}\sum_{n=1}^{\infty}e^{-pn}v_{n}(\overline{t}_{1})\leq\frac{D_{1}e^{H|\overline{t}_{1}|}}{\overline{t}_{1}}\sum_{n=1}^{\infty}e^{-(p-p_{1})n}<\infty.$

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

$\sum_{n=1}e^{-pn}\sum_{a\in
S_{n}}\rho_{a}\log\xi_{a}\geq\frac{1}{\underline{t}_{1}}\sum_{n=1}^{\infty}e^{-pn}v_{n}(\underline{t}_{1})\geq\frac{D_{1}e^{-H|\underline{t}_{1}|}}{\underline{t}_{1}}\sum_{n=1}^{\infty}e^{-(p-p_{1})n}>-\infty.$

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

We further set

$p_{t}:=P_{L}(\varphi+t\psi),$

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

$q_{t}=p+\lambda t,$ (3.10)

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

$p=P_{L}(\varphi)=h_{\mu_{\varphi}}(f)+\int\varphi d\mu_{\varphi}.$

Hence

$p_{t}=\sup_{\mu\in\mathcal{M}_{L}(f,Y)}\left\\{h_{\mu}(f)+\int(\varphi+t\psi)d\mu\right\\}\geq
h_{\mu_{\varphi}}(f)+\int(\varphi+t\psi)d\mu_{\varphi}=q_{t}.$

∎

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

$\underline{t}:=\inf\\{t>\underline{t}_{0}:\ \kappa_{1}(t)<0\\}\ \ \text{and}\
\ \overline{t}:=\sup\\{t<\overline{t}_{0}:\ \kappa_{1}(t)<0\\}.$ (3.11)

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

$u_{n}(t)\leq D_{t}e^{n(\kappa(t)+\varepsilon_{t})}\leq
D_{t}e^{n(q_{t}-\varepsilon_{t})}.$ (3.12)

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

$\begin{split}\sum_{a\in S}\sup_{x\in J_{a}}\exp(\overline{\varphi+t\psi-
q_{t}}(x))=\sum_{a\in
S}\rho_{a}(t)e^{-q_{t}\tau(a)}&=\sum_{n=1}^{\infty}e^{-nq_{t}}u_{n}(t)\\\
&\leq D_{t}\sum_{n=1}^{\infty}e^{-n\varepsilon_{t}}<\infty.\end{split}$

Thus Condition (P3) holds.

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

$p_{t}=P_{L}(\varphi+t\psi)=P_{L}(\varphi+t\psi-q_{t})+q_{t}<\infty.$

Also note that

$\begin{split}(\varphi+t\psi-q_{t})^{+}&=\overline{\varphi+t\psi-
q_{t}}-P_{L}(\varphi+t\psi-q_{t})\tau\\\ &=\overline{\varphi+t\psi}-q_{t}\tau-
P_{L}(\varphi+t\psi)\tau+q_{t}\tau\\\
&=\overline{\varphi+t\psi}-p_{t}\tau.\end{split}$

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

$\begin{split}\sum_{a\in S}\tau(a)\sup_{x\in J_{a}}\exp((\varphi+t\psi-
q_{t})^{+}(x)+\varepsilon\tau(x))=&\sum_{a\in
S}\tau(a)\rho_{a}(t)e^{\tau(a)(\varepsilon-p_{t})}\\\
=&\sum_{n=1}^{\infty}ne^{n(\varepsilon-p_{t})}u_{n}(t)\\\
\leq&D_{t}\sum_{n=1}^{\infty}ne^{-n\varepsilon_{t}/2}<\infty.\end{split}$

(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 $a\in
J_{a}$,

$\nu_{t}(J_{a})\leq K_{t}\sup_{x\in J_{a}}\exp(\varphi+t\psi-q_{t})^{+}(x)\leq
K_{t}\rho_{a}(t)e^{-p_{t}\tau(a)}$

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

$\begin{split}\nu_{t}(\\{x\in W:\ \tau(x)\geq N\\})=\sum_{n\geq N}\sum_{a\in
S_{n}}\nu_{t}(J_{a})&\leq K_{t}\sum_{n\geq N}e^{-np_{t}}u_{n}(t)\\\ &\leq
K_{t}D_{t}\sum_{n\geq N}e^{-n\varepsilon_{t}}\\\
&=C_{t}(e^{-\varepsilon_{t}})^{N},\end{split}$

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

$\varphi+(t+s)\psi-
q_{t+s}=\left[(\varphi-p)+t(\psi-\lambda)\right]+s(\psi-\lambda)$

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

$t\mapsto P_{L}(\varphi+t\psi)=P_{L}(\varphi+t\psi-q_{t})+p+\lambda t,$

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,

$\mathcal{G}_{m}:=\left\\{(A,n):\ n\geq m\ \text{and}\
A\in\mathcal{P}_{n}\right\\}.$

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

$M(Z,\varphi,\alpha,m):=\inf_{\mathcal{G}}\left\\{\sum_{(A,n)\in\mathcal{G}}\exp\left(-\alpha
n+\sup_{x\in A\cap Z}S_{n}\varphi(x)\right)\right\\},$ (4.1)

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

$M(Z,\varphi,\alpha):=\lim_{m\to\infty}M(Z,\varphi,\alpha,m).$

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

$P_{Z}(\varphi):=\inf\left\\{\alpha\in{\mathbb{R}}:\
M(Z,\varphi,\alpha)=0\right\\}=\sup\left\\{\alpha\in{\mathbb{R}}:\
M(Z,\varphi,\alpha)>0\right\\}.$

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

$P_{\mu}(\varphi)\leq P_{Z}(\varphi).$

###### 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 $x\in X_{0}$,

$S_{n}\varphi(x)\geq n\left(\int\varphi d\mu-\varepsilon\right)\ \ \text{and}\
\
\log\mu\left(\mathcal{P}_{n}(x)\right)\leq-n\left(h_{\mu}(f)-\varepsilon\right),$

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

$\begin{split}\sum_{(A,n)\in\mathcal{G}}\exp\left(-\alpha_{\varepsilon}n+\sup_{x\in
A\cap
Z}S_{n}\varphi(x)\right)&\geq\sum_{(A,n)\in\mathcal{G}}\exp\left(-\alpha_{\varepsilon}n+\sup_{x\in
A\cap Z_{0}}S_{n}\varphi(x)\right)\\\
&\geq\sum_{(A,n)\in\mathcal{G}}\exp\left(n\left(\int\varphi
d\mu-\varepsilon-\alpha_{\varepsilon}\right)\right)\\\
&=\sum_{(A,n)\in\mathcal{G}}\exp\left(-n\left(h_{\mu}(f)-\varepsilon\right)\right)\\\
&\geq\sum_{(A,n)\in\mathcal{G}}\mu(A)\geq\mu(Z_{0}).\end{split}$

Therefore,

$M(Z,\varphi,\alpha_{\varepsilon})=\lim_{m\to\infty}M(Z,\varphi,\alpha_{\varepsilon},m)\geq\lim_{m\to\infty}M(Z_{0},\varphi,\alpha_{\varepsilon},m)\geq\mu(Z_{0})>0,$

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

$\widehat{Y}:=\left\\{(x,k):x\in W,\ 0\leq k<\tau(x)\right\\}\subset
W\times{\mathbb{N}}$

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

$\widehat{f}(x,k):=\begin{cases}(x,k+1),&\ k<\tau(x)-1,\\\ (F(x),0),&\
k=\tau(x)-1.\end{cases}$

For any $N\geq 1$, we set

$\widehat{E}_{N}:=\left\\{(x,k)\in\widehat{Y}:\ 0\leq k\leq N\right\\}.$

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

$A_{n}^{N}(x):=\frac{1}{n}\\#\left\\{0\leq j<n:\
\widehat{f}^{j}((x,0))\in\widehat{E}_{N}\right\\}.$

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

$\lim_{k\to\infty}A_{n_{k}}^{N}(x)=0\ \ \text{uniformly on}\ Z\ \text{for
all}\ N\geq 1.$

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

${n\choose m}\leq\exp\left(nh\left(\frac{m}{n}\right)\right)<\exp(n),$ (4.2)

where

$h(t):=-t\log t-(1-t)\log(1-t),\ \text{for any}\ t\in[0,1].$ (4.3)

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

$\begin{split}U_{n}(\overline{\varphi})\leq De^{\mathcal{K}n}\ \text{for all}\
n\geq 1,\\\ U_{n}(\overline{\varphi})\leq e^{\mathcal{K}n}\ \ \text{for all}\
n\geq n_{0}.\end{split}$ (4.4)

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

$\mathcal{K}_{1}:=\mathcal{K}+3\delta+h(2\delta)+2\delta\log
D<P_{\mu}(\varphi),$ (4.5)

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

$\sup_{x\in Z}A_{n_{k}}^{N}(x)<\delta/2,\ \ \text{for all}\ k\geq 1.$ (4.6)

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

$\begin{split}(i)\ \ &n_{k}>\max\\{n_{0},100N/\delta\\},\ \text{where}\ n_{0}\
\text{is given by \eqref{est Un}};\\\ (ii)\ &n^{3}\leq e^{\delta n}\ \text{for
all}\ n\geq n_{k}.\end{split}$ (4.7)

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

$\mathbf{a}=(a_{0},\dots,a_{s-1})\in S^{s}$

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

$A_{\mathbf{a}}:=J_{a_{0}}\cap f^{-m_{1}}J_{a_{1}}\cap\dots\cap
f^{-m_{s-1}}J_{a_{s-1}},$

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

$\dfrac{\sum_{\tau_{i}\leq N}\ \tau_{i}}{n({\boldsymbol{\tau}})}<\delta,\
\text{or equivalently},\ \dfrac{\sum_{\tau_{i}>N}\
\tau_{i}}{n({\boldsymbol{\tau}})}\geq 1-\delta.$ (4.8)

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

$\dfrac{\sum_{\tau_{i}\leq N}\
\tau_{i}}{n({\boldsymbol{\tau}})}\leq\dfrac{n^{\prime}+N}{n({\boldsymbol{\tau}})}\leq\dfrac{n^{\prime}+N}{n_{k}}\leq\delta/2+\delta/100<\delta.$

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

$\begin{split}M(Z,\varphi,\alpha,n_{k})&\leq\sum_{\mathbf{a}\in\Gamma(Z,n_{k})}\exp\left(-\alpha
n(\mathbf{a})+\sup_{x\in A_{\mathbf{a}}\cap
Z}S_{n(\mathbf{a})}\varphi(x)\right)\\\
&=\sum_{{\boldsymbol{\tau}}\in\mathcal{A}(Z,n_{k})}e^{-\alpha
n({\boldsymbol{\tau}})}\sum_{\mathbf{a}\in\Gamma(Z,{\boldsymbol{\tau}})}\exp\left(\sup_{x\in
A_{\mathbf{a}}\cap Z}S_{n({\boldsymbol{\tau}})}\varphi(x)\right).\end{split}$

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

$\sup_{x\in A_{\mathbf{a}}\cap
Z}S_{n({\boldsymbol{\tau}})}\varphi(x)=\sup_{x\in A_{\mathbf{a}}\cap
Z}\sum_{i=0}^{s-1}\overline{\varphi}(f^{m_{i}}(x))\leq\sum_{i=0}^{s-1}\sup_{x\in
J_{a_{i}}}\overline{\varphi}(x).$

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

$\begin{split}\sum_{\mathbf{a}\in\Gamma(Z,{\boldsymbol{\tau}})}\exp\left(\sup_{x\in
A_{\mathbf{a}}\cap
Z}S_{n({\boldsymbol{\tau}})}\varphi(x)\right)&\leq\sum_{\mathbf{a}\in\Gamma(Z,{\boldsymbol{\tau}})}\prod_{i=0}^{s-1}\exp\left(\sup_{x\in
J_{a_{i}}}\overline{\varphi}(x)\right)\\\
&\leq\prod_{i=0}^{s-1}U_{\tau_{i}}(\overline{\varphi})\\\
&\leq\prod_{\tau_{i}\leq
N}De^{\mathcal{K}\tau_{i}}\prod_{\tau_{i}>N}e^{\mathcal{K}\tau_{i}}\\\
&\leq\exp\left(n({\boldsymbol{\tau}})\left(\mathcal{K}+\delta\log
D\right)\right).\end{split}$

Therefore,

$\begin{split}M(Z,\varphi,\alpha,n_{k})&\leq\sum_{{\boldsymbol{\tau}}\in\mathcal{A}(Z,n_{k})}\exp\left(n({\boldsymbol{\tau}})\left(\mathcal{K}-\alpha+\delta\log
D\right)\right)\\\
&=\sum_{n=n_{k}}^{\infty}\gamma_{n}\exp\left(n\left(\mathcal{K}-\alpha+\delta\log
D\right)\right),\end{split}$

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

$n=\tau_{0}+\tau_{1}+\dots+\tau_{s-1}$

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 $n\geq n_{k}$,

$\begin{split}\gamma_{n}&\leq\sum_{s_{1}=0}^{\lfloor\delta
n\rfloor}\sum_{s_{2}=1}^{\lfloor\delta n\rfloor}{n-s_{2}N-1\choose
s_{1}+s_{2}-1}{s_{1}+s_{2}\choose s_{1}}\\\ &\leq\sum_{s_{1}=0}^{\lfloor\delta
n\rfloor}\sum_{s_{2}=1}^{\lfloor\delta n\rfloor}{n\choose
s_{1}+s_{2}-1}{s_{1}+s_{2}\choose s_{1}}\\\ &\leq\sum_{s_{1}=0}^{\lfloor\delta
n\rfloor}\sum_{s_{2}=1}^{\lfloor\delta
n\rfloor}\exp\left(nh\left(\frac{s_{1}+s_{2}-1}{n}\right)\right)\exp\left(s_{1}+s_{2}\right)\\\
&\leq\sum_{s_{1}=0}^{\lfloor\delta n\rfloor}\sum_{s_{2}=1}^{\lfloor\delta
n\rfloor}\exp\left(nh\left(2\delta\right)\right)\exp\left(2\delta n\right)\\\
&\leq 2(\delta
n)^{2}\cdot\exp\left(n\left(2\delta+h(2\delta)\right)\right)<\exp\left(n(3\delta+h(2\delta))\right).\end{split}$

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

$\begin{split}M(Z,\varphi,\alpha,n_{k})&\leq\sum_{n=n_{k}}^{\infty}\exp\left(n\left(\mathcal{K}-\alpha+3\delta+h(2\delta)+\delta\log
D\right)\right)\\\
&\leq\sum_{n=n_{k}}^{\infty}\exp\left(n\left(\mathcal{K}_{1}-\alpha\right)\right)\leq\dfrac{\exp\left(n_{k}(\mathcal{K}_{1}-\alpha)\right)}{1-\exp(\mathcal{K}_{1}-\alpha)},\end{split}$

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

$-\infty<\mathcal{K}\left(\overline{\varphi+t\psi}\right)=\kappa(t)<q_{t}\leq
p_{t}=P_{L}(\varphi+t\psi)<\infty.$

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

$\begin{split}P_{\mu_{*}}(\varphi+t\psi)&=P^{W}(\varphi+t\psi)\\\
&=\sup\left\\{P_{\mu}(\varphi+t\psi):\ \mu\in\mathcal{M}^{W}(f,X)\ \text{and}\
\int|\varphi+t\psi|d\mu<\infty\right\\},\end{split}$

we have

$\mathcal{K}\left(\overline{\varphi+t\psi}\right)<P_{L}(\varphi+t\psi)\leq
P_{\mu_{*}}(\varphi+t\psi)\leq h(f)+\int|\varphi+t\psi|d\mu_{*}<\infty.$

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).

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