Density of periodic measures and large deviation principle for generalized ($\alpha,\beta$)- transformations

Let $\mathbb{N}_{0}$ be the set of non-negative integers. For a finite or countable set $A$, we denote by $A^{\mathbb{N}_{0}}$ the one-sided infinite product of $A$ equipped with the product topology of the discrete topology of $A$. To simplify the notation, given integers $i\leq j$ and $x_{i},x_{i+1},\ldots,x_{j}\in A$, we set $x_{[i,j]}:=x_{i}x_{i+1}\cdots x_{j}$. We also write $x_{[i,j)}:=x_{[i,j-1]}$ and similarly for $x_{(i,j]}$ and $x_{(i,j)}$. A sequence $\underline{x}\in A^{\mathbb{N}_{0}}$ will be defined by all of its coordinates $x_{n}\in A$ with $n\geq 0$. Let $\sigma$ be the shift map on $A^{\mathbb{N}_{0}}$ (i.e., $(\sigma(\underline{x}))_{n}=x_{n+1}$ for each $n\geq 0$ and $\underline{x}\in A^{\mathbb{N}_{0}}$). When a subset $\Sigma^{+}$ of $A^{\mathbb{N}_{0}}$ is $\sigma$-invariant and closed, we call it a subshift and call $A$ the alphabet of $\Sigma^{+}$. For a matrix $M=(M_{ij})_{(i,j)\in A^{2}}$, each entry of which is $0$ or $1$, we define a subshift $\Sigma^{+}_{M}\subset A^{\mathbb{N}_{0}}$ by

and call $\Sigma^{+}_{M}$ a Markov shift with an adjacency matrix $M$.

For a subshift $\Sigma^{+}$ on an alphabet $A$, we set $\mathcal{L}(\Sigma^{+}):=\{x_{[0,n]}:\underline{x}\in\Sigma^{+},\ n\geq 0\}$ and $[u]:=\left\{\underline{x}\in\Sigma^{+}:u=x_{[0,|u|)}\right\}$ for each $u\in\mathcal{L}(\Sigma^{+})$, where $|u|$ denotes the length of $u$. A word $v\in\mathcal{L}(\Sigma^{+})$ is called a subword of $u=u_{0}\cdots u_{n}\in\mathcal{L}(\Sigma^{+})$ if $v=u_{[i,j]}$ for some $0\leq i\leq j\leq n$. For $u,v\in\mathcal{L}(\Sigma^{+})$, we use juxtaposition $uv$ to denote the word obtained by the concatenation and $u^{\infty}$ means a one-sided infinite sequence $uuu\cdots\in A^{\mathbb{N}_{0}}$. Moreover, we set $\mathcal{S}(u):=\{i\in A:ui\in\mathcal{L}(\Sigma^{+})\}$ for $u\in\mathcal{L}(\Sigma^{+})$. Finally, we say that $\Sigma^{+}$ is transitive if for any $u,v\in\mathcal{L}(\Sigma^{+})$, we can find $w\in\mathcal{L}(\Sigma^{+})$ such that $uwv\in\mathcal{L}(\Sigma^{+})$ holds. For the rest of this paper, we denote by $h_{\sigma}(\mu)$ the metric entropy of $\mu\in\mathcal{M}_{\sigma}(\Sigma^{+})$.

Let $X=[0,1]$ and $T\colon X\to X$ be a piecewise monotonic map with critical points $0=c_{0}<c_{1}<\cdots<c_{k}=1$. Let $X_{T}:=\bigcap_{n=0}^{\infty}T^{-n}(\bigcup_{j=1}^{k}(c_{j-1},c_{j}))$, and define the coding map $\mathcal{I}\colon X_{T}\to\{1,\ldots,k\}^{\mathbb{N}_{0}}$ by

which is injective since $T$ is transitive (see [27, Proposition 6.1]). We denote the closure of $\mathcal{I}(X_{T})$ in $\{1,\ldots,k\}^{\mathbb{N}}$ by $\Sigma_{T}^{+}$. Then, $\Sigma_{T}^{+}$ is a subshift, and $(\Sigma_{T}^{+},\sigma)$ is called the coding space of $(X,T)$. We use the following notations:

• •

  $\displaystyle\underline{a}^{(i)}:=\lim_{x\rightarrow c_{i}+0}\mathcal{I}(x)$, $\displaystyle\underline{b}^{(i)}:=\lim_{x\rightarrow c_{i}-0}\mathcal{I}(x)$ for $1\leq i\leq k-1$.

• •

  ${\rm adj}(\underline{a}^{(i)}):=\underline{b}^{(i)}$, ${\rm adj}(\underline{b}^{(i)}):=\underline{a}^{(i)}$ for $1\leq i\leq k-1$.

• •

  $\displaystyle\underline{a}:=\lim_{x\rightarrow+0}\mathcal{I}(x)$, $\displaystyle\underline{b}:=\lim_{x\rightarrow 1-0}\mathcal{I}(x)$.

We also set $\mathcal{CR}:=\{\underline{a}^{(i)},\underline{b}^{(i)}:1\leq i\leq k-1\}$ and call it a critical set.

In what follows, we define the Markov diagram, introduced by Hofbauer ([10]), which is a countable oriented graph with subsets of $\Sigma_{T}^{+}$ as vertices. Let $C\subset\Sigma_{T}^{+}$ be a closed subset with $C\subset[j]$ for some $1\leq j\leq k$. We say that a non-empty closed subset $D\subset\Sigma_{T}^{+}$ is a successor of $C$ if $D=[l]\cap\sigma(C)$ for some $1\leq l\leq k$. The expression $C\rightarrow D$ denotes that $D$ is a successor of $C$. Now, we define a set $\mathcal{D}_{T}$ of vertices by induction. First, we set $\mathcal{D}_{0}:=\{[1],\ldots,[k]\}$. If $\mathcal{D}_{n}$ is defined for $n\geq 0$, then we set

We note that $\mathcal{D}_{n}$ is a finite set for each $n\geq 0$ since the number of successors of any closed subset of $\Sigma_{T}^{+}$ is at most $k$ by definition. Finally, we set

The oriented graph $(\mathcal{D}_{T},\rightarrow)$ is called the Markov diagram of $T$. For notational simplicity, we use the expression $\mathcal{D}$ instead of $\mathcal{D}_{T}$ if no confusion arises.

For $\underline{x}\in\Sigma_{T}^{+}$ and $n\geq 0$, we set $D_{n}^{\underline{x}}:=\sigma^{n}(x_{[0,n]})$. We define a sequence $\{R_{m}^{\underline{x}}\}_{m\geq 0}$ of integers inductively as follows. First, we set $R_{0}^{\underline{x}}:=0$. If $R_{m}^{\underline{x}}$ is defined for $m\geq 0$, then let

We also set $r_{m}^{\underline{x}}:=R_{m}^{\underline{x}}-R_{m-1}^{\underline{x}}$. Now, we summarize properties of Hofbauer’s Markov Diagram, which are appeared in [12] (see also [13, Page 224]).

For a subset $\mathcal{C}\subset\mathcal{D}$, we define a matrix $M(\mathcal{C})=(M(\mathcal{C})_{C,D})_{(C,D)\in\mathcal{C}^{2}}$ by

Then, $\Sigma^{+}_{M(\mathcal{C})}=\{\underline{C}\in\mathcal{C}^{\mathbb{N}_{0}}:C_{n}\rightarrow C_{n+1},n\geq 0\}$ is a one-sided Markov shift with a countable alphabet $\mathcal{C}$ and an adjacency matrix $M(\mathcal{C})$. For notational simplicity, we denote $\Sigma_{\mathcal{C}}^{+}$ instead of $\Sigma^{+}_{M(\mathcal{C})}$.We say that $\mathcal{C}$ is irreducible if for any $C,D\in\mathcal{C}$, there are finite vertices $C_{0},\ldots,C_{n}\in\mathcal{C}$ such that $C_{i}\rightarrow C_{i+1}$ for $0\leq i\leq n-1$ (i.e., $C_{[0,n]}\in\mathcal{L}(\Sigma^{+}_{\mathcal{C}})$), $C_{0}=C$ and $C_{n}=D$, and if every subset of $\mathcal{D}$, which contains $\mathcal{C}$ does not have this property. It is clear that $\Sigma^{+}_{\mathcal{C}}$ is transitive if and only if $\mathcal{C}$ is irreducible. We define a map $\Psi\colon\Sigma_{\mathcal{D}}^{+}\to\{1,\ldots,k\}^{\mathbb{N}_{0}}$ by

where $1\leq x_{n}\leq k$ is a unique integer such that $C_{n}\subset[x_{n}]$ holds for each $n\in\mathbb{N}_{0}$. Then it is not difficult to see that $\Psi$ is continuous, countable to one and $\Psi(\Sigma_{\mathcal{D}}^{+})=\Sigma_{T}^{+}$. We say that $\mu\in\mathcal{M}^{e}_{\sigma}(\Sigma_{T}^{+})$ is liftable if there is $\overline{\mu}\in\mathcal{M}^{e}_{\sigma}(\Sigma_{\mathcal{D}}^{+})$ such that $\mu=\overline{\mu}\circ\Psi^{-1}$ holds. It is known that not every $\mu\in\mathcal{M}^{e}_{\sigma}(\Sigma_{T}^{+})$ is liftable in general although $\Psi$ is surjective (see [14]). In [10], Hofbauer provided a sufficient condition for the liftability.

We recall two important facts for transitive piecewise monotonic maps.

Hereafter, let $T\colon[0,1]\to[0,1]$ be a generalized $(\alpha,\beta)$-transformation and $k$ be a smallest integer not less than $\alpha+\beta$. To simplifies the notation, we set $A_{n}:=\sigma^{n}([a_{[0,n]}])$ and $B_{n}:=\sigma^{n}([b_{[0,n]}])$ for each $n\geq 0$ and set $R_{m}:=R_{m}^{\underline{a}}$, $S_{m}:=R_{m}^{\underline{b}}$, $r_{m+1}:=r_{m+1}^{\underline{a}}=R_{m+1}-R_{m}$ and $s_{m+1}:=r_{m+1}^{\underline{b}}=S_{m+1}-S_{m}$ for each $m\geq 0$. We also denote

In this subsection, we give a proof of Proposition A with the assumption that $T$ is transitive. As we mentioned in §1, this theorem appears as [5, Theorem A] with the stronger assumption that $\mathcal{D}$ is irreducible. In [5], the hypothesis of the irrecducibility for $\mathcal{D}$ is used only in [5, Proposition 3.1]. The other part of the proof of [5, Theorem A] can be shown similarly by using the transitivity of $T$ instead of the irreducibility. Hence to prove Proposition A, it is sufficient to show the following proposition, which is analogous to [5, Proposition 3.1].