# Level-2 Large deviation principle for
countable Markov shifts without Gibbs states

Hiroki Takahasi  Keio Institute of Pure and Applied Sciences (KiPAS),
Department of Mathematics, Keio University, Yokohama, 223-8522, JAPAN
<EMAIL_ADDRESS>
###### Abstract.

We consider level-2 large deviations for the one-sided countable full shift
without assuming the existence of Bowen’s Gibbs state. To deal with non-
compact closed sets, we provide a sufficient condition in terms of inducing
which ensures the exponential tightness of a sequence of Borel probability
measures constructed from periodic configurations. Under this condition we
establish the level-2 Large Deviation Principle. We apply our results to the
continued fraction expansion of real numbers in $[0,1)$ generated by the Rényi
map, and obtain the level-2 Large Deviation Principle, as well as a weighted
equidistribution of a set of quadratic irrationals to equilibrium states of
the Rényi map.

###### 2020 Mathematics Subject Classification:

37A44, 37A50, 37A60, 60F10

Keywords: thermodynamic formalism; Gibbs state; Large Deviation Principle;
periodic points; equidistribution

##  1\. Introduction

Dynamical systems (iterated maps) equipped with finite Markov partitions are
represented as finite Markov shifts, and the construction of relevant
invariant measures and the investigation of their statistical properties are
done on the symbolic level, with adaptations of ideas in statistical mechanics
(see e.g., [4, 5, 30, 33, 40]). This thermodynamic formalism initiated in the
60s has been successfully extended to maps with infinite Markov partitions and
shift spaces with countably infinite number of states (see e.g., [1, 6, 11,
12, 19, 31, 32, 42]). This paper is concerned with level-2 large deviations
for such countable Markov shifts, and its application to a dynamical system
related to number theory.

The theory of large deviations aims to characterize limit behaviors of
probability measures in terms of rate functions. Let $\mathcal{X}$ be a
topological space, and let $\mathcal{M}(\mathcal{X})$ denote the space of
Borel probability measures on $\mathcal{X}$ endowed with the weak* topology.
We say a sequence $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ in
$\mathcal{M}(\mathcal{X})$ satisfies the Large Deviation Principle (LDP) if
there exists a lower semicontinuous function $I\colon\mathcal{X}\to[0,\infty]$
such that

(1.1)
$\liminf_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{G})\geq-\inf_{\mathcal{G}}I\
\text{ for any open set }\mathcal{G}\subset\mathcal{X},$

and

(1.2)
$\limsup_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{C})\leq-\inf_{\mathcal{C}}I\
\text{for any closed set }\mathcal{C}\subset\mathcal{X},$

where $\inf\emptyset=\infty$, $\log 0=-\infty$ by convention. We call
$x\in\mathcal{X}$ a minimizer if $I(x)=0$ holds. The set of minimizers is a
closed set. The LDP means that in the limit $n\to\infty$ the measure
$\tilde{\mu}_{n}$ assigns all but exponentially small mass to the set of
minimizers. The function $I$ is called a rate function, and called a good rate
function if its level set $\\{x\in\mathcal{X}\colon I(x)\leq\alpha\\}$ is
compact for any $\alpha>0$. If $\mathcal{X}$ is a metric space and
$\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ satisfies the LDP, the rate function is
unique. The setup in our mind is that $\mathcal{X}$ is the space of Borel
probability measures on a topological space $X$ on which a Borel map $\sigma$
acts, and each $\tilde{\mu}_{n}\in\mathcal{M}(\mathcal{X})$ is given in terms
of empirical measures

$V_{n}(x)=\frac{1}{n}(\delta_{x}+\delta_{\sigma
x}+\cdots\delta_{\sigma^{n-1}x}),$

where $\delta_{\sigma^{k}x}\in\mathcal{X}$ denotes the unit point mass at
$\sigma^{k}x\in X$. We refer to the LDP in this setup as level-2 [8, Chapter
1].

For topologically mixing finite Markov shifts together with Hölder continuous
potentials, the level-2 LDP for empirical distributions and that for sequences
constructed from empirical measures on periodic orbits were established in
[16, 23, 34] and [17] respectively. A key ingredient in these classical cases
is the existence of Bowen’s Gibbs states [4]. With the aid of Bowen’s Gibbs
states, one can deduce the lower bound (1.1) by combining Birkhoff’s and
Shannon-McMillan-Breiman’s theorems, and the upper bound (1.2) by modifying
the standard proof of the variational principle [41] (see [34]). For countable
Markov shifts, Bowen’s Gibbs states were constructed under the assumption of a
good regularity of potentials and a strong connectivity of transition matrices
defining the shift spaces (see e.g., [1, 6, 12, 19, 31]). Several level-2 LDPs
were established in [35] under the existence of Bowen’s Gibbs states. For a
Markov chain on a countable state space, a ‘weak level-2 LDP’ restricting the
upper bound to compact closed sets was obtained in [10].

It has been realized that not all dynamically relevant invariant probability
measures correspond to Bowen’s Gibbs states. One of the best known examples is
the absolutely continuous invariant probability measure of an interval map of
Manneville-Pomeau type, with finitely many branches and a neutral fixed point.
Such a measure still retains a weak form of Bowen’s Gibbs state [42], and has
the weak Gibbsian property in statistical mechanics sense [9, 18, 43]. For a
thermodynamic formalism and level-2 large deviations for a class of this map,
see [13, 29, 42] and [26, 27] respectively. With these historical developments
and the abundance of interesting dynamical systems modeled by countable Markov
shifts without Bowen’s Gibbs states (see e.g., [14, 44]), it is important to
establish the level-2 LDP for countable Markov shifts without assuming the
existence of Bowen’s Gibbs states.

A main new difficulty for countable Markov shifts is a treatment of non-
compact closed sets. We say a sequence $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$
of Borel probability measures on a non-compact space $\mathcal{X}$ is
exponentially tight if for any $L>0$ there exists a compact set
$\mathcal{K}\subset\mathcal{X}$ such that

$\limsup_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{X}\setminus\mathcal{K})\leq-L.$

If $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ is exponentially tight, then one
only has to consider compact closed sets in verifying the upper bound (1.2),
see e.g., [7] for details. The proof of the level-2 LDPs in [35] relies on the
existence of Bowen’s Gibbs states in order to verify the exponential
tightness.

In the lack of Bowen’s Gibbs states, our strategy is to use inducing to verify
the exponential tightness. Inducing is a familiar procedure in ergodic theory
originally considered in works by Kakutani, Rohlin and others, and was used in
the construction of absolutely continuous invariant measures or Gibbs-
equilibrium states (see e.g., [2, 3, 24, 25]). An inducing scheme we use here
is given by the first return map to an a priori fixed domain. In terms of this
inducing, we will formulate a sufficient condition which ensures the
exponential tightness for the original system. A key concept is that of local
Gibbs states, to be introduced in Section 2.2.

###  1.1. Statements of results

Throughout the rest of this paper, let $\mathbb{N}$ denote the discrete set of
positive integers. Let $X$ denote the one-sided infinite Cartesian product
topological space of $\mathbb{N}$, called a countable full shift. The topology
of $X$ has a base that consists of cylinders

$[p_{1}\cdots p_{n}]=\\{x=(x_{n})_{n=1}^{\infty}\in X\colon x_{k}=p_{k}\text{
for }k\in\\{1,2,\ldots,n\\}\\},$

where $n\geq 1$ and $p_{1}\cdots p_{n}\in\mathbb{N}^{n}$. This topology is
metrizable with the metric $d_{X}(x,y)=\exp\left(-\inf\\{n\geq 1\colon
x_{n}\neq y_{n}\\}\right)$ where $\exp(-\infty)=0$ by convention. Let $\sigma$
denote the left shift acting on $X$ continuously: $(\sigma x)_{n}=x_{n+1}$ for
$n\geq 1$.

Let $\phi\colon X\to\mathbb{R}$ be a function, called a potential. We say
$\phi$ is acceptable if it is uniformly continuous and satisfies

$\sup_{p\in\mathbb{N}}\left(\sup_{[p]}\phi-\inf_{[p]}\phi\right)<\infty.$

We say $\phi$ is locally Hölder continuous if there exist $C>0$ and
$\alpha\in(0,1]$ such that for any $p\in\mathbb{N}$ and all $x,y\in[p]$,

$|\phi(x)-\phi(y)|\leq Cd_{X}(x,y)^{\alpha}.$

Clearly, if $\phi$ is locally Hölder continuous then it is acceptable. For
each $n\geq 1$ we write $S_{n}\phi$ for the Birkhoff sum
$\sum_{k=0}^{n-1}\phi\circ\sigma^{k}$, and introduce a pressure

$P(\phi)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{p_{1}\cdots
p_{n}\in\mathbb{N}^{n}}\sup_{[p_{1}\cdots p_{n}]}\exp S_{n}\phi.$

This limit exists by the sub-additivity, which is never $-\infty$ since $X$ is
the full shift.

Let $\phi\colon X\to\mathbb{R}$ be acceptable and satisfy $P(\phi)<\infty$. We
consider a sequence $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ of Borel
probability measures on $\mathcal{M}(X)$ given by

(1.3) $\tilde{\mu}_{n}=\frac{1}{Z_{n}(\phi)}\sum_{x\in
E_{n}}\exp(S_{n}\phi(x))\delta_{V_{n}(x)},$

where

$E_{n}=\\{x\in X\colon\sigma^{n}x=x\\},$

and $\delta_{V_{n}(x)}$ denotes the unit point mass at $V_{n}(x)$, and
$Z_{n}(\phi)$ the normalizing constant. In dynamical systems terms, $E_{n}$ is
the set of periodic points of period $n$. In statistical mechanics terms, the
measure $\tilde{\mu}_{n}$ is closely related to the canonical ensemble subject
to a periodic boundary condition.

An inducing scheme consists of a subset $X^{*}$ of $X$ of the form

(1.4) $X^{*}=X\setminus\bigcup_{p\in\mathbb{N}\cap[1,p^{*})}[p],$

where $p^{*}\geq 2$, and a function $R\colon
X^{*}\to\mathbb{N}\cup\\{\infty\\}$ given by

(1.5) $R(x)=\inf\\{n\geq 1\colon\sigma^{n}x\in X^{*}\\}.$

Given an inducing scheme $(X^{*},R)$, for each $k\in\mathbb{N}$ we write

$\\{R=k\\}=\\{x\in X^{*}\colon R(x)=k\\},$

and define an induced map

(1.6) $\tau\colon\bigcup_{k=1}^{\infty}\\{R=k\\}\mapsto\sigma^{R(x)}x\in
X^{*},$

and define an inducing domain

(1.7)
$\Sigma=\bigcap_{n=0}^{\infty}\tau^{-n}\left(\bigcup_{k=1}^{\infty}\\{R=k\\}\right).$

In other words, $\tau$ is the first return map to $X^{*}$ and $\Sigma$ is the
domain on which $\tau$ can be iterated infinitely many times. We call
$(\Sigma,\tau|_{\Sigma})$ an induced system. Given a potential $\phi\colon
X\to\mathbb{R}$, we introduce a parametrized family of induced potentials
$\Phi_{\gamma}\colon\Sigma\to\mathbb{R}$ ($\gamma\in\mathbb{R}$) by

(1.8) $\Phi_{\gamma}(x)=S_{R(x)}\phi(x)-\gamma R(x).$

As shown in Section 2.1, there exists a countably infinite partition that
topologically conjugates the induced system to the countable full shift. The
local Hölder continuity of the induced potential $\Phi_{\gamma}$ and its
pressure $P(\Phi_{\gamma})$ are well-defined in terms of this conjugacy. Our
main result is stated as follows.

######  Theorem A (the level-2 LDP).

Let $\phi\colon X\to\mathbb{R}$ be acceptable and satisfy $P(\phi)<\infty$.
Suppose there exists an induced system for which the induced potentials
$\Phi_{\gamma}$, $\gamma\in\mathbb{R}$ are locally Hölder continuous, and
there exists $\gamma_{0}\in\mathbb{R}$ such that $P(\Phi_{\gamma_{0}})=0$.
Then $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ is exponentially tight and
satisfies the LDP with the good rate function.

Let us define the rate function in Theorem A. Let $\mathcal{M}(X,\sigma)$
denote the set of $\sigma$-invariant elements of $\mathcal{M}(X)$ and let
$\mathcal{M}_{\phi}(X,\sigma)=\\{\mu\in\mathcal{M}(X,\sigma)\colon\int\phi{\rm
d}\mu>-\infty\\}$. Define $F_{\phi}\colon\mathcal{M}(X)\to[-\infty,0]$ by

$F_{\phi}(\mu)=\begin{cases}h(\mu)+\int\phi{\rm d}\mu-P(\phi)&\text{ if
$\mu\in\mathcal{M}_{\phi}(X,\sigma)$},\\\ -\infty&\text{
otherwise,}\end{cases}$

where $h(\mu)\in[0,\infty]$ denotes the measure-theoretic entropy of $\mu$
with respect to $\sigma$. Since $X$ is the full shift, $\phi$ is acceptable
and $P(\phi)<\infty$, $\sup\phi$ is finite (see [19, Proposition 2.1.9]). By
[19, Theorem 2.1.7], for each $\mu\in\mathcal{M}_{\phi}(X,\sigma)$ we have
$h(\mu)+\int\phi{\rm d}\mu\leq P(\phi)<\infty$, and so $h(\mu)<\infty$. The
variational principle [19, Theorem 2.1.8] holds:

$P(\phi)=\sup\left\\{h(\mu)+\int\phi{\rm
d}\mu\colon\mu\in\mathcal{M}_{\phi}(X,\sigma)\right\\},$

A measure $\mu\in\mathcal{M}_{\phi}(X,\sigma)$ which attains this supremum is
called an equilibrium state for the potential $\phi$. The rate function
$I_{\phi}\colon\mathcal{M}(X)\to[0,\infty]$ in Theorem A is given by

(1.9) $I_{\phi}(\mu)=-\inf_{\mathcal{G}\ni\mu}\sup_{\mathcal{G}}F_{\phi},$

where the infimum is taken over all open subsets $\mathcal{G}$ of
$\mathcal{M}(X)$ containing $\mu$. Since the entropy is not upper
semicontinuous on $\mathcal{M}(X,\sigma)$, $I_{\phi}$ may not equal
$-F_{\phi}$.

Since the sequence $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ in Theorem A is
exponentially tight, it is tight. By Prohorov’s theorem, it has a limit point.
Since the rate function $I_{\phi}$ in Theorem A is the good rate function,
there exists at least one minimizer. If the minimizer is unique, we obtain a
‘level-2 weighted equidistribution of elements of
$\bigcup_{n=1}^{\infty}E_{n}$ toward minimizers’.

######  Theorem B (level-2 weighted equidistribution).

Let $\phi\colon X\to\mathbb{R}$ be acceptable and satisfy $P(\phi)<\infty$.
Suppose there exists an induced system for which the induced potentials
$\Phi_{\gamma}$, $\gamma\in\mathbb{R}$ are locally Hölder continuous, and
there exists $\gamma_{0}\in\mathbb{R}$ such that $P(\Phi_{\gamma_{0}})=0$.
Assume that the minimizer of the rate function $I_{\phi}$ is unique, denoted
by $\mu_{\rm min}$. For any bounded continuous function
$\tilde{\varphi}\colon\mathcal{M}(X)\to\mathbb{R}$ we have

$\lim_{n\to\infty}\frac{1}{Z_{n}(\phi)}\sum_{x\in
E_{n}}\exp(S_{n}\phi(x))\tilde{\varphi}(V_{n}(x))=\tilde{\varphi}(\mu_{\rm
min}).$

Under the assumption of Theorem A, minimizers are not always unique, and not
always an equilibrium state. A sufficient condition was given in [36] which
ensures that minimizers are equilibrium states.

Taking various continuous functions $\tilde{\varphi}$ in Theorem B, we obtain
convergences of various time averages over the elements of $E_{n}$. Let $C(X)$
denote the set of real-valued bounded continuous functions on $X$.

######  Corollary (Inspired by Olsen [22, Section 1.1]).

Under the assumption of Theorem B, assume moreover the minimizer is unique,
denoted by $\mu_{\rm min}$.

  * (a)

For all $\varphi,\psi\in C(X)$,

$\begin{split}\lim_{n\to\infty}\frac{1}{Z_{n}(\phi)}\sum_{x\in
E_{n}}\exp(S_{n}\phi(x))\frac{1}{n^{2}}&S_{n}\varphi(x)S_{n}\psi(x)=\int\varphi{\rm
d}\mu_{\rm min}\int\psi{\rm d}\mu_{\rm min}.\end{split}$

  * (b)

For $\varphi,\psi\in C(X)$ with $\inf\psi>0$,

$\lim_{n\to\infty}\frac{1}{Z_{n}(\phi)}\sum_{x\in
E_{n}}\exp(S_{n}\phi(x))\frac{S_{n}\varphi(x)}{S_{n}\psi(x)}=\frac{\int\varphi{\rm
d}\mu_{\rm min}}{\int\psi{\rm d}\mu_{\rm min}}.$

  * (c)

For $\pi_{1},\pi_{2}\in C(X)$ and a bounded continuous function
$f\colon\mathbb{R}\to\mathbb{R}$,

$\begin{split}\lim_{n\to\infty}\frac{1}{Z_{n}(\phi)}\sum_{x\in
E_{n}}\exp(S_{n}\phi(x))\frac{1}{n^{2}}&\sum_{k_{1},k_{2}=0}^{n-1}f(\pi_{1}(\sigma^{k_{1}}x)+\pi_{2}(\sigma^{k_{2}}x))\\\
&=\int f{\rm d}(\mu_{\rm min}\circ\pi_{1}^{-1}\otimes\mu_{\rm
min}\circ\pi_{2}^{-1}),\end{split}$

where $\otimes$ denotes the convolution.

###### Proof.

Apply Theorem B to the bounded continuous functions
$\mu\in\mathcal{M}(X)\mapsto\int\varphi{\rm d}\mu\int\psi{\rm d}\mu$,
$\mu\in\mathcal{M}(X)\mapsto\int\varphi{\rm d}\mu/\int\psi{\rm d}\mu$,
$\mu\in\mathcal{M}(X)\mapsto\int f{\rm
d}(\mu\circ\pi_{1}^{-1}\otimes\mu\circ\pi_{2}^{-1})$ respectively. ∎

Figure 1. The graph of the Rényi map $T$.

###  1.2. Applications

Our results can be applied to dynamical systems modeled by the countable full
shift without Bowen’s Gibbs state. The assumption in Theorem A can be
verified, for example, for the infinite Manneville-Pomeau map [14, Section
2.2], and the two-dimensional conformal maps in [44, Section 5] related to
number theory. Minimizers of the associated rate functions are not unique, and
so Theorem B does not apply. Further applications of different taste will be
given in our forthcoming paper.

A prime example to which our results apply is the Rényi map
$T\colon[0,1)\to[0,1)$ given by

(1.10) $T(\xi)=\frac{1}{1-\xi}-\left\lfloor\frac{1}{1-\xi}\right\rfloor,$

where $\lfloor\cdot\rfloor$ denotes the floor function. The graph of $T$ is
obtained by reversing the graph of the well-known Gauss map $\xi\in(0,1]\to
1/\xi-\lfloor 1/\xi\rfloor\in[0,1)$ around the axis $\\{\xi=1/2\\}$, as shown
in Figure 1. The map $T$ leaves invariant the absolutely continuous infinite
measure ${\rm d}x/x$, and $x=0$ is its neutral fixed point: $T(0)=0$,
$T^{\prime}(0)=1$. The asymptotic distribution of typical orbits, in the
Lebesgue measure sense, are concentrated on this neutral fixed point.

The iteration of $T$ generates an infinite continued fraction expansion of
each number $\xi\in[0,1)$ of the form

(1.11) $\xi=1-\cfrac{1}{d_{1}(\xi)-\cfrac{1}{d_{2}(\xi)-\ddots}},$

where $d_{n}(\xi)=\lfloor 1/(1-T^{n-1}(\xi))\rfloor+1\geq 2$ for $n\geq 1$.
The intervals

$J_{p}=\left[1-\frac{1}{p},1-\frac{1}{p+1}\right),\ p\in\mathbb{N}$

form an infinite Markov partition of $[0,1)$, which allows us to represent $T$
as the left shift acting on $X$ [14, 38]. The coding map

(1.12) $\pi\colon(x_{n})_{n=1}^{\infty}\in
X\mapsto\pi((x_{n})_{n=1}^{\infty})\in\bigcap_{n=1}^{\infty}\overline{T^{-n+1}(J(x_{n}))}\subset[0,1)$

is well-defined and satisfies $T\circ\pi=\pi\circ\sigma$. We consider the
potential $\phi=-\log|T^{\prime}\circ\pi|$, where $T^{\prime}$ denotes the
derivative of $T$ which is one-sided at boundary points of the Markov
partition. From the mean value theorem applied to the inverse branches of $T$,
for any $p\geq 1$ and all $\xi,\eta\in J_{p}$ we have

$\log\frac{|T^{\prime}(\xi)|}{|T^{\prime}(\eta)|}\leq 2|T(\xi)-T(\eta)|<2.$

In particular, $\phi$ is acceptable. Since $\sup_{[p]}e^{\phi}$ is comparable
to $p^{-2}$, $P(\beta\phi)<\infty$ holds if and only if
$\sum_{p\in\mathbb{N}}p^{-2\beta}$ is finite, which is equivalent to
$\beta>1/2$. It is easy to see that for $n\geq 1$, $T^{n}$ maps $[0,1/(n+1))$
diffeomorphically onto $[0,1)$. The mean value theorem implies
$\lim_{n\to\infty}\sup_{[0,1/(n+1))}|(T^{n})^{\prime}|=\infty$, while
$|(T^{n})^{\prime}(0)|=1$ for $n\geq 1$. It follows that for any $\beta>1/2$
there is no Bowen’s Gibbs state for the potential $\beta\phi$. Meanwhile, it
is known [14] that for $\beta>1/2$, the equilibrium state for $\beta\phi$ is
unique, which we denote by $\mu_{\beta\phi}$. For $1/2<\beta<1$,
$\mu_{\beta\phi}$ has positive entropy and fully supported. For $\beta\geq 1$,
$\mu_{\beta\phi}$ is the unit point mass at $\pi^{-1}(0)$.

Let $\mathbb{I}$ denote the set of irrational numbers in $(0,1)$. The set
$E_{n}$ corresponds to the set of numbers in $\mathbb{I}\cup\\{0\\}$ for which
the continued fraction (1.11) is periodic of period $n$. As in the proof of
[15, Theorem 28], one can show that any number in
$\bigcup_{n=1}^{\infty}\\{\xi\in\mathbb{I}\colon T^{n}(\xi)=\xi\\}$ is a
quadratic irrational, i.e., an irrational root of a quadratic polynomial with
integer coefficients. Conversely, any quadratic irrational in $\mathbb{I}$ has
an eventually periodic continued fraction of the form (1.11), see [21, Theorem
3].

An induced system as in Theorem A is obtained from the first return map to the
interval $[1/2,1)$ not containing the neutral fixed point. From Theorems A and
B we obtain the following. For $\xi\in[0,1)$ and $n\geq 1$, let $V_{n}(\xi)$
denote the empirical measure $(1/n)\sum_{k=0}^{n-1}\delta_{T^{k}(\xi)}$ on
$[0,1)$.

######  Theorem C.

For any $\beta\in(1/2,1]$, the sequence of Borel probability measures on
$\mathcal{M}(\pi(X))$ given by

$\frac{1}{Z_{n}(\beta\phi)}\sum_{\stackrel{{\scriptstyle\xi\in\mathbb{I}\cup\\{0\\}}}{{T^{n}(\xi)=\xi}}}|(T^{n})^{\prime}(\xi)|^{-\beta}\delta_{V_{n}(\xi)}\
\text{ for }n=1,2,\ldots$

satisfies the LDP. The minimizer is unique and it is the unit point mass at
$\mu_{\beta\phi}\circ\pi^{-1}$. Moreover, for any bounded continuous function
$\tilde{\varphi}\colon\mathcal{M}(\pi(X))\to\mathbb{R}$ we have

$\lim_{n\to\infty}\frac{1}{Z_{n}(\beta\phi)}\sum_{\stackrel{{\scriptstyle\xi\in\mathbb{I}\cup\\{0\\}}}{{T^{n}(\xi)=\xi}}}|(T^{n})^{\prime}(\xi)|^{-\beta}\tilde{\varphi}(V_{n}(\xi))=\tilde{\varphi}(\mu_{\beta\phi}\circ\pi^{-1}).$

###  1.3. Structure of the paper

The rest of this paper consists of two sections. In Section 2 we verify the
exponential tightness of the sequence in (1.3) under the assumption of Theorem
A. In Section 3 we complete proofs of all the theorems. We close with a remark
on possible generalizations of the main results.

##  2\. Exponential tightness

The aim of this section is to verify the exponential tightness of the sequence
in (1.3). In Section 2.1 we start with a symbolic representation of the
induced system. In Section 2.2 we introduce the notion of local Gibbs states
together with examples. In Section 2.3 we prove a main technical estimate
assuming the existence of a local Gibbs state. Using this estimate, we verify
the exponential tightness in Section 2.4. In Section 2.5 we show that the
assumption of Theorem A implies the existence of a local Gibbs state.

###  2.1. Symbolic representation of the induced system

For a set $\mathbb{S}$ and an integer $j\geq 1$, let $\mathbb{S}^{j}$ denote
the set of words of elements of $\mathbb{S}$ of word length $j$. We introduce
an empty word $\emptyset$ and set $\mathbb{S}^{0}=\\{\emptyset\\}$,
$a\emptyset=a=\emptyset a$, $a\emptyset b=ab$ for $a,b\in\mathbb{S}$. We set
$W(\mathbb{S})=\bigcup_{j=1}^{\infty}\mathbb{S}^{j}$,
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$ and
$W_{0}(\mathbb{S})=W(\mathbb{S})\cup\mathbb{S}^{0}.$

Let $(X^{*},R)$ be an inducing scheme. Set

$\mathbb{N}^{*}=\mathbb{N}\cap[p^{*},\infty)\ \text{ and }\
\mathbb{N}_{*}=\mathbb{N}\cap[1,p^{*}).$

For each $p\in\mathbb{N}^{*}$ and $\omega\in W_{0}(\mathbb{N}_{*})$, the set
$\bigcup_{q\in\mathbb{N}^{*}}[p\omega q]$ is mapped by the induced map $\tau$
in (1.6) bijectively onto $X^{*}$. Since the domain
$\bigcup_{k=1}^{\infty}\\{R=k\\}$ of definition of $\tau$ is partitioned into
countably infinite sets of this form, the induced system $\tau|_{\Sigma}$ is
represented as the countable full shift over the infinite alphabet

(2.1) $\mathbb{A}=\left\\{\bigcup_{q\in\mathbb{N}^{*}}[p\omega q]\colon
p\in\mathbb{N}^{*}\text{ and }\omega\in W_{0}(\mathbb{N}_{*})\right\\}.$

To make the above statement into a rigorous one, we endow $\mathbb{A}$ with
the discrete topology, and consider the countable full shift

${\mathbb{A}}^{\mathbb{N}}=\\{z=(z_{n})_{n=1}^{\infty}\colon
z_{n}\in\mathbb{A}\text{ for }n\geq 1\\}.$

We will often use bold letters to denote elements of $W(\mathbb{A})$, and a
double square bracket $[\\![\cdot]\\!]$ to denote cylinders in
$\mathbb{A}^{\mathbb{N}}$: the $n$-cylinder $(n\geq 1)$ spanned by
$\mathbb{a}=a_{1}\cdots a_{n}\in\mathbb{A}^{n}$ is

$[\\![\mathbb{a}]\\!]=\\{z=(z_{k})_{k=1}^{\infty}\in\mathbb{A}^{\mathbb{N}}\colon
z_{k}=a_{k}\text{ for }k\in\\{1,\ldots,n\\}\\}.$

By definition, for each $k\in\\{1,\ldots,n\\}$ we have
$a_{k}=\bigcup_{q\in\mathbb{N}^{*}}[p_{k}\omega^{(j_{k})}q]$ where
$p_{k}\in\mathbb{N}^{*}$, $j_{k}\in\mathbb{N}_{0}$,
$\omega^{(j_{k})}\in\mathbb{N}_{*}^{j_{k}}$. Let $|\mathbb{a}|$ denote the
word length of the word $p_{1}\omega^{(j_{1})}p_{2}\omega^{(j_{2})}\cdots
p_{n}\omega^{(j_{n})}$ in $W(\mathbb{N})$:

$|\mathbb{a}|=n+\sum_{k=1}^{n}j_{k}.$

Let $[\mathbb{a}]$ denote the corresponding $|\mathbb{a}|$-cylinder in $X$:

$[\mathbb{a}]=[p_{1}\omega^{(j_{1})}p_{2}\omega^{(j_{2})}\cdots
p_{n}\omega^{(j_{n})}]\subset X.$

It is easy to check that a coding map
$\Pi\colon\mathbb{A}^{\mathbb{N}}\to\Sigma$ is defined by

(2.2)
$\Pi\colon(z_{n})_{n=1}^{\infty}\in\mathbb{A}^{\mathbb{N}}\mapsto\Pi((z_{n})_{n=1}^{\infty})\in\bigcap_{n=1}^{\infty}[z_{1}\cdots
z_{n}]\subset\Sigma,$

and becomes a homeomorphism. Let $\theta$ denote the left shift acting on
${\mathbb{A}}^{\mathbb{N}}$. Clearly we have
$\Pi\circ\theta=\tau|_{\Sigma}\circ\Pi$,
$\Pi([\\![\mathbb{a}]\\!])=[\mathbb{a}]$ for any $\mathbb{a}\in
W(\mathbb{A})$. Let $d_{\Sigma}$ denote the metric on $\Sigma$ given by
$d_{\Sigma}(x,y)=d_{\mathbb{A}^{\mathbb{N}}}(\Pi^{-1}(x),\Pi^{-1}(y))$.

For $\mathbb{a}\in\mathbb{A}^{n}$ as above and $q\in\mathbb{N}^{*}$, we will
often write

$\mathbb{a}q=p_{1}\omega^{(j_{1})}p_{2}\omega^{(j_{2})}\cdots
p_{n}\omega^{(j_{n})}q,$

and interpret $\mathbb{a}q$ as a word in $W(\mathbb{N})$ of word length
$|\mathbb{a}|+1$.

######  Lemma 2.1.

Let $(\Sigma,\tau|_{\Sigma})$ be an induced system and let $\Pi$ be the coding
map in (2.2).

  * (a)

For every $\mathbb{a}\in W(\mathbb{A})$,

$\Pi[\\![\mathbb{a}]\\!]=\Sigma\cap\bigcup_{q\in\mathbb{N}^{*}}[\mathbb{a}q].$

  * (b)

If $\mathbb{a}$, $\mathbb{b}\in W(\mathbb{A})$ satisfy
$|\mathbb{a}|=|\mathbb{b}|$, then $\mathbb{a}=\mathbb{b}$ or
$[\\![\mathbb{a}]\\!]\cap[\\![\mathbb{b}]\\!]=\emptyset.$

###### Proof.

If $n\geq 1$, $\mathbb{a}\in\mathbb{A}^{n}$ then the sum
$\sum_{k=0}^{n-1}R\circ\tau^{k}$ of return times equals $|\mathbb{a}|$ on
$\Pi[\\![\mathbb{a}]\\!]$, which implies (a). If $\mathbb{a}$, $\mathbb{b}\in
W(\mathbb{A})$ satisfy $|\mathbb{a}|=|\mathbb{b}|$ and
$\mathbb{a}\neq\mathbb{b}$, then (a) implies
$\Pi[\\![\mathbb{a}]\\!]\cap\Pi[\\![\mathbb{b}]\\!]=\emptyset$, and so
$[\\![\mathbb{a}]\\!]\cap[\\![\mathbb{b}]\\!]=\emptyset$, verifying (b). ∎

###  2.2. Local Gibbs states

Let $\phi\colon X\to\mathbb{R}$ and let $(\Sigma,\tau|_{\Sigma})$ be an
induced system. A Borel probability measure $\lambda_{\phi}$ on
${\mathbb{A}}^{\mathbb{N}}$ is called a local Gibbs state for the potential
$\phi$ associated with $(\Sigma,\tau|_{\Sigma})$, if there exist constants
$C\geq 1$, $\gamma_{0}\in\mathbb{R}$ such that for any $\mathbb{a}\in
W(\mathbb{A})$ and any $x\in\Pi[\\![\mathbb{a}]\\!]$ we have

(2.3)
$C^{-1}\leq\frac{\lambda_{\phi}[\\![\mathbb{a}]\\!]}{\exp\left(S_{|\mathbb{a}|}\phi(x)-\gamma_{0}|\mathbb{a}|\right)}\leq
C.$

We do not require the $\theta$-invariance of $\lambda_{\phi}$. We will simply
call $\lambda_{\phi}$ a local Gibbs state (associated with
$(\Sigma,\tau|_{\Sigma})$) if the context is clear.

######  Remark 2.2 (Examples of local Gibbs states).

The Rényi map $T$ in (1.10) is modeled on the countable full shift $X$ via the
coding map $\pi$ in (1.12). In consideration of the neutral fixed point $0$ of
$T$, we set $p^{*}=2$ and define an inducing scheme $(X^{*},R)$ by (1.4),
(1.5), and the induced system $(\Sigma,\tau|_{\Sigma})$ by (1.6), (1.7). We
also define the infinite alphabet $\mathbb{A}$ and the coding map $\Pi$ by
(2.1), (2.2) respectively. Note that
$\Sigma=\left(1/2,1\right)\cap\mathbb{I}$. Let $\mu_{1}$ denote the normalized
restriction of the Lebesgue measure on $\mathbb{R}$ to $(1/2,1)$. Let
$\mu_{2}$ denote the normalized restriction of ${\rm d}x/x$ to $(1/2,1)$. Both
$\mu_{1}\circ\pi\circ\Pi$ and $\mu_{2}\circ\pi\circ\Pi$ are local Gibbs states
for the potential $\phi=-\log|T^{\prime}\circ\pi|$ with $\gamma_{0}=0$. The
latter is a $\theta$-invariant measure.

If $\lambda_{\phi}$ is a local Gibbs state, then for any $\mathbb{a}\in
W(\mathbb{A})$, the $\lambda_{\phi}$-measure of the cylinder
$[\\![\mathbb{a}]\\!]$ in $\mathbb{A}^{\mathbb{N}}$ is given (up to
multiplicative constants) by the Birkhoff sum of $\phi$ along the orbit of $x$
of length $|\mathbb{a}|$ and the word length $|\mathbb{a}|$. Since the word
length of $\mathbb{a}$ as a word in $W(\mathbb{A})$ does not appear in the
formula (2.3), $\lambda_{\phi}$ well captures part of the original dynamics
$(X,\sigma)$.

######  Lemma 2.3.

Let $\phi\colon X\to\mathbb{R}$, let $(\Sigma,\tau|_{\Sigma})$ be an induced
system and let $\lambda_{\phi}$ be a local Gibbs state for the potential
$\phi$ associated with $(\Sigma,\tau|_{\Sigma})$. For any $\mathbb{a}\in
W(\mathbb{A})$ we have

$\lambda_{\phi}[\\![\mathbb{a}]\\!]=\lambda_{\phi}\circ\Pi^{-1}(\Sigma\cap[\mathbb{a}]).$

###### Proof.

We write $\nu_{\phi}$ for $\lambda_{\phi}\circ\Pi^{-1}$. We have
$\nu_{\phi}\\{R=n\\}>0$ for every $n\geq 1$. Let $\nu_{\phi}|_{\\{R=n\\}}$
denote the restriction of $\nu_{\phi}$ to $\\{R=n\\}$. The measure

$\mu_{\phi}=\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\nu_{\phi}|_{\\{R=n\\}}\circ\sigma^{-k}$

is a finite measure if and only if $\int R{\rm d}\nu_{\phi}<\infty$. Since the
restriction of $R$ to $\Sigma$ is the first return time to $\Sigma$,
$\\{R=n\\}$ is disjoint from
$\Sigma\cap\bigcup_{k=1}^{n-1}\sigma^{-k}(\Sigma)$ for $n\geq 2$ and we have

(2.4)
$\mu_{\phi}|_{\Sigma}=\sum_{n=1}^{\infty}\nu_{\phi}|_{\\{R=n\\}}=\nu_{\phi}=\lambda_{\phi}\circ\Pi^{-1}.$

For any $\mathbb{a}\in W(\mathbb{A})$ we have
$\Pi[\\![\mathbb{a}]\\!]\subset\Sigma$, and so

(2.5) $\lambda_{\phi}[\\![\mathbb{a}]\\!]=\mu_{\phi}\Pi[\\![\mathbb{a}]\\!].$

By $\mu_{\phi}|_{\Sigma}=\nu_{\phi}$ in (2.4) and Lemma 2.1(a), for any
$\mathbb{a}\in W(\mathbb{A})$ we have

(2.6)
$\mu_{\phi}\Pi[\\![\mathbb{a}]\\!]=\nu_{\phi}\Pi[\\![\mathbb{a}]\\!]=\sum_{q\in\mathbb{N}^{*}}\nu_{\phi}(\Sigma\cap[\mathbb{a}q]).$

Let $j\geq 1$ satisfy $\mathbb{a}\in\mathbb{A}^{j}$. Then
$\sigma^{|\mathbb{a}|}$ and $\tau^{j}$ coincide on $\Sigma\cap[\mathbb{a}]$.
By $\tau(\Sigma)\subset\Sigma$ we have
$\bigcup_{q\in\mathbb{N}_{*}}(\Sigma\cap[\mathbb{a}q])\subset(\tau|_{\Sigma})^{-j}(\bigcup_{q\in\mathbb{N}_{*}}[q])=\emptyset$,
and so $\nu_{\phi}(\bigcup_{q\in\mathbb{N}_{*}}\Sigma\cap[\mathbb{a}q])=0.$
Combining this with (2.5), (2.6) we obtain

$\lambda_{\phi}[\\![\mathbb{a}]\\!]=\sum_{q\in\mathbb{N}_{*}}\nu_{\phi}(\Sigma\cap[\mathbb{a}q])+\sum_{q\in\mathbb{N}^{*}}\nu_{\phi}(\Sigma\cap[\mathbb{a}q])=\nu_{\phi}(\Sigma\cap[\mathbb{a}]),$

as required. ∎

###  2.3. Exponential decay on partition functions

The next proposition provides a main technical estimate under the existence of
a local Gibbs state.

######  Proposition 2.4.

Let $\phi\colon X\to\mathbb{R}$, let $(\Sigma,\tau|_{\Sigma})$ be an induced
system and let $\lambda_{\phi}$ be a local Gibbs state for the potential
$\phi$ associated with $(\Sigma,\tau|_{\Sigma})$. There exist
$\delta^{\prime}\in(0,1/5]$ and $n_{0}\geq 1$ such that if
$\delta\in(0,\delta^{\prime}]$ and $\\{N_{i}\\}_{i=1}^{\infty}$ is a non-
decreasing sequence of integers such that

(2.7) $\max\mathbb{N}_{*}\leq N_{1}\text{ and }$ (2.8)
$\sum_{k=N_{i}+1}^{\infty}\sum_{\stackrel{{\scriptstyle\mathbb{a}\in\mathbb{A}}}{{\Pi[\\![\mathbb{a}]\\!]\subset[k]}}}\lambda_{\phi}[\\![\mathbb{a}]\\!]\leq\delta^{2i}\text{
for every }i\geq 1,$

then for every $n\geq n_{0}$ and every $m\in\\{1,\ldots,n\\}$ we have

(2.9) $\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp
S_{n}\phi(x)\leq\frac{2^{n}ne^{\gamma_{0}n}(4\delta)^{m}}{1-4\delta},$

where

(2.10) $\Gamma=\\{x=(x_{i})_{i=1}^{\infty}\in X\colon x_{i}\leq N_{i}\text{
for every }i\geq 1\\}.$

Proposition 2.4 asserts that the contribution of elements of $E_{n}$ to
$Z_{n}(\phi)$ whose orbits escape from the compact set $\Gamma$ exactly $m$
times within period $n$ is exponentially small in $m$. Similar estimates were
obtained in [35] under the existence of Bowen’s Gibbs states.

###### Proof of Proposition 2.4.

Since $\lambda_{\phi}$ is a local Gibbs state, there exist constants $C\geq 1$
and $\gamma_{0}\in\mathbb{R}$ such that for any $\mathbb{a}\in W(\mathbb{A})$
and any $x\in\Pi[\\![\mathbb{a}]\\!]$ we have

(2.11)
$C^{-1}\leq\frac{\lambda_{\phi}[\\![\mathbb{a}]\\!]}{\exp\left(S_{|\mathbb{a}|}\phi(x)-\gamma_{0}|\mathbb{a}|\right)}\leq
C.$

For the rest of the proof of Proposition 2.4, we will use the notation $a\ll
b$ for two positive reals $a$, $b$ to indicate that $a/b$ is bounded from
above by a constant which depends only on $C$ in (2.11). If $a\ll b$ and $b\ll
a$, we will write $a\asymp b$.

The first inequality in (2.11) will be used to bound a partial sum of
$Z_{n}(\phi)$ from above by a sum of $\lambda_{\phi}$-measures of cylinders in
${\mathbb{A}}^{\mathbb{N}}$. Further, we will bound this sum using the
following product property which is a consequence of (2.11):

(2.12)
$\lambda_{\phi}[\\![\mathbb{a}\mathbb{b}]\\!]\asymp\lambda_{\phi}[\\![\mathbb{a}]\\!]\lambda_{\phi}[\\![\mathbb{b}]\\!]\text{
for }\mathbb{a},\mathbb{b}\in W(\mathbb{A}).$

Let $\delta\in(0,1/5]$ and let $\\{N_{i}\\}_{i=1}^{\infty}$ be a non-
decreasing integer sequence satisfying (2.7) and (2.8). Let
$\Gamma=\Gamma(\\{N_{i}\\}_{i=1}^{\infty})$ be the compact subset of $X$ given
by (2.10). Let $n\geq 1$. If $x\in
E_{n}\setminus\bigcup_{i=0}^{n-1}\sigma^{-i}(\Sigma)$ then
$x_{i}\leq\max\mathbb{N}_{*}\leq N_{1}\leq N_{i}$ for $1\leq i\leq n$, and so
$V_{n}(x)(\Gamma)=1$. By this and the periodicity of elements of $E_{n}$, for
$1\leq m\leq n$ we have

(2.13) $\begin{split}\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp
S_{n}\phi(x)=&\sum_{\stackrel{{\scriptstyle x\in
E_{n}\cap\bigcup_{i=0}^{n-1}\sigma^{-i}(\Sigma)}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp
S_{n}\phi(x)\\\ &+\sum_{\stackrel{{\scriptstyle x\in
E_{n}\setminus\bigcup_{i=0}^{n-1}\sigma^{-i}(\Sigma)}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp
S_{n}\phi(x)\\\ =&\sum_{\stackrel{{\scriptstyle x\in
E_{n}\cap\bigcup_{i=0}^{n-1}\sigma^{-i}(\Sigma)}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp
S_{n}\phi(x)\\\ \leq&n\cdot\sum_{\stackrel{{\scriptstyle x\in
E_{n}\cap\Sigma}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp
S_{n}\phi(x).\end{split}$

In order to bound the last sum in (2.13), we decompose the set $\\{x\in
E_{n}\cap\Sigma\colon V_{n}(x)(X\setminus\Gamma)=m/n\\}$ into subsets sharing
the same itinerary ‘up to time $n$’, and estimate a contribution from each
subset separately, and finally unify all these estimates counting the total
number of possible itineraries.

Define a function $r\colon X\setminus\Gamma\to\mathbb{N}$ by

$r(x)=\min\\{i\geq 1\colon x_{i}>N_{i}\\}.$

Let $x\in E_{n}\cap\Sigma$ satisfy $V_{n}(x)(X\setminus\Gamma)>0$. By an
itinerary of $x$ we mean a pair of sequences $\\{n_{j}(x)\\}_{j=1}^{\infty}$
and $\\{r_{j}(x)\\}_{j=1}^{\infty}$ in $\mathbb{N}_{0}$ given by the recursion
formulas

$\begin{split}n_{1}(x)&=\min\\{i\geq 0\colon\sigma^{i}x\notin\Gamma\\},\text{
and }\\\ r_{j}(x)&=r(\sigma^{n_{j}(x)}x),\ n_{j+1}(x)=\min\\{i\geq
n_{j}(x)+r_{j}(x)\colon\sigma^{i}x\notin\Gamma\\}\text{ for }j\geq
1.\end{split}$

Note that

$0\leq n_{1}(x)\leq\cdots\leq n_{j}(x)\leq
n_{j}(x)+r_{j}(x)-1<n_{j+1}(x)\leq\cdots$

######  Lemma 2.5.

Let $x\in E_{n}\cap\Sigma$ satisfy $V_{n}(x)(X\setminus\Gamma)>0$.

  * (a)

$\\{i\geq
0\colon\sigma^{i}x\notin\Gamma\\}=\bigcup_{j=1}^{\infty}[n_{j}(x),n_{j}(x)+r_{j}(x)-1]\cap\mathbb{N}_{0}.$

  * (b)

$x_{n_{j}(x)+r_{j}(x)}\in\mathbb{N}^{*}$ for every $j\geq 1$.

  * (c)

If $n_{j}(x)\leq n-1$ then $n_{j}(x)+r_{j}(x)\leq 2n-1$.

###### Proof.

Since $\\{N_{i}\\}_{i=1}^{\infty}$ is non-decreasing, if $x\notin\Gamma$ then
$\sigma^{i}x\notin\Gamma$ for $0\leq i\leq r(x)-1$. This implies (a). Since
$\sigma^{n_{j}(x)}x=x_{n_{j}(x)+1}x_{n_{j}(x)+2}\cdots$ and
$\sigma^{n_{j}(x)}x\notin\Gamma$ with $r(\sigma^{n_{j}(x)}x)=r_{j}(x)$, we
obtain $x_{n_{j}(x)+r_{j}(x)}>N_{r_{j}(x)}\geq N_{1}$. This together with
(2.7) yields $x_{n_{j}(x)+r_{j}(x)}\in\mathbb{N}^{*}$ as in (b). Since $x\in
E_{n}$, (c) holds. ∎

For each $j\in\\{1,\ldots,m\\}$, $n_{1}\cdots n_{j}\in\mathbb{N}_{0}^{j}$ with
$0\leq n_{1}<\cdots<n_{j}<n$ and $r_{1}\cdots r_{j}\in\mathbb{N}^{j}$, we
consider the set

(2.14) $\Delta_{n_{1}\cdots n_{j}}^{r_{1}\cdots r_{j}}=\\{x\in
E_{n}\cap\Sigma\colon(n_{i}(x),r_{i}(x))=(n_{i},r_{i})\ \text{ for every
}i\in\\{1,\ldots,j\\}\\}.$

######  Lemma 2.6.

If $\delta\in(0,1/5]$ is sufficiently small, then for all
$j\in\\{1,\ldots,m\\}$, $n_{1}\cdots n_{j}\in\mathbb{N}_{0}^{j}$ with $0\leq
n_{1}<\cdots<n_{j}<n$ and $r_{1}\cdots r_{j}\in\mathbb{N}^{j}$ we have

$\sum_{x\in\Delta_{n_{1}\cdots n_{j}}^{r_{1}\cdots r_{j}}}\exp
S_{n}\phi(x)\leq e^{\gamma_{0}n}\delta^{r_{1}+\cdots+r_{j}}.$

###### Proof.

We may assume $\Delta_{n_{1}\cdots n_{j}}^{r_{1}\cdots r_{j}}\neq\emptyset$
for otherwise there is nothing to prove. Starting with the case $j=1$, we
introduce two sets of induced words

$\begin{split}W_{0,1}&=\\{\mathbb{a}\in
W(\mathbb{A})\colon|\mathbb{a}|=n_{1}+r_{1}-1\\}\ \text{ and }\\\
W_{0,2}&=\\{\mathbb{b}\in W(\mathbb{A})\colon|\mathbb{b}|=n-n_{1}-r_{1}+1,\
\Pi[\\![\mathbb{b}]\\!]\subset\cup_{k=N_{r_{1}}+1}^{\infty}[k]\\}.\end{split}$

For each $x\in\Delta_{n_{1}}^{r_{1}}$ we have $x_{n+1}=x_{1}\in\mathbb{N}^{*}$
by the definition (2.14), and $x_{n_{1}+r_{1}}\in\mathbb{N}^{*}$ by Lemma
2.5(b). Hence there exist $\mathbb{a}\in W_{0,1}$ and $\mathbb{b}\in W_{0,2}$
such that $[\mathbb{a}]=[x_{1}\cdots x_{n_{1}+r_{1}-1}]$ and
$[\mathbb{b}]=[x_{n_{1}+r_{1}}\cdots x_{n}]$, and
$x\in\Pi[\\![\mathbb{a}\mathbb{b}]\\!]$. By (2.11) and (2.12) we have

$\exp S_{n}\phi(x)\ll
e^{\gamma_{0}n}\lambda_{\phi}[\\![\mathbb{a}\mathbb{b}]\\!]\ll
e^{\gamma_{0}n}\lambda_{\phi}[\\![\mathbb{a}]\\!]\lambda_{\phi}[\\![\mathbb{b}]\\!].$

Summing this inequality over all $x\in\Delta_{n_{1}}^{r_{1}}$, and then using
$\sum_{\mathbb{a}\in W_{0,1}}\lambda_{\phi}[\\![\mathbb{a}]\\!]\leq 1$ and
$\sum_{\mathbb{b}\in
W_{0,2}}\lambda_{\phi}[\\![\mathbb{b}]\\!]\leq\delta^{2r_{1}}$ which follow
from Lemma 2.1(b) and (2.8) respectively, we obtain

(2.15) $\begin{split}\sum_{x\in\Delta_{n_{1}}^{r_{1}}}\exp
S_{n}\phi(x)\ll&e^{\gamma_{0}n}\sum_{\mathbb{a}\in
W_{0,1}}\lambda_{\phi}[\\![\mathbb{a}]\\!]\sum_{\mathbb{b}\in
W_{0,2}}\lambda_{\phi}[\\![\mathbb{b}]\\!]\leq
e^{\gamma_{0}n}\delta^{2r_{1}}\leq e^{\gamma_{0}n}\delta^{r_{1}},\end{split}$

provided $\delta$ is small enough. In case $m=1$ we are done.

To proceed, suppose $m\geq 2$. let $j,j+1\in\\{1,\ldots,m\\}$ and let
$n_{1}\cdots n_{j}n_{j+1}\in\mathbb{N}_{0}^{j+1}$, $r_{1}\cdots
r_{j}r_{j+1}\in\mathbb{N}^{j+1}$ satisfy $\Delta_{n_{1}\cdots
n_{j}n_{j+1}}^{r_{1}\cdots r_{j}r_{j+1}}\neq\emptyset$. Define

$\begin{split}W_{j,1}&=\left\\{\mathbb{a}\in
W(\mathbb{A})\colon|\mathbb{a}|=n_{j+1}+r_{j+1}-1\right\\},\\\
W_{j,2}&=\\{\mathbb{b}\in
W(\mathbb{A})\colon|\mathbb{b}|=n-n_{j+1}-r_{j+1}+1,\
\Pi[\\![\mathbb{b}]\\!]\subset\cup_{k=N_{r_{j+1}}+1}^{\infty}[k]\\},\\\
W_{j,3}&=\\{\mathbb{c}\in W(\mathbb{A})\colon|\mathbb{c}|=n_{j}+r_{j},\
\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots n_{j+1}}^{r_{1}\cdots
r_{j+1}}\neq\emptyset\\},\\\ W_{j,4}&=\left\\{\mathbb{d}\in
W(\mathbb{A})\colon|\mathbb{d}|=n-n_{j}-r_{j}\right\\}.\end{split}$

Let $\mathbb{c}\in W_{j,3}$. For each $\mathbb{d}\in W_{j,4}$ we have
$|\mathbb{c}\mathbb{d}|=n$. Since $X$ is the full shift,
$\Pi[\\![\mathbb{c}\mathbb{d}]\\!]$ contains a unique element of
$E_{n}\cap\Sigma$ which we denote by $\overline{\mathbb{c}\mathbb{d}}$. We
have
$\overline{\mathbb{c}\mathbb{d}}\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots
n_{j}}^{r_{1}\cdots r_{j}}$, and by (2.11) and (2.12),

$\exp S_{n}\phi(\overline{\mathbb{c}\mathbb{d}})\gg
e^{\gamma_{0}n}\lambda_{\phi}[\\![\mathbb{c}]\\!]\lambda_{\phi}[\\![\mathbb{d}]\\!],$

which implies

(2.16) $\begin{split}\sum_{x\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots
n_{j}}^{r_{1}\cdots r_{j}}}\exp S_{n}\phi(x)&\gg
e^{\gamma_{0}n}\lambda_{\phi}[\\![\mathbb{c}]\\!]\sum_{\mathbb{d}\in
W_{j,4}}\lambda_{\phi}[\\![\mathbb{d}]\\!].\end{split}$

By Lemma 2.3, for each $\mathbb{d}\in W_{j,4}$ we have
$\lambda_{\phi}[\\![\mathbb{d}]\\!]=\lambda_{\phi}\circ\Pi^{-1}(\Sigma\cap[\mathbb{d}]).$
Since the sets $\Sigma\cap[\mathbb{d}]$, $\mathbb{d}\in W_{j,4}$ are pairwise
disjoint and their union equals $\Sigma$, we have

(2.17) $\sum_{\mathbb{d}\in W_{j,4}}\lambda_{\phi}[\\![\mathbb{d}]\\!]=1.$

Combining (2.16) and (2.17) yields

(2.18) $\begin{split}\sum_{x\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots
n_{j}}^{r_{1}\cdots r_{j}}}\exp S_{n}\phi(x)\gg
e^{\gamma_{0}n}\lambda_{\phi}[\\![\mathbb{c}]\\!].\end{split}$

Similarly to the case $j=1$, for each
$x\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots n_{j+1}}^{r_{1}\cdots
r_{j+1}}$ we have $x_{n+1}=x_{1}\in\mathbb{N}^{*}$ by the definition (2.14)
and $x_{n_{j+1}+r_{j+1}}\in\mathbb{N}^{*}$ by Lemma 2.5(b). Hence there exist
$\mathbb{a}\in W_{j,1}$, $\mathbb{b}\in W_{j,2}$ such that
$[\mathbb{a}]=[x_{1}\cdots x_{n_{j+1}+r_{j+1}-1}]$ and
$[\mathbb{b}]=[x_{n_{j+1}+r_{j+1}}\cdots x_{n}]$. We have
$[\\![\mathbb{a}]\\!]\subset[\\![\mathbb{c}]\\!]$ and
$x\in\Pi[\\![\mathbb{a}\mathbb{b}]\\!]$, and by (2.11) and (2.12),

$\exp S_{n}\phi(x)\ll
e^{\gamma_{0}n}\lambda_{\phi}[\\![\mathbb{a}]\\!]\lambda_{\phi}[\\![\mathbb{b}]\\!].$

Summing this inequality over all
$x\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots n_{j+1}}^{r_{1}\cdots
r_{j+1}}$, and then using $\sum_{\stackrel{{\scriptstyle\mathbb{a}\in
W_{j,1}}}{{[\\![\mathbb{a}]\\!]\subset[\\![\mathbb{c}]\\!]}}}\lambda_{\phi}[\\![\mathbb{a}]\\!]\leq\lambda_{\phi}[\\![\mathbb{c}]\\!]$
and $\sum_{\mathbb{b}\in
W_{j,2}}\lambda_{\phi}[\\![\mathbb{b}]\\!]\leq\delta^{2r_{j+1}}$ which follow
from Lemma 2.1(b) and (2.8) respectively, we obtain

(2.19) $\begin{split}\sum_{x\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots
n_{j+1}}^{r_{1}\cdots r_{j+1}}}\exp S_{n}\phi(x)&\ll
e^{\gamma_{0}n}\sum_{\stackrel{{\scriptstyle\mathbb{a}\in
W_{j,1}}}{{[\\![\mathbb{a}]\\!]\subset[\\![\mathbb{c}]\\!]}}}\lambda_{\phi}[\\![\mathbb{a}]\\!]\sum_{\mathbb{b}\in
W_{j,2}}\lambda_{\phi}[\\![\mathbb{b}]\\!]\\\ &\leq
e^{\gamma_{0}n}\lambda_{\phi}[\\![\mathbb{c}]\\!]\delta^{2r_{j+1}}.\end{split}$

Combining (2.18) and (2.19) yields

$\frac{\sum_{x\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots
n_{j+1}}^{r_{1}\cdots r_{j+1}}}\exp
S_{n}\phi(x)}{\sum_{x\in\Pi[\\![\mathbb{c}]\\!]\cap\Delta_{n_{1}\cdots
n_{j}}^{r_{1}\cdots r_{j}}}\exp S_{n}\phi(x)}\leq\delta^{r_{j+1}},$

provided $\delta$ is small enough. Rearranging this inequality and summing the
result over all $\mathbb{c}\in W_{j,3}$ yields

$\sum_{x\in\Delta_{n_{1}\cdots n_{j+1}}^{r_{1}\cdots r_{j+1}}}\exp
S_{n}\phi(x)\leq\delta^{r_{j+1}}\sum_{x\in\Delta_{n_{1}\cdots
n_{j}}^{r_{1}\cdots r_{j}}}\exp S_{n}\phi(x).$

Applying this inequality recursively and combining the final result with
(2.15) yields the desired inequality in Lemma 2.6.∎

Returning to the proof of Proposition 2.4, for two integers $R\geq m$ and
$s\in\\{1,\ldots,m\\}$ we denote by $K_{R,s}$ the cardinality of the set of
elements $(n_{1}\cdots n_{s},r_{1}\cdots r_{s})$ of
$\mathbb{N}_{0}^{s}\times\mathbb{N}^{s}$ with $0\leq n_{1}<\cdots<n_{s}<n$ and
$r_{1}+\cdots+r_{s}=R$. The number of ways of locating $n_{1},\ldots,n_{s}$ in
$[0,n]$ does not exceed $\left(\begin{smallmatrix}n\\\
s\end{smallmatrix}\right)$, and for each location $(n_{1},\ldots,n_{s})$ the
number of all feasible combinations of $(r_{1},\ldots,r_{s})$ with
$r_{1}+\cdots+r_{s}=R$ is bounded by the number of ways of dividing $R$
objects into $s$ groups, not exceeding $\left(\begin{smallmatrix}R+s-1\\\
s-1\end{smallmatrix}\right)\leq 2^{R+s-1}$. This yields
$K_{R,s}\leq\left(\begin{smallmatrix}n\\\
s\end{smallmatrix}\right)\left(\begin{smallmatrix}R+s-1\\\
s-1\end{smallmatrix}\right)\leq 2^{n}2^{R+s-1}$. From Lemma 2.5(a)(c), for
each $x\in E_{n}\cap\Sigma$ satisfying $V_{n}(x)(X\setminus\Gamma)=m/n$ there
exist $R\in\\{m,\ldots,2n\\}$, $s\in\\{1,\ldots,m\\}$, $(n_{1}\cdots
n_{s},r_{1}\cdots r_{s})\in K_{R,s}$ such that $x\in\Delta_{n_{1}\cdots
n_{s}}^{r_{1}\cdots r_{s}}$. If $\delta\in(0,1/5]$ is sufficiently small, then
together with Lemma 2.6 we obtain

$\begin{split}\sum_{\stackrel{{\scriptstyle x\in
E_{n}\cap\Sigma}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp
S_{n}\phi(x)&\leq\sum_{s=1}^{m}\sum_{R=m}^{2n}\sum_{(n_{1}\cdots
n_{s},r_{1}\cdots r_{s})\in K_{R,s}}\sum_{x\in\Delta_{n_{1}\cdots
n_{s}}^{r_{1}\cdots r_{s}}}\exp S_{n}\phi(x)\\\ &\leq
2^{n}e^{\gamma_{0}n}\sum_{s=1}^{m}\sum_{R=m}^{2n}2^{R+s-1}\delta^{R}\leq
2^{n}e^{\gamma_{0}n}\sum_{R=m}^{2n}(4\delta)^{R}\\\
&\leq\frac{2^{n}e^{\gamma_{0}n}(4\delta)^{m}}{1-4\delta}.\end{split}$

From this and (2.13), the desired inequality (2.9) follows and the proof of
Proposition 2.4 is complete. ∎

###  2.4. Verifying exponential tightness

We now use Proposition 2.4 to show the exponential tightness.

######  Proposition 2.7.

Let $\phi\colon X\to\mathbb{R}$ be acceptable such that $P(\phi)<\infty$ and
let $(\Sigma,\tau|_{\Sigma})$ be an induced system. If there exists a local
Gibbs state for the potential $\phi$ associated with
$(\Sigma,\tau|_{\Sigma})$, then $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ is
exponentially tight.

###### Proof.

The argument below is an adaptation of the proof of Sanov’s theorem (see e.g.,
[7]) to our setup. For each integer $\ell\geq 1$, we fix
$\delta_{\ell}\in(0,1/5]$ such that

(2.20)
$\frac{1}{1-4\delta_{\ell}}\sum_{m=1}^{\infty}e^{2\ell^{2}m}(4\delta_{\ell})^{m}\leq
1.$

We fix a non-decreasing integer sequence $\\{N_{i}\\}_{i=1}^{\infty}$ such
that (2.7) and (2.8) with $\delta=\delta_{\ell}$ hold. We define a compact
subset $\Gamma_{\ell}=\Gamma(\\{N_{i}\\}_{i=1}^{\infty})$ of $X$ by (2.10),
and set

$\mathcal{K}^{\ell}=\left\\{\nu\in\mathcal{M}(X)\colon\nu(\Gamma_{\ell})\geq
1-\frac{1}{\ell}\right\\}.$

Since $\mathcal{M}(X)$ is a Polish space and $\Gamma_{\ell}$ is a closed set,
the weak* convergence $\mu_{k}\to\mu$ for a sequence
$\\{\mu_{k}\\}_{k=1}^{\infty}$ in $\mathcal{K}^{\ell}$ implies
$\limsup_{k\to\infty}\mu_{k}(\Gamma_{\ell})\leq\mu(\Gamma_{\ell})$. Hence,
$\mathcal{K}^{\ell}$ is a closed set. For an integer $L\geq 1$ we define

$\mathcal{K}_{L}=\bigcap_{\ell=L}^{\infty}\mathcal{K}^{\ell}.$

This set is tight, and by Prohorov’s theorem any sequence in $\mathcal{K}_{L}$
has a limit point. Hence $\mathcal{K}_{L}$ is sequentially compact. Since the
weak* topology is metrizable with the bounded Lipschitz metric [7],
$\mathcal{K}_{L}$ is compact. By Chebyshev’s inequality, for $n\geq 1$ we have

$\begin{split}\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{\exp\left(\ell^{2}nV_{n}(x)(X\setminus\Gamma_{\ell})\right)\geq
e^{\ell n}}}}\exp S_{n}\phi(x)\leq&e^{-2\ell n}\times\\\
\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)(X\setminus\Gamma_{\ell})\geq
1/n}}}&\exp\left(2\ell^{2}nV_{n}(x)(X\setminus\Gamma_{\ell})\right)\exp
S_{n}\phi(x)\\\ =&e^{-2\ell
n}\sum_{m=1}^{n}e^{2\ell^{2}m}\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)(X\setminus\Gamma)=m/n}}}\exp S_{n}\phi(x).\end{split}$

Combining this inequality with Proposition 2.4 and (2.20), we have

$\begin{split}\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)\notin\mathcal{K}^{\ell}}}}\exp
S_{n}\phi(x)&=\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)(X\setminus\Gamma_{\ell})\geq 1/\ell}}}\exp
S_{n}\phi(x)=\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{\exp\left(\ell^{2}nV_{n}(x)(X\setminus\Gamma_{\ell})\right)\geq
e^{\ell n}}}}\exp S_{n}\phi(x)\\\
&\leq\frac{2^{n}ne^{\gamma_{0}n}}{1-4\delta_{\ell}}e^{-2\ell
n}\sum_{m=1}^{n}e^{2\ell^{2}m}(4\delta_{\ell})^{m}\leq
2^{n}ne^{\gamma_{0}n}e^{-2\ell n}.\end{split}$

If $L\geq 1$ is large enough, then

$\begin{split}\tilde{\mu}_{n}(\mathcal{M}(X)\setminus\mathcal{K}_{L})&\leq\sum_{\ell=L}^{\infty}\tilde{\mu}_{n}(\mathcal{M}(X)\setminus\mathcal{K}^{\ell})=\sum_{\ell=L}^{\infty}\frac{1}{Z_{n}(\phi)}\sum_{\stackrel{{\scriptstyle
x\in E_{n}}}{{V_{n}(x)\notin\mathcal{K}^{\ell}}}}\exp S_{n}\phi(x)\\\
&\leq\frac{2^{n}ne^{\gamma_{0}n}}{Z_{n}(\phi)}\sum_{\ell=L}^{\infty}e^{-2\ell
n}\leq\frac{2^{n}e^{\gamma_{0}n}}{Z_{n}(\phi)}e^{-Ln}.\end{split}$

Combining this with the equality $\lim_{n\to\infty}(1/n)\log
Z_{n}(\phi)=P(\phi)<\infty$ which follows from the acceptability of $\phi$, we
obtain

$\limsup_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{M}(X)\setminus\mathcal{K}_{L})\leq-L+\gamma_{0}+\log
2.$

Since $L\geq 1$ can be made arbitrarily large, the exponential tightness of
$\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ follows. ∎

###  2.5. Existence of a local Gibbs state

The next proposition ensures the existence of a local Gibbs state under the
assumption of Theorem A.

######  Proposition 2.8.

Let $\phi\colon X\to\mathbb{R}$, and let $(\Sigma,\tau|_{\Sigma})$ be an
induced system for which the induced potentials $\Phi_{\gamma}$,
$\gamma\in\mathbb{R}$ associated with $\phi$ are locally Hölder continuous,
and there exists $\gamma_{0}\in\mathbb{R}$ such that $P(\Phi_{\gamma_{0}})=0$.
There exists a $\theta$-invariant local Gibbs state for the potential $\phi$
associated with $(\Sigma,\tau|_{\Sigma})$.

###### Proof.

Note that $P(\Phi_{\gamma_{0}}\circ\Pi)=P(\Phi_{\gamma_{0}})=0$. Since
$\mathbb{A}^{\mathbb{N}}$ is the countable full shift, the finiteness of
$P(\Phi_{\gamma_{0}}\circ\Pi)$ implies the summability of the potential
$\Phi_{\gamma_{0}}\circ\Pi$. By [19, Corollary 2.7.5] together with the
summability and the local Hölder continuity of $\Phi_{\gamma_{0}}\circ\Pi$,
there exists a unique $\theta$-invariant Bowen’s Gibbs state for the potential
$\Phi_{\gamma_{0}}\circ\Pi$, which we denote by $\lambda_{\phi}$. There exists
$C\geq 1$ such that for every $m\geq 1$, any $\mathbb{a}\in{\mathbb{A}}^{m}$
and any $z\in[\\![\mathbb{a}]\\!]$ we have

$\begin{split}C^{-1}\leq\frac{\lambda_{\phi}[\\![\mathbb{a}]\\!]}{\exp\left(-P(\Phi_{\gamma_{0}}\circ\Pi)m+\sum_{k=0}^{m-1}\Phi_{\gamma_{0}}\circ\Pi(\theta^{k}z)\right)}&\leq
C.\end{split}$

For the series in the denominator of the fraction, for
$x\in\Pi[\\![\mathbb{a}]\\!]$ we have

$\begin{split}\sum_{k=0}^{m-1}\Phi_{\gamma_{0}}\circ\Pi(\theta^{k}\Pi^{-1}(x))&=S_{\sum_{k=0}^{m-1}R(\tau^{k}x)}\phi(x)-\gamma_{0}\sum_{k=0}^{m-1}R(\tau^{k}x)\\\
&=S_{|\mathbb{a}|}\phi(x)-\gamma_{0}|\mathbb{a}|.\end{split}$

Substituting this and $P(\Phi_{\gamma_{0}}\circ\Pi)=0$ into the denominator of
the fraction implies that $\lambda_{\phi}$ is a local Gibbs state for the
potential $\phi$. ∎

######  Remark 2.9.

Under the assumption and notation of Proposition 2.8 and its proof, if
$\gamma_{0}=P(\phi)$ and $\int R{\rm d}(\lambda_{\phi}\circ\Pi^{-1})<\infty$,
then the measure

$\frac{1}{\int R{\rm
d}(\lambda_{\phi}\circ\Pi^{-1})}\sum_{n=0}^{\infty}(\lambda_{\phi}\circ\Pi^{-1})|_{\\{R>n\\}}\circ\sigma^{-n}$

is in $\mathcal{M}_{\phi}(X,\sigma)$, and it is an equilibrium state for the
potential $\phi$.

##  3\. Proofs of the main results

In this section we complete the proofs of all the theorems. In Sections 3.1
and 3.2, we prove lower and upper bounds for certain fundamental open and
closed subsets of $\mathcal{M}(X)$ respectively. In Section 3.3, we combine
these bounds and the exponential tightness verified in Section 2 to complete
the proof of Theorem A. In Section 3.4 we complete the proof of Theorem B. In
view of applications, in Section 3.5 we give a sufficient condition for the
vanishing of the pressure of the induced potential that is assumed in Theorem
A. Using this condition, we complete the proof of Theorem C in Section 3.6. In
Section 3.7 we mention some generalizations.

###  3.1. Lower bound for fundamental open sets

We introduce notations in this and the next two subsections. Let $C_{u}(X)$
denote the set of real-valued bounded uniformly continuous functions on $X$.
For an integer $\ell\geq 1$ we define

$C_{u}(X)^{\ell}=\\{\vec{\varphi}=(\varphi_{1},\ldots,\varphi_{\ell})\colon\varphi_{j}\in
C_{u}(X)\text{ for every }j\in\\{1,\ldots,\ell\\}\\}.$

For $\vec{\varphi}=(\varphi_{1},\ldots,\varphi_{\ell})\in C_{u}(X)^{\ell}$,
$\vec{\alpha}=(\alpha_{1},\ldots,\alpha_{\ell})\in\mathbb{R}^{\ell}$ and
$\mu\in\mathcal{M}(X)$, the expression $\int\vec{\varphi}{\rm
d}\mu>\vec{\alpha}$ indicates that $\int\varphi_{j}{\rm d}\mu>\alpha_{j}$
holds for all $j\in\\{1,\ldots,\ell\\}$. The meaning of $\int\vec{\varphi}{\rm
d}\mu\geq\vec{\alpha}$ is analogous. Put $\|\vec{\alpha}\|=\max_{1\leq
j\leq\ell}|\alpha_{j}|$. For $\varepsilon\in\mathbb{R}$ we write
$\vec{\varepsilon}=(\varepsilon,\ldots,\varepsilon)\in\mathbb{R}^{\ell}.$ For
$n\geq 1$ and $p_{1}\cdots p_{n}\in\mathbb{N}^{n}$, let $\overline{p_{1}\cdots
p_{n}}$ denote the element of $E_{n}$ that is contained in $[p_{1}\cdots
p_{n}]\subset X$.

######  Proposition 3.1.

Let $\phi\colon X\to\mathbb{R}$ be acceptable and satisfy $P(\phi)<\infty$.
Let $\ell\geq 1$, $\vec{\varphi}\in C_{u}(X)^{\ell}$ and
$\vec{\alpha}\in\mathbb{R}^{\ell}$. Let $\mathcal{G}\subset\mathcal{M}(X)$ be
an open set of the form

$\mathcal{G}=\left\\{\mu\in\mathcal{M}(X)\colon\int\vec{\varphi}{\rm
d}\mu>\vec{\alpha}\right\\}.$

For any measure $\mu\in\mathcal{M}_{\phi}(X,\sigma)\cap\mathcal{G}$, we have

$\liminf_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{G})\geq
F_{\phi}(\mu).$

###### Proof.

By virtue of the definition of the pressure $P(\phi)$, it suffices to show
that

(3.1) $\liminf_{n\to\infty}\frac{1}{n}\log\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)\in\mathcal{G}}}}\exp S_{n}\phi(x)\geq h(\mu)+\int\phi{\rm
d}\mu.$

The proof of [37, Main Theorem] works verbatim to show the next lemma that
approximates non-ergodic measures with ergodic ones in a particular sense.

######  Lemma 3.2.

For any $\mu\in\mathcal{M}_{\phi}(X,\sigma)$ and any $\varepsilon>0$, there
exists an ergodic measure $\mu^{\prime}\in\mathcal{M}_{\phi}(X,\sigma)$ which
is supported on a compact set and satisfies

$|h(\mu)-h(\mu^{\prime})|<\varepsilon,\ \left\|\int\vec{\varphi}{\rm
d}\mu-\int\vec{\varphi}{\rm d}\mu^{\prime}\right\|<\varepsilon\ \text{ and }\
\left|\int\phi{\rm d}\mu-\int\phi{\rm d}\mu^{\prime}\right|<\varepsilon.$

By Lemma 3.2, it suffices to show (3.1) for all
$\mu\in\mathcal{M}_{\phi}(X,\sigma)$ which is ergodic. Let $\varepsilon>0$ be
such that

(3.2) $\int\vec{\varphi}{\rm d}\mu>\vec{\alpha}+\vec{\varepsilon}.$

Since each component of $\vec{\varphi}$ is bounded uniformly continuous and
$\phi$ is acceptable, one can use Birkhoff’s ergodic theorem and Shannon-
McMillan-Breiman’s theorem to show that for any sufficiently large $n\geq 1$
there is a finite subset $G_{n}$ of $\mathbb{N}^{n}$ such that

(3.3) $\left|\frac{1}{n}\log\\#G_{n}-h(\mu)\right|<\frac{\varepsilon}{2},$

and for every $p_{1}\cdots p_{n}\in G_{n}$,

(3.4) $\sup_{x\in[p_{1}\cdots p_{n}]}\left\|\int\vec{\varphi}{\rm
d}V_{n}(x)-\int\vec{\varphi}{\rm d}\mu\right\|<\frac{\varepsilon}{2}\ \text{
and}$ (3.5) $\sup_{x\in[p_{1}\cdots
p_{n}]}\left|\frac{1}{n}S_{n}\phi(x)-\int\phi{\rm
d}\mu\right|<\frac{\varepsilon}{2}.$

Then (3.2) and (3.4) yield $\int\vec{\varphi}{\rm
d}V_{n}(\overline{p_{1}\cdots p_{n}})>\vec{\alpha}$, and (3.5) yields
$(1/n)S_{n}\phi(\overline{p_{1}\cdots p_{n}})>\int\phi{\rm
d}\mu-\varepsilon/2.$ Therefore

$\begin{split}\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)\in\mathcal{G}}}}\exp S_{n}\phi(x)&\geq\sum_{p_{1}\cdots
p_{n}\in G_{n}}\exp S_{n}\phi(\overline{p_{1}\cdots
p_{n}})\geq\\#G_{n}\exp\left(n\int\phi{\rm d}\mu-\frac{\varepsilon
n}{2}\right).\end{split}$

Taking logarithms, dividing by a sufficiently large $n$ and using (3.3) we
have

$\begin{split}\frac{1}{n}\log\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)\in\mathcal{G}}}}\exp
S_{n}\phi(x)&\geq\frac{1}{n}\log\\#G_{n}+\int\phi{\rm
d}\mu-\frac{\varepsilon}{2}>h(\mu)+\int\phi{\rm d}\mu-\varepsilon.\end{split}$

Letting $n\to\infty$ and then $\varepsilon\to 0$ yields (3.1). The proof of
Proposition 3.1 is complete. ∎

###  3.2. Upper bound for fundamental closed sets

We proceed to upper bounds on fundamental closed sets which are not
necessarily compact.

######  Proposition 3.3.

Let $\phi\colon X\to\mathbb{R}$ be acceptable and satisfy $P(\phi)<\infty$.
Let $\ell\geq 1$, $\vec{\varphi}\in C_{u}(X)^{\ell}$,
$\vec{\alpha}\in\mathbb{R}^{\ell}$ and let $\mathcal{C}\subset\mathcal{M}(X)$
be a non-empty closed set of the form

$\mathcal{C}=\left\\{\mu\in\mathcal{M}(X)\colon\int\vec{\varphi}{\rm
d}\mu\geq\vec{\alpha}\right\\}.$

For any $\varepsilon>0$ there exists $\mu\in\mathcal{M}_{\phi}(X,\sigma)$ such
that $\int\vec{\varphi}{\rm d}\mu>\vec{\alpha}-\vec{\varepsilon}$ and

$\limsup_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{C})\leq
F_{\phi}(\mu).$

###### Proof.

A main ingredient is the next lemma, the proof of which is analogous to the
standard proof of the variational principle [41]. For $n\geq 1$ we put

$D_{n}(\phi)=\sup_{p_{1}\cdots p_{n}\in\mathbb{N}^{n}}\sup_{x,y\in[p_{1}\cdots
p_{n}]}S_{n}\phi(x)-S_{n}\phi(y).$

######  Lemma 3.4.

For any $\varepsilon>0$ there exists $n_{0}\geq 1$ such that if $n\geq n_{0}$
then for any non-empty finite subset $H_{n}$ of $\mathbb{N}^{n}$ satisfying
$V_{n}(\overline{p_{1}\cdots p_{n}})\in\mathcal{C}$ for all $p_{1}\cdots
p_{n}\in H_{n}$, there exists a measure
$\mu_{0}\in\mathcal{M}_{\phi}(X,\sigma)$ such that

$\log\sum_{p_{1}\cdots p_{n}\in H_{n}}\sup_{[p_{1}\cdots p_{n}]}\exp
S_{n}\phi\leq\left(h(\mu_{0})+\int\phi{\rm
d}\mu_{0}\right)n+D_{n}(\phi)\quad\text{and}\quad\int\vec{\varphi}{\rm
d}\mu_{0}>\vec{\alpha}-\vec{\varepsilon}.$

###### Proof.

Since all components of $\vec{\varphi}$ are bounded uniformly continuous
functions on $X$, for any $\varepsilon>0$ there exists $n_{0}\geq 1$ such that
if $n\geq n_{0}$ then for any $p_{1}\cdots p_{n}\in\mathbb{N}^{n}$ satisfying
$V_{n}(\overline{p_{1}\cdots p_{n}})\in\mathcal{C}$, $\int\vec{\varphi}{\rm
d}V_{n}(x)\geq\vec{\alpha}-(1/2)\vec{\varepsilon}$ holds for any
$x\in[p_{1}\cdots p_{n}]$. In what follows we assume $n\geq n_{0}$.

We set

$\Lambda=\bigcap_{k=0}^{\infty}\sigma^{-nk}\left(\bigcup_{p_{1}\cdots p_{n}\in
H_{n}}[p_{1}\cdots p_{n}]\right).$

Then $\sigma^{n}|_{\Lambda}\colon\Lambda\to\Lambda$ is topologically conjugate
to the left shift acting on the finite full shift space

$H_{n}^{\mathbb{N}}=\\{(q_{m})_{m=1}^{\infty}\colon q_{m}\in H_{n}\text{ for
every }m\geq 1\\}.$

Since the function $\hat{\phi}=S_{n}\phi$ induces a continuous potential on
$H_{n}^{\mathbb{N}}$, the variational principle [41] yields

$\begin{split}\sup_{\hat{\mu}\in\mathcal{M}(\Lambda,\sigma^{n}|_{\Lambda})}\left(h(\hat{\mu})+\int\hat{\phi}{\rm
d}\hat{\mu}\right)&=\lim_{m\to\infty}\frac{1}{m}\log\sum_{q_{1}\cdots q_{m}\in
H_{n}^{m}}\sup_{[q_{1}\cdots
q_{m}]}\left(\exp{\sum_{k=0}^{m-1}\hat{\phi}\circ\sigma^{nk}}\right),\end{split}$

where $\mathcal{M}(\Lambda,\sigma^{n}|_{\Lambda})$ denotes the space of
$\sigma^{n}|_{\Lambda}$-invariant Borel probability measures endowed with the
weak* topology, and $h(\hat{\mu})$ denotes the measure-theoretic entropy of
$\hat{\mu}\in\mathcal{M}(\Lambda,\sigma^{n}|_{\Lambda})$ with respect to
$\sigma^{n}|_{\Lambda}$. For the right-hand side, we have

$\begin{split}\sum_{q_{1}\cdots q_{m}\in H_{n}^{m}}\sup_{[q_{1}\cdots
q_{m}]}\exp\left({\sum_{k=0}^{m-1}\hat{\phi}\circ\sigma^{nk}}\right)&\geq\left(\sum_{p_{1}\cdots
p_{n}\in H_{n}}\inf_{[p_{1}\cdots p_{n}]}\exp S_{n}\phi\right)^{m}\\\
&\geq\left(\exp(-D_{n}(\phi))\sum_{p_{1}\cdots p_{n}\in
H_{n}}\sup_{[p_{1}\cdots p_{n}]}\exp S_{n}\phi\right)^{m}.\end{split}$

Taking logarithms of both sides, dividing by $m$ and then letting $m\to\infty$
gives

$\lim_{m\to\infty}\frac{1}{m}\log\sum_{q_{1}\cdots q_{m}\in
H_{n}^{m}}\sup_{[q_{1}\cdots
q_{m}]}\exp\left(\sum_{k=0}^{m-1}\hat{\phi}\circ\sigma^{nk}\right)\geq\log\sum_{p_{1}\cdots
p_{n}\in H_{n}}\sup_{[p_{1}\cdots p_{n}]}\exp S_{n}\phi-D_{n}(\phi).$

Plugging this into the previous inequality yields

$\sup_{\hat{\mu}\in\mathcal{M}(\Lambda,\sigma^{n}|_{\Lambda})}\left(h(\hat{\mu})+\int\hat{\phi}{\rm
d}\hat{\mu}\right)\geq\log\sum_{p_{1}\cdots p_{n}\in H_{n}}\sup_{[p_{1}\cdots
p_{n}]}\exp S_{n}\phi-D_{n}(\phi).$

By the compactness of the space $\mathcal{M}(\Lambda,\sigma^{n}|_{\Lambda})$
and the upper semicontinuity of the map $\hat{\mu}\mapsto
h(\hat{\mu})+\int\hat{\phi}{\rm d}\hat{\mu}$ on this space, the supremum is
attained, say by $\hat{\mu}_{0}$. The measure
$\mu_{0}=(1/n)\sum_{j=0}^{n-1}\hat{\mu}_{0}\circ\sigma^{-j}$ is in
$\mathcal{M}_{\phi}(X,\sigma)$ and satisfies

$\left(h(\mu_{0})+\int\phi{\rm
d}\mu_{0}\right)n=\sup_{\hat{\mu}\in\mathcal{M}(\Lambda,\sigma^{n}|_{\Lambda})}\left(h(\hat{\mu})+\int\hat{\phi}{\rm
d}\hat{\mu}\right).$

Since the support of $\mu_{0}$ is contained in set $\\{x\in
X\colon\int\vec{\varphi}{\rm d}V_{n}(x)>\vec{\alpha}-\vec{\varepsilon}/2\\}$
by the choice of $n_{0}$ and the assumption $n\geq n_{0}$, we obtain
$\int\vec{\varphi}{\rm d}\mu_{0}>\vec{\alpha}-\vec{\varepsilon}$ as required.
∎

Continuing the proof of Proposition 3.3, we note that the acceptability of
$\phi$ and $P(\phi)<\infty$ implies $Z_{n}(\phi)<\infty$ for every $n\geq 1$.
Hence it is possible to choose a finite subset $H_{n}$ of the countable set
$\left\\{p_{1}\cdots p_{n}\in\mathbb{N}^{n}\colon V_{n}(\overline{p_{1}\cdots
p_{n}})\in\mathcal{C}\right\\}$ such that

$\sum_{\stackrel{{\scriptstyle p_{1}\cdots
p_{n}\in\mathbb{N}^{n}}}{{V_{n}(\overline{p_{1}\cdots
p_{n}})\in\mathcal{C}}}}\exp S_{n}\phi(\overline{p_{1}\cdots p_{n}})\leq
2\sum_{p_{1}\cdots p_{n}\in H_{n}}\exp S_{n}\phi(\overline{p_{1}\cdots
p_{n}}).$

By this inequality and Lemma 3.4, there exists
$\mu_{0}\in\mathcal{M}_{\phi}(X,\sigma)$ such that $\int\vec{\varphi}{\rm
d}\mu_{0}>\vec{\alpha}-\vec{\varepsilon}$ and

$\begin{split}\log\sum_{\stackrel{{\scriptstyle x\in
E_{n}}}{{V_{n}(x)\in\mathcal{C}}}}\exp
S_{n}\phi(x)&=\log\sum_{\stackrel{{\scriptstyle p_{1}\cdots
p_{n}\in\mathbb{N}^{n}}}{{V_{n}(\overline{p_{1}\cdots
p_{n}})\in\mathcal{C}}}}\exp S_{n}\phi(\overline{p_{1}\cdots p_{n}})\\\
&\leq\log\sum_{p_{1}\cdots p_{n}\in H_{n}}\exp S_{n}\phi(\overline{p_{1}\cdots
p_{n}})+\log 2\\\ &\leq\log\sum_{p_{1}\cdots p_{n}\in H_{n}}\sup_{[p_{1}\cdots
p_{n}]}\exp S_{n}\phi+\log 2\\\ &\leq\left(h(\mu_{0})+\int\phi{\rm
d}\mu_{0}\right)n+D_{n}(\phi)+\log 2.\end{split}$

Since $\phi$ is acceptable, we have $D_{n}(\phi)=o(n)$ $(n\to\infty)$.
Dividing both sides of the last inequality by $n$, letting $n\to\infty$ and
combining the result with $P(\phi)=\lim_{n\to\infty}(1/n)\log Z_{n}(\phi)$
yields the desired inequality. ∎

###  3.3. Proof of Theorem A

Let $\phi\colon X\to\mathbb{R}$ be acceptable and satisfy $P(\phi)<\infty$.
Suppose there exists an induced system for which the induced potentials
$\Phi_{\gamma}$, $\gamma\in\mathbb{R}$ associated with $\phi$ are locally
Hölder continuous, and there exists $\gamma_{0}\in\mathbb{R}$ such that
$P(\Phi_{\gamma_{0}})=0$.

Let $\mathcal{G}$ be a non-empty open subset of $\mathcal{M}(X)$. Since
subsets of $\mathcal{M}(X)$ of the form
$\left\\{\mu\in\mathcal{M}(X)\colon\int\vec{\varphi}{\rm
d}\mu>\vec{\alpha}\right\\}$ with $\ell\geq 1$, $\vec{\varphi}\in
C_{u}(X)^{\ell}$, $\vec{\alpha}\in\mathbb{R}^{\ell}$ constitute a base of the
weak* topology of $\mathcal{M}(X)$, $\mathcal{G}$ is written as the union
$\mathcal{G}=\bigcup_{\lambda}\mathcal{G}_{\lambda}$ of sets of this form. For
each $\mathcal{G}_{\lambda}$, Proposition 3.1 gives

$\liminf_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{G}_{\lambda})\geq\sup_{\mathcal{G}_{\lambda}}F_{\phi},$

and hence

$\liminf_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{G})\geq\sup_{\lambda}\sup_{\mathcal{G}_{\lambda}}F_{\phi}=\sup_{\mathcal{G}}F_{\phi}=-\inf_{\mathcal{G}}I_{\phi},$

as required in (1.1).

Let $\mathcal{C}$ be a compact closed subset of $\mathcal{M}(X)$. Let
$\mathcal{G}$ be an arbitrary open set containing $\mathcal{C}$. Since
$\mathcal{M}(X)$ is metrizable by the bounded Lipschitz metric and
$\mathcal{C}$ is compact, we can choose $\varepsilon>0$ and finitely many
closed sets $\mathcal{C}_{1},\ldots,\mathcal{C}_{s}$ of the form
$\mathcal{C}_{k}=\left\\{\mu\in\mathcal{M}(X)\colon\int\vec{\varphi}_{k}{\rm
d}\mu\geq\vec{\alpha}_{k}\right\\}$ with $\ell_{k}\geq 1$,
$\vec{\varphi}_{k}\in C_{u}(X)^{\ell_{k}}$,
$\vec{\alpha}_{k}\in\mathbb{R}^{\ell_{k}}$, so that
$\mathcal{C}\subset\bigcup_{k=1}^{s}\mathcal{C}_{k}\subset\bigcup_{k=1}^{s}\mathcal{C}_{k}(\varepsilon)\subset\mathcal{G}$
where
$\mathcal{C}_{k}(\varepsilon)=\\{\mu\in\mathcal{M}(X)\colon\int\vec{\varphi}_{k}{\rm
d}\mu>\vec{\alpha}_{k}-\vec{\varepsilon}\\}$. By Lemma 3.4 and $F_{\phi}\leq-
I_{\phi}$, for $1\leq k\leq s$ we have

$\limsup_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{C}_{k})\leq-\inf_{\mathcal{C}_{k}(\varepsilon)}I_{\phi}+\varepsilon.$

Then we have

$\limsup_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{C})\leq\max_{1\leq
k\leq
s}\left(-\inf_{\mathcal{C}_{k}(\varepsilon)}I_{\phi}\right)+\varepsilon\leq-\inf_{\mathcal{G}}I_{\phi}+\varepsilon.$

Since $\varepsilon>0$ is arbitrary and $\mathcal{G}$ is an arbitrary open set
containing $\mathcal{C}$, it follows that

$\limsup_{n\to\infty}\frac{1}{n}\log\tilde{\mu}_{n}(\mathcal{C})\leq\inf_{\mathcal{G}\supset\mathcal{C}}\left(-\inf_{\mathcal{G}}I_{\phi}\right)=-\inf_{\mathcal{C}}I_{\phi},$

as required in (1.2). The last equality is due to the lower semicontinuity of
$I_{\phi}$.

Since $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ is exponentially tight by
Proposition 2.7, by the standard arguments as in [7], the upper bound (1.2)
holds for any non-compact closed subset of $\mathcal{M}(X)$, and $I_{\phi}$ is
a good rate function. This completes the proof of Theorem A.∎

###  3.4. Proof of Theorem B

Let $\phi\colon X\to\mathbb{R}$ be acceptable and satisfy $P(\phi)<\infty$.
Suppose there exists an induced system for which the associated induced
potentials $\Phi_{\gamma}$, $\gamma\in\mathbb{R}$ are locally Hölder
continuous, and there exists $\gamma_{0}\in\mathbb{R}$ such that
$P(\Phi_{\gamma_{0}})=0$. Assume that the minimizer of the rate function
$I_{\phi}$ in (1.9) is unique, denoted by $\mu_{\rm min}$. Let
$\\{\tilde{\mu}_{n(j)}\\}_{j=1}^{\infty}$ be an arbitrary convergent
subsequence of $\\{\tilde{\mu}_{n}\\}_{n=1}^{\infty}$ with the limit measure
$\tilde{\mu}$. It suffices to show that $\tilde{\mu}$ is the unit point mass
at $\mu_{\min}$.

We fix a metric that generates the weak* topology on $\mathcal{M}(X)$. Since
the rate function $I_{\phi}$ in (1.9) is the good rate function by Theorem A,
for any $\alpha>0$ the level set

$\mathcal{L}(\alpha)=\\{\mu\in\mathcal{M}(X)\colon I_{\phi}(\mu)\leq\alpha\\}$

is a compact set. Let $\mu\in\mathcal{M}(X)\setminus\\{\mu_{\rm min}\\}$. By
the lower semicontinuity of the rate function and $I_{\phi}(\mu)>0$, there
exists $r>0$ such that the closure of the open ball $\mathcal{B}(r;\mu)$ of
radius $r$ about $\mu$ in $\mathcal{M}(X)$ does not intersect the level set
$\mathcal{L}(I_{\phi}(\mu)/2)$. The weak* convergence
$\tilde{\mu}_{n(j)}\to\tilde{\mu}$ gives

$\tilde{\mu}(\mathcal{B}(r;\mu))\leq\liminf_{j\to\infty}\tilde{\mu}_{n(j)}(\mathcal{B}(r;\mu))\leq\limsup_{j\to\infty}\tilde{\mu}_{n(j)}(\overline{\mathcal{B}(r;\mu)}).$

By this and the large deviations upper bound for closed sets (1.2), we have

$\tilde{\mu}(\mathcal{B}(r;\mu))\leq\limsup_{j\to\infty}\exp\left(-n(j)\inf_{\overline{\mathcal{B}(r;\mu)}}I_{\phi}\right)\leq\limsup_{j\to\infty}\exp\left(-\frac{n(j)}{2}I_{\phi}(\mu)\right)=0.$

Hence, the support of $\tilde{\mu}$ does not contain $\mu$. Since $\mu$ is an
arbitrary element of $\mathcal{M}(X)\setminus\\{\mu_{\rm min}\\}$, it follows
that $\tilde{\mu}$ is the unit point mass at $\mu_{\min}$. This completes the
proof of Theorem B. ∎

###  3.5. Sufficient condition for vanishing of pressure

A direct check of the condition $P(\Phi_{\gamma_{0}})=0$ in Theorem A may be
cumbersome, while checking the finiteness of induced pressures is considered
to be easier. In view of applications, we give a sufficient condition for the
second assumption in Theorem A on the induced potential.

######  Lemma 3.5.

Let $\phi\colon X\to\mathbb{R}$, let $(\Sigma,\tau|_{\Sigma})$ be an induced
system and let $\Phi_{\gamma}\colon\Sigma\to\mathbb{R}$
$(\gamma\in\mathbb{R})$ be the associated family of induced potentials. If
$\Phi_{\gamma}$, $\gamma\in\mathbb{R}$ are acceptable and there exists
$\delta\in\mathbb{R}$ such that $0<P(\Phi_{\delta})<\infty$, then there exists
$\gamma_{0}\in\mathbb{R}$ such that $P(\Phi_{\gamma_{0}})=0$.

###### Proof.

Let $n\geq 1$. Since $|\mathbb{a}|\geq n$ for any
$\mathbb{a}\in\mathbb{A}^{n}$ for any $\gamma>\delta$ we have

$\sum_{\mathbb{a}\in\mathbb{A}^{n}}\sup_{[\mathbb{a}]}\exp(S_{|\mathbb{a}|}\phi-\gamma|\mathbb{a}|)\leq\exp(-(\gamma-\delta)n)\sum_{\mathbb{a}\in\mathbb{A}^{n}}\sup_{[\mathbb{a}]}\exp(S_{|\mathbb{a}|}\phi-\delta|\mathbb{a}|).$

Taking logarithms, dividing by $n$ and letting $n\to\infty$ yields
$P(\Phi_{\gamma})\leq-\gamma+\delta+P(\Phi_{\delta})$. Hence, the function
$\gamma\in[\delta,\infty)\mapsto P(\Phi_{\gamma})$ is strictly decreasing. The
variational principle [19, Theorem 2.1.8] implies that this function is
continuous. Hence there exists $\gamma_{0}>\delta$ such that
$P(\Phi_{\gamma_{0}})=0$. ∎

###  3.6. Proof of Theorem C

Recall that the Rényi map $T$ in (1.10) is modeled on the countable full shift
$X$ via the coding map $\pi$ in (1.12). As in Remark 2.2, we set $p^{*}=2$ and
define an inducing scheme $(X^{*},R)$ by (1.4), (1.5), and the induced system
$(\Sigma,\tau|_{\Sigma})$ by (1.6), (1.7). We also define the infinite
alphabet $\mathbb{A}$ and the coding map $\Pi$ by (2.1), (2.2) respectively,
keeping the notation in Section 2.1. We have

$\Sigma=\left[\frac{1}{2},1\right)\cap\mathbb{I}\ \text{ and }\
\mathbb{A}=\left\\{\bigcup_{q=2}^{\infty}[p1^{n-1}q]\colon p\geq 2\text{ and
}n\geq 1\right\\}.$

Recall that $\mathbb{I}$ denotes the set of irrational numbers in $(0,1)$. For
a bounded interval $J\subset\mathbb{R}$, let $|J|$ denote its Euclidean
length. Various positive constants which depend only on $T$ will be simply
denoted by $C$.

For each $\mathbb{a}\in\mathbb{A}$ with
$\mathbb{a}=\bigcup_{q=2}^{\infty}[p1^{n-1}q]$, we put

$J(\mathbb{a})=T^{-1}\left(\left[\frac{1}{|\mathbb{a}|+1},\frac{1}{|\mathbb{a}|}\right)\right)\cap
J_{p}.$

Then $R$ equals $|\mathbb{a}|$ on the set
$\Pi[\\![\mathbb{a}]\\!]=\pi^{-1}(J(\mathbb{a}))$, where $\pi$ denotes the
coding map in (1.12). There is $C\geq 1$ such that for any $\mathbb{a}\in
W(\mathbb{A})$ we have

(3.6) $C^{-1}\leq|J(\mathbb{a})|\cdot|\mathbb{a}|^{2}\leq C.$

Define an induced map
$U\colon\bigcup_{\mathbb{a}\in\mathbb{A}}J(\mathbb{a})\to[0,1)$ by
$U|_{J(\mathbb{a})}=T^{|\mathbb{a}|}|_{J(\mathbb{a})}$. Recall that
$\phi=-\log|T^{\prime}\circ\pi|$. For $\beta,\gamma\in\mathbb{R}$ define
$\Phi_{\beta,\gamma}\colon\Sigma\to\mathbb{R}$ by

$\Phi_{\beta,\gamma}(x)=\beta S_{R(x)}\phi(x)-\gamma R(x),$

which is the induced potential associated with $\beta\phi$.

######  Lemma 3.6.

For all $\beta,\gamma\in\mathbb{R}$, $\Phi_{\beta,\gamma}$ is locally Hölder
continuous with respect to the metric $d_{\Sigma}$.

###### Proof.

From the bounded distortion near the neutral fixed point [20, Lemma 2.2],
there is $C>0$ such that for any $\mathbb{a}\in\mathbb{A}$ and all $x$,
$y\in[\mathbb{a}]$ we have

(3.7) $\Phi_{\beta,\gamma}(x)-\Phi_{\beta,\gamma}(y)\leq
C\beta|U(\xi)-U(\eta)|\leq C\beta,$

where $\xi=\pi(x)$ and $\eta=\pi(y)$. If $x\neq y$ then
$d_{\Sigma}(x,y)=e^{-n}$ holds for some $n\geq 2$, and there exists
$a_{1}\cdots a_{n}\in\mathbb{A}^{n}$ such that $x,y\in\Pi[\\![a_{1}\cdots
a_{n}]\\!]$. Since $\inf_{[0,1)\setminus J_{1}}|T^{\prime}|\geq\rho$ holds for
some $\rho>1$, if $n\geq 3$ then we have

(3.8)
$|U(\xi)-U(\eta)|\leq\frac{|U^{n-1}(\xi)-U^{n-1}(\eta)|}{\inf_{\bigcap_{k=2}^{n-1}U^{-k}(J(a_{k}))}|(U^{n-2})^{\prime}|}\leq\rho^{2-n}.$

The local Hölder continuity of $\Phi_{\beta,\gamma}$ follows from (3.7) and
(3.8). ∎

######  Lemma 3.7.

For any $\beta\in(1/2,1]$ there exists $\gamma\in\mathbb{R}$ such that $0\leq
P(\Phi_{\beta,\gamma})<\infty$.

###### Proof.

From Lemma 3.6, $|T(J_{p})|=1$ for $p\geq 2$ and (3.6), there is $C>0$ such
that for $\mathbb{a}=\bigcup_{q=2}^{\infty}[p1^{n-1}q]\in\mathbb{A}$ and all
$\beta,\gamma\in\mathbb{R}$ we have

$\frac{1}{|J_{p}|^{\beta}}\sup_{[\mathbb{a}]}\exp\Phi_{\beta,\gamma}=e^{-\gamma
n}\sup_{[\mathbb{a}]}\frac{\exp(\beta S_{n}\phi)}{|J_{p}|^{\beta}}\leq
Ce^{-\gamma n}n^{-2\beta}.$

Summing the result over all $\mathbb{a}\in\mathbb{A}$, we have

$\sum_{n=1}^{\infty}\sum_{\stackrel{{\scriptstyle\mathbb{a}\in\mathbb{A}}}{{|\mathbb{a}|=n}}}\sup_{[\mathbb{a}]}\exp\Phi_{\beta,\gamma}\leq
C\sum_{n=1}^{\infty}e^{-\gamma
n}n^{-2\beta}\sum_{p=2}^{\infty}|J_{p}|^{\beta}\leq
C\sum_{n=1}^{\infty}e^{-\gamma n}n^{-2\beta}\sum_{p=2}^{\infty}p^{-2\beta}.$

Let $\beta\in(1/2,1)$. Then we have $P(\beta\phi)>0$ [14], and the above
series is finite for all $\gamma\in(0,P(\beta\phi)]$. In particular,
$P(\Phi_{\beta,P(\beta\phi)})$ is finite. Since any measure in
$\mathcal{M}(X,\sigma)$ other than the unit point mass at
$1^{\infty}=111\cdots$ charges $\Sigma$, the equilibrium state
$\mu_{\beta\phi}$ for the potential $\beta\phi$ satisfies
$\mu_{\beta\phi}(\Sigma)>0$. Let $\hat{\mu}_{\beta\phi}$ denote the normalized
restriction of $\mu_{\beta\phi}$ to $\Sigma$. Since $\tau$ is the first return
map to $\Sigma$, $\hat{\mu}_{\beta\phi}$ is $\tau|_{\Sigma}$-invariant and
satisfies $\int R{\rm d}\hat{\mu}_{\beta\phi}<\infty$. By the variational
principle for $\Phi_{\beta,P(\beta\phi)}$ and Abramov-Kac’s formula [45,
Theorem 5.1], we obtain

$\begin{split}\infty>P(\Phi_{\beta,P(\beta\phi)})&\geq
h(\hat{\mu}_{\beta\phi})+\int(\Phi_{\beta}-P(\beta\phi)R){\rm
d}\hat{\mu}_{\beta\phi}\\\
&=\left(F_{\beta\phi}(\mu_{\beta\phi})+P(\beta\phi)-P(\beta\phi)\right)\int
R{\rm d}\hat{\mu}_{\beta\phi}=0.\end{split}$

We have verified111In fact, one can show $P(\Phi_{\beta,P(\beta\phi)})=0$. See
[24] for example. that $0\leq P(\Phi_{\beta,P(\beta\phi)})<\infty$ as required
in Lemma 3.7.

For the remaining case $\beta=1$, we have $P(\phi)=0$ [14]. From Lemma 3.6
there is $C\geq 1$ such that for $n\geq 1$ and $a_{1}\cdots
a_{n}\in{\mathbb{A}}^{n}$,

$C^{-1}\leq\frac{\left|\bigcap_{k=1}^{n}U^{-k}(J(a_{k}))\right|}{\sup_{[a_{1}\cdots
a_{n}]}\exp\left(\sum_{k=0}^{n-1}\Phi_{1,0}\circ\tau^{k}\right)}\leq C.$

Therefore, the sum of the denominator over all elements of $\mathbb{A}^{n}$ is
uniformly bounded from both sides. This implies $P(\Phi_{1,0})=0$ as required
in Lemma 3.7. ∎

Lemma 3.6 and Lemmas 3.5, 3.7 altogether verify the assumption in Theorem A
for the potential $\beta\phi$, $\beta\in(1/2,1]$. It follows from [36] that
for any $\beta\in(1/2,1]$, any minimizer of the rate function $I_{\beta\phi}$
is an equilibrium state for $\beta\phi$. Since the equilibrium state for
$\beta\phi$ is unique [14], the minimizer of the rate function $I_{\beta\phi}$
is unique. Since the map $\pi$ in (1.12) is continuous, the assertions in
Theorem C follow from Theorems A and B. ∎

###  3.7. Some generalizations

We have mainly worked on the two full shift spaces $X$ and $\Sigma$ (or
$\mathbb{A}^{\mathbb{N}}$), the latter obtained from the former via inducing
procedure as detailed in Section 2.1. The assumption that $X$ is the full
shift has been used to construct sets of periodic points of the same period,
in the proofs of exponential tightness (Lemma 2.6) and the lower large
deviation bound (Proposition 3.1). For the induced system
$(\Sigma,\tau|_{\Sigma})$, we have effected the thermodynamic formalism for
countable Markov shifts [19].

The setup in this paper can be slightly generalized. The above-mentioned
constructions of sets of periodic points can be done even if $X$ is replaced
by a finitely primitive shift (see [19] for the definition). Then the induced
shift space becomes finitely irreducible, for which the thermodynamic
formalism works too [28].

### Acknowledgments

I thank the referees for their careful readings of the manuscript and giving
useful suggestions for improvements. This research was partially supported by
the JSPS KAKENHI 19K21835 and 20H01811.

The conflict of interest statement: We have no conflict of interest to
disclose.

The date availability statement: This article has no associated data.

## References

  * [1] Aaronson, J., Denker, M., Urbański, M.: Ergodic Theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337, 495–548 (1993)
  * [2] Adler, R.L.: $F$-expansions revisited. Recent advances in topological dynamics, pp.1–5. Lecture Notes in Mathematics, 318, Springer-Verlag, Berlin 1973
  * [3] Bowen, R.: Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69, 1–17 (1979)
  * [4] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Second revised edition. Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin 2008
  * [5] Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)
  * [6] Buzzi, J., Sarig, O.: Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergodic Theory and Dynamical Systems 23, 1383–1400 (2003)
  * [7] Dembo, A., Zeitouni, O.: Large deviations techniques and applications, Applications of Mathematics 38, Springer, second edition (1998)
  * [8] Ellis, R.S.: Entropy, large deviations, and statistical mechanics, Grundlehren der Mathematischen Wissenschaften 271,  Springer (1985)
  * [9] van Enter, A.C.D., Fernández, R., Sokal, A.D.: Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72, 879–1167 (1993)
  * [10] Fayolle, G., de La Fortelle, A.: Entropy and large deviations for discrete-time Markov chains. Probl. Inf. Transm. 38, 354–367 (2002)
  * [11] Fiebig, D., Fiebig, U.-R., Yuri, M.: Pressure and equilibrium states for countable state Markov shifts. Isr. J. Math. 131, 221–257 (2002)
  * [12] Gurevič, B.M., Savchenko, S.V.: Thermodynamic formalism for countable symbolic Markov chains. Russ. Math. Surv. 53, 245–344 (1998)
  * [13] Hofbauer, F.: Examples for the nonuniqueness of the equilibrium state. Trans. Amer. Math. Soc. 228, 223–241 (1977)
  * [14] Iommi, G.: Multifractal analysis of the Lyapunov exponent for the backward continued fraction map. Ergodic Theory and Dynamical Systems 30, 211-232 (2010)
  * [15] Khinchin, A.Y.: Continued fractions. University of Chicago Press, III.-London, 1964
  * [16] Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321, 505–524 (1990)
  * [17] Kifer, Y.: Large deviations, averaging and periodic orbits of dynamical systems. Commun. Math. Phys. 162, 33–46 (1994)
  * [18] Maes, C., Redig, F., Takens, F., van Moffaert, A., Verbitski, E.: Intermittency and weak Gibbs states. Nonlinearity 13, 1681–1698 (2000)
  * [19] Mauldin, R.D., Urbański, M.: Graph directed Markov systems: Geometry and Dynamics of Limit Sets. Cambridge Tracts in Mathematics 148, Cambridge University Press (2003)
  * [20] Nakaishi, K.: Multifractal formalism for some parabolic maps. Ergodic Theory and Dynamical Systems 20, 843–857 (2000)
  * [21] Northshield, S.: A short proof and generalization of Lagrange’s theorem on continued fractions. The American Mathematical Monthly 118, 171–175 (2011)
  * [22] Olsen, L.: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. 82, 1591–1649 (2003)
  * [23] Orey, S., Pelikan, S.: Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms. Trans. Amer. Math. Soc. 315, 741–753 (1989)
  * [24] Pesin, Y., Senti, S.: Equilibrium measures for maps with inducing schemes. J. Mod. Dyn. 2, 397–430 (2008)
  * [25] Pianigiani, G.: First return map and invariant measures. Isr. J. Math. 35, 32–48 (1980)
  * [26] Pollicott, M., Sharp, R.: Large deviations for intermittent maps. Nonlinearity 22, 2079–2092 (2009)
  * [27] Pollicott, M., Sharp, R., Yuri, M.: Large deviations for maps with indifferent fixed points. Nonlinearity 11, 1173–1184 (1998)
  * [28] Pollicott, M., Urbański, M.: Asymptotic counting in conformal dynamical systems. Mem. Amer. Math. Soc., 271 (2021), v+139.
  * [29] Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66, 503–514 (1992)
  * [30] Ruelle, D.: Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics. Second edition. Cambridge University Press (2004)
  * [31] Sarig, O.: Thermodynamic formalism for countable Markov shifts, Ergodic Theory and Dynamical Systems 19, 1565–1593 (1999)
  * [32] Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131, 1751–1758 (2003)
  * [33] Sinaĭ, J.G.:  Gibbs measures in ergodic theory. Uspehi Mat. Nauk. 27, 21–64 (1972)
  * [34] Takahashi, Y.: Entropy functional (free energy) for dynamical systems and their random perturbations. Stochastic analysis (Katata/Kyoto, 1982), 437–467, North-Holland Math. Library, 32, North-Holland, Amsterdam, 1984
  * [35] Takahasi, H.: Large deviation principles for countable Markov shifts. Trans. Amer. Math. Soc. 372, 7831–7855 (2019)
  * [36] Takahasi, H.: Uniqueness of minimizer for countable Markov shifts and equidistribution of periodic points, J. Stat. Phys. 181, 2415–2431 (2020)
  * [37] Takahasi, H.: Entropy-approchability for transitive Markov shifts over infinite alphabet. Proc. Amer. Math. Soc. 148, 3847–3857 (2020)
  * [38] Takahasi, H.: Large deviation principle for the backward continued fraction expansion, Stochastic Processes and Their Applications 144, 153–172 (2022)
  * [39] Thaler, M.: Transformations on $[0,1]$ with infinite invariant measures. Isr. J. Math. 46, 67–96 (1983)
  * [40] Walters, P.: Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236, 121–153 (1978)
  * [41] Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79, Springer-Verlag, New York-Berlin 1982
  * [42] Yuri, M.: Thermodynamic formalism for certain nonhyperbolic maps. Ergodic Theory and Dynamical Systems 19, 1365–1378 (1999)
  * [43] Yuri, M.: Weak Gibbs measures for intermittent systems and weakly Gibbsian states in statistical mechanics. Commun. Math. Phys. 241, 453–466 (2003)
  * [44] Yuri, M.: Large deviations for countable to one Markov systems. Commun. Math. Phys. 258, 455–474 (2005)
  * [45] Zweimüller, R.: Invariant measures for general(ized) induced transformations. Proc. Amer. Math. Soc. 133 (2005) 2283–2295.