Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz--Krieger algebras

TL;DR

This paper investigates the relationship between continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz--Krieger algebras, revealing connections with topological entropy.

Contribution

It establishes a link between continuous orbit equivalence and KMS states, and explores how topological entropy behaves under this equivalence.

Findings

KMS states are studied for gauge actions with potential functions.

A relationship between topological entropy and orbit equivalence is identified.

The work enhances understanding of the structure of Cuntz--Krieger algebras in relation to topological dynamics.

Abstract

We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.

Equations

$$\psi (y{\alpha }_{i\log \beta }(x))=\psi (xy),x\in {\mathcal{O}}_A^a,y\in {\mathcal{O}}_A.$$

$${\gamma }_t^f(S_j)=V_t(f)S_j,j=1,\dots ,N.$$

$${\lambda }_{\varphi }(a)=\sum _{i=1}^N{S}_i^{\ast }a{e}^{\varphi }{S}_i,a\in C(X_A)$$

$${\sigma }_B^{k_1(x)}(h({\sigma }_A(x)))$$

$${\sigma }_A^{k_2(y)}(h^{-1}({\sigma }_B(y)))$$

$$h_{top}({\sigma }_A)$$

$$h_{top}({\sigma }_B)$$

$$A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],B=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$$

$$f^n(x)=\sum _{i=0}^{n-1}f({\sigma }_A^i(x)),x\in X_A.$$

$${\lambda }_{\varphi }(f)(x)=\sum _{{\sigma }_A(y)=x}{e}^{\varphi (y)}f(y),x\in X_A.$$

$$({\lambda }_{\varphi }{)}^n(a)={\lambda }_A^n(a{e}^{{\varphi }^n}),a\in {\mathcal{D}}_A$$

$${\lambda }_{\varphi }(g_{\varphi })=r_{\varphi }{g}_{\varphi },{\phi }_{\varphi }\circ {\lambda }_{\varphi }=r_{\varphi }{\phi }_{\varphi },{\phi }_{\varphi }(g_{\varphi })=1.$$

$${\gamma }_t^f(S_j)=V_t(f)S_j,j=1,\dots ,N,\text{ where }{V}_t(f)=\exp (itf).$$

$$S_i{V}_t(f)=\sum _{j=1}^N{S}_j{V}_t(f)S_j^{\ast }{S}_i=V_t(f\circ {\sigma }_A)S_i,i=1,\dots ,N$$

$${\gamma }_t^f(S_{\mu })=V_t(f^n)S_{\mu },\mu =({\mu }_1,\dots ,{\mu }_n)\in B_n(X_A),t\in \mathbb{R}$$

$${\lambda }_{\varphi }(a)=\sum _{j=1}^N{S}_j^{\ast }a{e}^{\varphi }{S}_j=\beta \sum _{j=1}^N{S}_j^{\ast }a{\beta }^{-f}{S}_j=\beta \sum _{j=1}^N{S}_j^{\ast }a{\gamma }_{i\log \beta }^f(S_j),a\in {\mathcal{D}}_A.$$

$$(\phi \circ {\lambda }_{\varphi })(a)=\beta \sum _{j=1}^N\psi (S_j^{\ast }a{\gamma }_{i\log \beta }^f(S_j))=\beta \sum _{j=1}^N\psi (S_j{S}_j^{\ast }a)=\beta \phi (a),a\in {\mathcal{D}}_A.$$

$$\psi (y{\gamma }_{i\log \beta }^f(x))$$

$$=\frac{1}{{\beta }^n}\phi (S_{\nu }^{\ast }{S}_{\nu }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{f^k\log \beta }{S}_{\nu }^{\ast }{e}^{f^n\log \beta }{e}^{{\varphi }^n}{S}_{\nu })$$

$$=\frac{1}{{\beta }^n}\phi (S_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{f^k\log \beta }{S}_{\nu }^{\ast }{e}^{f^n\log \beta +{\varphi }^n}{S}_{\nu })$$

$$=\phi (S_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{f^k\log \beta }{S}_{\nu }^{\ast }{S}_{\nu })$$

$$=\frac{1}{{\beta }^k}\phi ({\lambda }_{\varphi }^k(S_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{f^k\log \beta }{S}_{\nu }^{\ast }{S}_{\nu }))$$

$$=\frac{1}{{\beta }^k}\phi (S_{\eta }^{\ast }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{f^k\log \beta }{S}_{\nu }^{\ast }{S}_{\nu }{e}^{{\varphi }^k}{S}_{\eta })$$

$$=\frac{1}{{\beta }^k}\phi (S_{\eta }^{\ast }{e}^{f^k\log \beta }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\eta }{S}_{\eta }^{\ast }{e}^{{\varphi }^k}{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi })$$

$$=\frac{1}{{\beta }^k}\phi (S_{\eta }^{\ast }{e}^{f^k\log \beta +{\varphi }^k}{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi })$$

$$=\phi (S_{\eta }^{\ast }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta }{S}_{\xi })$$

$$=\phi (S_{\xi }^{\ast }{S}_{\xi }{S}_{\eta }^{\ast }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta -{\varphi }^k}{e}^{{\varphi }^k}{S}_{\xi })$$

$$=\phi ({\lambda }_{\varphi }^k(S_{\xi }{S}_{\eta }^{\ast }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-f^k\log \beta -{\varphi }^k}))$$

$$=\phi ({\lambda }_{\varphi }^k(S_{\xi }{S}_{\eta }^{\ast }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{-k\log \beta }))=\psi (xy).$$

$$\psi (y{\gamma }_{i\log \beta }^f(x))$$

$$=\frac{1}{{\beta }^n}\phi ({\lambda }_{\varphi }^n(S_{\nu }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{f^{k+m}\log \beta }{S}_{\nu }^{\ast }{e}^{f^n\log \beta }))$$

$$=\frac{1}{{\beta }^n}\phi (S_{\nu }^{\ast }{S}_{\nu }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{f^{k+m}\log \beta }{S}_{\nu }^{\ast }{e}^{f^n\log \beta +{\varphi }^n}{S}_{\nu })$$

$$=\phi (S_{\xi }{S}_{\eta }^{\ast }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{f^{k+m}\log \beta }{S}_{\nu }^{\ast }{S}_{\nu })$$

$$=\frac{1}{{\beta }^{k+m}}\phi ({\lambda }_{\varphi }^{k+m}(S_{\xi }{S}_{\eta }^{\ast }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{f^{k+m}\log \beta }{S}_{\nu }^{\ast }{S}_{\nu }))$$

$$=\frac{1}{{\beta }^{k+m}}\phi (S_{\zeta }^{\ast }{S}_{\xi }^{\ast }{S}_{\xi }{S}_{\eta }^{\ast }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{f^{k+m}\log \beta }{S}_{\nu }^{\ast }{S}_{\nu }{e}^{{\varphi }^{m+k}}{S}_{\xi }{S}_{\zeta })$$

$$=\phi (S_{\zeta }^{\ast }{S}_{\eta }^{\ast }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\xi }{S}_{\zeta })$$

$$=\phi ({\lambda }_{\varphi }^{k+m}(e^{-{\varphi }^{k+m}-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\xi }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\eta }^{\ast }))$$

$$=\phi (S_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{S}_{\nu }^{\ast }{S}_{\nu }{S}_{\xi }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\eta }^{\ast })=\psi (xy).$$

$$\psi (y{\gamma }_{i\log \beta }^f(x))$$

$$=\frac{1}{{\beta }^{k+m}}\phi ({\lambda }_{\varphi }^{k+m}(S_{\xi }{S}_{\eta }^{\ast }{S}_{\nu }^{\ast }{e}^{-f^n\log \beta }{S}_{\nu }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{f^{k+m}\log \beta }))$$

$$=\frac{1}{{\beta }^{k+m}}\phi (S_{\zeta }^{\ast }{S}_{\xi }^{\ast }{S}_{\xi }{S}_{\eta }^{\ast }{S}_{\nu }^{\ast }{e}^{-f^n\log \beta }{S}_{\nu }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{e}^{f^{k+m}\log \beta +{\varphi }^{k+m}}{S}_{\xi }{S}_{\zeta })$$

$$=\phi (S_{\zeta }^{\ast }{S}_{\xi }^{\ast }{S}_{\xi }{S}_{\eta }^{\ast }{S}_{\nu }^{\ast }{e}^{-f^n\log \beta }{S}_{\nu }{e}^{-f^{k+m}\log \beta }{S}_{\eta }{S}_{\zeta }{S}_{\zeta }^{\ast }{S}_{\xi }^{\ast }{S}_{\xi }{S}_{\zeta })$$

Full text

1 Introduction

Let $A=[A(i,j)]_{i,j=1}^{N}$ be a square matrix with entries in $\{0,1\}$, where $1<N\in{\mathbb{N}}$. Throughout the paper, we assume that $A$ is irreducible and not a permutation. The shift space $X_{A}$ of a one-sided topological Markov shift $(X_{A},\sigma_{A})$ consists of one-sided sequences $(x_{n})_{n\in\mathbb{N}}\in\{1,\dots,N\}^{\mathbb{N}}$ satisfying $A(x_{n},x_{n+1})=1$ for all $n\in{\mathbb{N}}$. Take and fix $\theta\in\mathbb{R}$ satisfying $0<\theta<1$. Define a metric $d_{\theta}$ on $X_{A}$ by $d_{\theta}(x,y)=\theta^{k}$ for $x=(x_{n})_{n\in\mathbb{N}},y=(y_{n})_{n\in\mathbb{N}}$ with $x\neq y$ where $k$ is the largest non-negative integer such that $x_{n}=y_{n},n<k$. The space $X_{A}$ is a zero-dimensional compact Hausdorff space by the metric $d_{\theta}$. The shift transformation $\sigma_{A}:X_{A}\longrightarrow X_{A}$ is defined by a continuous surjection satisfying $\sigma_{A}((x_{n})_{n\in\mathbb{N}})=(x_{n+1})_{n\in\mathbb{N}}$. The two-sided topological Markov shift $(\bar{X}_{A},\bar{\sigma}_{A})$ is a topological dynamical system of a homeomorphism $\bar{\sigma}_{A}((x_{n})_{n\in\mathbb{Z}}))=(x_{n+1})_{n\in\mathbb{Z}}$ on the zero-dimensional compact Hausdorff space $\bar{X}_{A}$ consisting of two-sided sequences $(x_{n})_{n\in\mathbb{Z}}$ satisfying $A(x_{n},x_{n+1})=1$ for all $n\in{\mathbb{Z}}$.

The Cuntz–Krieger algebra ${{\mathcal{O}}_{A}}$ for the matrix $A$ is defined to be a universal unique $C^{*}$-algebra generated by partial isometries $S_{1},\dots,S_{N}$ satisfying the relations: $\sum_{j=1}^{N}S_{j}S_{j}^{*}=1,\,S_{i}^{*}S_{i}=\sum_{j=1}^{N}A(i,j)S_{j}S_{j}^{*},\,i=1,\dots,N(\cite[cite]{[\@@bibref{}{CK}{}{}]}).$ The gauge automorphisms $\gamma^{A}_{t},t\in{\mathbb{R}}$ on ${{\mathcal{O}}_{A}}$ are defined by $\gamma^{A}_{t}(S_{j})=e^{\sqrt{-1}t}S_{j},j=1,\dots,N.$ They yield an action of $\mathbb{R}$ on ${{\mathcal{O}}_{A}}$ which we call the gauge action. A word $\mu=(\mu_{1},\dots,\mu_{k})$ for $\mu_{i}\in\{1,\dots,N\}$ is said to be admissible for the topological Markov shift $(X_{A},\sigma_{A})$ if $\mu$ appears somewhere in an element $x$ in $X_{A}$. The length of $\mu$ is $k$, which is denoted by $|\mu|$. We denote by $B_{k}(X_{A})$ the set of all admissible words of length $k$. We set $B_{*}(X_{A})=\cup_{k=0}^{\infty}B_{k}(X_{A})$ where $B_{0}(X_{A})$ denotes the empty word $\emptyset$. Denote by $U_{\mu}$ the cylinder set $\{(x_{n})_{n\in\mathbb{N}}\in X_{A}\mid x_{1}=\mu_{1},\dots,x_{k}=\mu_{k}\}$ for $\mu=(\mu_{1},\dots,\mu_{k})\in B_{k}(X_{A})$. The set of cylinder sets form a basis of the topology of $X_{A}$. Let us denote by ${{\mathcal{D}}_{A}}$ the $C^{*}$-subalgebra of ${{\mathcal{O}}_{A}}$ generated by the projections of the form $S_{\mu_{1}}\cdots S_{\mu_{n}}S_{\mu_{n}}^{*}\cdots S_{\mu_{1}}^{*}$ for ${\mu_{1}\cdots\mu_{n}}\in B_{*}(X_{A})$. It is well-known that the commutative $C^{*}$-algebra $C(X_{A})$ of complex valued continuous functions on $X_{A}$ is regarded as the $C^{*}$-subalgebra ${{\mathcal{D}}_{A}}$ by identifying the characteristic functions $\chi_{U_{\mu_{1}\cdots\mu_{n}}}\in C(X_{A})$ of the cylinder sets $U_{\mu_{1}\cdots\mu_{n}}$ with the projections $S_{\mu_{1}}\cdots S_{\mu_{n}}S_{\mu_{n}}^{*}\cdots S_{\mu_{1}}^{*}\in{{\mathcal{D}}_{A}}.$

Let $\alpha$ be an action of $\mathbb{R}$ on ${{\mathcal{O}}_{A}}$. Denote by ${{\mathcal{O}}^{a}_{A}}$ the set of analytic elements of the action. For a positive real number $\beta\in\mathbb{R}$, a state $\psi$ on ${{\mathcal{O}}_{A}}$ is called a $\log{\beta}$-KMS state for the action $\alpha$ if $\psi$ satisfies the condition

[TABLE]

The studies of KMS states on operator algebras are very crucial from the viewpoints of quantum statistical mechanics and the structure theory of $C^{*}$-algebras and von Neumann algebras. There have been many important and interesting studies as in the text book by Bratteli–Robinson ([3]). For Cuntz–Krieger algebras, Enomoto–Fujii–Watatani proved that the gauge action $\gamma^{A}$ on ${{\mathcal{O}}_{A}}$ has a $\log\beta$-KMS state if and only if $\beta=r_{A}$ the Perron–Frobenius eigenvalue of the matrix $A$, and the admitted KMS state is faithful and unique ([7], see [19] for Cuntz algebras). The reciprocal of the eigenvalue $r_{A}$ is the radius of convergence of the zeta function $\zeta_{A}(z)={{\operatorname{exp}}}\left(\sum_{n=1}^{\infty}\frac{{{\operatorname{Per}}}_{n}(\bar{X}_{A})}{n}z^{n}\right)$ of the topological Markov shift $(\bar{X}_{A},\bar{\sigma}_{A})$, and the value $\log r_{A}$ is topological entropy of the topological Markov shift $(\bar{X}_{A},\bar{\sigma}_{A})$ ([20], cf, [11]).

In [13], the author studied generalized gauge actions from the viewpoints of continuous orbit equivalence and flow equivalence of topological Markov shifts (see also [4], [5], [15], [16], [18], etc.). We regard a function $f$ in $C(X_{A},\mathbb{R})$ as an element of the subalgebra $C(X_{A},\mathbb{C})(={{\mathcal{D}}_{A}})$ of ${{\mathcal{O}}_{A}}$. For $f\in C(X_{A},\mathbb{R})$, define one-parameter unitaries $V_{t}(f)\in{{\mathcal{D}}_{A}},t\in\mathbb{R}$ by setting $V_{t}(f)={{\operatorname{exp}}}({\sqrt{-1}tf})$ and an automorphism $\gamma_{t}^{f}$ on ${{\mathcal{O}}_{A}}$ for each $t\in\mathbb{R}$ by

[TABLE]

The automorphisms $\gamma_{t}^{f},t\in\mathbb{R}$ yield an action of $\mathbb{R}$ on ${{\mathcal{O}}_{A}}$ such that $\gamma^{f}_{t}(a)=a$ for all $a\in{{\mathcal{D}}_{A}}$. Such an action on ${{\mathcal{O}}_{A}}$ is called a generalized gauge action in [9]. If the function $f$ is constantly $1$, the action $\gamma^{1}$ is the gauge action $\gamma^{A}$. Let $F_{\theta}(X_{A})$ be the set of real valued $\theta$-Hölder continuous functions on $X_{A}$. In [8], [9] and [10], R. Exel and Exel–Lopes have studied KMS states for generalized gauge actions on the $C^{*}$-algebras constructed from crossed products by endomorphisms including Cuntz-Krieger algebras from the viewpoint of thermodynamic formalism of dynamical systems and shown that there exists a bijective correspondence between KMS states and eigenvectors of Ruelle operators (cf. [23]). As a result, Exel [9] showed that there exists a unique KMS state for a generalized gauge action on Cuntz–Krieger algebras. For $\phi\in F_{\theta}(X_{A})$, the Ruelle operator $\lambda_{\phi}:C(X_{A})\rightarrow C(X_{A})$ on a topological Markov shift $(X_{A},\sigma_{A})$ is defined by

[TABLE]

where $e^{\phi}\in{{\mathcal{D}}_{A}}$ is defined by $(e^{\phi})(x)=e^{\phi(x)},x\in X_{A}$ (cf. [21], [24]). The Ruelle operators have been playing a key role in Ruelle’s thermodynamic formalism in topological Markov shifts ([24], [25], cf. [1], [2], [21], etc.).

In the first part of the paper, we will give a direct proof for Exel’s result above in the case of Cuntz–Krieger algebras which says that there exists a bijective correspondence between KMS states and eigenvectors of the Ruelle operators (Proposition 2.2).

In the second part of the paper, which is a main part of the paper, we will find a certain relationship between topological entropy of continuously orbit equivalent topological Markov shifts, by using Exel’s result. The author in [12] has introduced a notion of continuous orbit equivalence of one-sided topological Markov shifts. It is weaker than one-sided topological conjugacy and gives rise to isomorphic Cuntz–Krieger algebras ([15], see also [17], [4], etc.). Let $A$ and $B$ be irreducible square matrices with entries in $\{0,1\}$. If there exists a homeomorphism $h:X_{A}\rightarrow X_{B}$ such that

[TABLE]

for some continuous functions $k_{1},l_{1}\in C(X_{A},{\mathbb{Z}}_{+}),\,k_{2},l_{2}\in C(X_{B},{\mathbb{Z}}_{+}),$ the one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are said to be continuously orbit equivalent, where ${\mathbb{Z}}_{+}=\{0,1,2,\dots\}$. The functions $c_{1}(x)=l_{1}(x)-k_{1}(x),x\in X_{A}$ and $c_{2}(y)=l_{2}(y)-k_{2}(y),y\in X_{B}$ are called the cocycle function for $h$ and the cocycle function for $h^{-1}$, respectively. In [18], Matui and the author have shown that the zeta functions of continuous orbit equivalent topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ have a certain relationship by using the above cocycle functions (see also [14, Theorem 4.6]). It suggests that the topological entropy of the topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ have some relation, because the topological entropy are maximum poles of their zeta functions. In this paper, we will show the following theorem that tells us a relationship between topological entropy of continuously orbit equivalent topological Markov shifts.

Theorem 1.1** (Theorem 3.4).**

Let $A$ and $B$ be irreducible, non-permutation matrices with entries in $\{0,1\}$. Suppose that one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are continuously orbit equivalent. Let $\varphi_{A}$ and $\varphi_{B}$ be the unique KMS states for the gauge actions on ${{\mathcal{O}}_{A}}$ and on ${{\mathcal{O}}_{B}}$, respectively. Let $r_{A}$ and $r_{B}$ be the Perron–Frobenius eigenvalues of the matrix $A$ and of the matrix $B$, respectively. Denote by $h_{top}(\sigma_{A})$ and $h_{top}(\sigma_{B})$ the topological entropy of $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$, respectively. Then we have

[TABLE]

where $r_{B}^{-c_{1}^{n}}\in C(X_{A})$ is defined by $r_{B}^{-c_{1}^{n}}(x)=r_{B}^{-{\sum_{k=0}^{n-1}c_{1}(\sigma_{A}^{k}(x))}},\,x\in X_{A}$, and $r_{A}^{-c_{2}^{n}}\in C(X_{B})$ is similarly defined.

In [13], a notion of strongly continuous orbit equivalence between one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ was introduced. It is defined as the cases where the cocycle function $c_{1}$ is cohomologous to $1$ in $C(X_{A},\mathbb{Z})$. Although we have already known that strongly continuous orbit equivalence implies topological conjugacy of their two-sided topological Markov shifts ([13, Theorem 5.5]) and hence $h_{top}(\sigma_{A})=h_{top}(\sigma_{B})$, as a direct corollary of the above theorem, we have

Corollary 1.2** (Corollary 3.5).**

If one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are strongly continuously orbit equivalent, then we have $h_{top}(\sigma_{A})=h_{top}(\sigma_{B})$.

In the final section, we will concretely calculate the formulas (1.6), (1.7) for two matrices

[TABLE]

for which $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are continuously orbit equivalent.

2 Ruelle operators and KMS states

In this section, we fix a topological Markov shift $(X_{A},\sigma_{A})$. For a function $f\in C(X_{A},\mathbb{Z})$ and $n\in\mathbb{N}$, we define a function $f^{n}$ on $X_{A}$ by

[TABLE]

We fix generating partial isometries $S_{1},\dots,S_{N}$ of the Cuntz–Krieger algebra ${{\mathcal{O}}_{A}}$ satisfying the relations: $\sum_{j=1}^{N}S_{j}S_{j}^{*}=1,\,S_{i}^{*}S_{i}=\sum_{j=1}^{N}A(i,j)S_{j}S_{j}^{*},\,i=1,\dots,N(\cite[cite]{[\@@bibref{}{CK}{}{}]}).$ For a word $\mu=(\mu_{1},\dots,\mu_{n})\in B_{n}(X_{A})$, denote by $S_{\mu}$ the partial isometry $S_{\mu_{1}}\cdots S_{\mu_{n}}$. The $C^{*}$-subalgebra ${{\mathcal{D}}_{A}}$ generated by the projections $S_{\mu}S_{\mu}^{*},\mu\in B_{*}(X_{A})$ is identified with the commutative $C^{*}$-algebra $C(X_{A})$ of the complex valued continuous functions on $X_{A}$ through the correspondence $S_{\mu}S_{\mu}^{*}\leftrightarrow\chi_{U_{\mu}}$, where $\chi_{U_{\mu}}\in C(X_{A})$ is the characteristic function of of the cylinder set $U_{\mu}$. Recall that $0<\theta<1$ is a real number defining the metric $d_{\theta}(x,y),\,x,y\in X_{A}$ on $X_{A}$. A continuous function $\phi:X_{A}\rightarrow\mathbb{R}$ is said to be $\theta$-Hölder continuous if there exists a constant $C$ such that $|\phi(x)-\phi(y)|\leq Cd_{\theta}(x,y)$. We denote by $F_{\theta}(X_{A})$ the set of $\theta$-Hölder continuous functions on $X_{A}$. It is easy to see that an integer valued continuous function on $X_{A}$ is $\theta$-Hölder continuous, and hence so is a scalar multiple of an integer valued continuous function on $X_{A}$. Under our identification between ${{\mathcal{D}}_{A}}$ and $C(X_{A})$ above, the Ruelle operator $\lambda_{\phi}:{{\mathcal{D}}_{A}}\rightarrow{{\mathcal{D}}_{A}}$ for $\phi\in F_{\theta}(X_{A})$ is defined by the formula (1.3). It may be written

[TABLE]

It is also called the transfer operator (cf. [1]). One easily sees that $\lambda_{\phi}(f)\in F_{\theta}(X_{A})$ for $f\in F_{\theta}(X_{A})$. For $\phi\equiv 0$, we write $\lambda_{\phi}$ as $\lambda_{A}$. We note that the identities

[TABLE]

hold, where $\phi^{n}(x)=\sum_{i=0}^{n-1}\phi(\sigma_{A}^{i}(x)),x\in X_{A}$. The following well-known result is known as Ruelle–Perron–Frobenius Theorem.

Lemma 2.1** ([21, Theorem 2.2]).**

For $\phi\in F_{\theta}(X_{A})$, we have the following assertions.

1. (i)

There exists a unique positive eigenvalue $r_{\phi}$ of $\lambda_{\phi}$, a strictly positive function $g_{\phi}\in{{\mathcal{D}}_{A}}$ and a faithful state $\varphi_{\phi}$ on ${{\mathcal{D}}_{A}}$ satisfying

[TABLE] 2. (ii)

$\frac{(\lambda_{\phi})^{n}(a)}{r_{\phi}^{n}}$* uniformly converges to $\varphi_{\phi}(a)g_{\phi}$ for each $a\in{{\mathcal{D}}_{A}}$.*

Let us denote by ${{\mathcal{F}}_{A}}$ the $C^{*}$-subalgebra of ${{\mathcal{O}}_{A}}$ consisting of fixed elements under the automorphisms $\gamma^{A}_{t},t\in\mathbb{R}$ of the gauge action. The algebra is an AF-algebra whose diagonal elements give rise to the algebra ${{\mathcal{D}}_{A}}$. There is a canonical conditional expectation $E_{D}:{{\mathcal{F}}_{A}}\rightarrow{{\mathcal{D}}_{A}}$ by taking diagonal elements. Let us denote by $E_{A}:{{\mathcal{O}}_{A}}\rightarrow{{\mathcal{D}}_{A}}$ the canonical conditional expectation defined by $E_{A}(T)=E_{D}(\frac{1}{2\pi}\int_{0}^{2\pi}\gamma^{A}_{t}(T)dt)$ for $T\in{{\mathcal{O}}_{A}}$ where $dt$ is the Lebesgue measure on $[0,2\pi]$.

We regard a function $f$ in $C(X_{A},\mathbb{R})$ as an element of the subalgebra $C(X_{A},\mathbb{C})(={{\mathcal{D}}_{A}})$ of ${{\mathcal{O}}_{A}}$. For $f\in C(X_{A},\mathbb{R})$, recall that a one-parameter automorphism group $\gamma_{t}^{f},t\in\mathbb{R}$ on ${{\mathcal{O}}_{A}}$ is defined by

[TABLE]

It is easy to see that the automorphisms $\gamma_{t}^{f},t\in\mathbb{R}$ yield an action of $\mathbb{R}$ on ${{\mathcal{O}}_{A}}$ such that $\gamma^{f}_{t}(a)=a$ for all $a\in{{\mathcal{D}}_{A}}$. If the function $f$ is constantly $1$, the action $\gamma^{1}$ is the gauge action $\gamma^{A}$. Through the identification between $\chi_{U_{\mu}}\in C(X_{A})$ and $S_{\mu}S_{\mu}^{*}\in{{\mathcal{D}}_{A}}$, we have $\sum_{i=1}^{N}S_{i}fS_{i}^{*}=f\circ\sigma_{A}$ so that the equalities

[TABLE]

hold. The identity

[TABLE]

is directly shown (cf. [15, Lemma 3.1]) and useful in our proof of Proposition 2.2 below.

Let us denote by ${\mathfrak{S}}({{\mathcal{O}}_{A}})$ the state space of ${{\mathcal{O}}_{A}}$ which is the set of continuous positive linear functionals $\varphi$ on ${{\mathcal{O}}_{A}}$ satisfying $\varphi(1)=1$. We similarly define the state space ${\mathfrak{S}}({{\mathcal{D}}_{A}})$ of ${{\mathcal{D}}_{A}}$. Let $\alpha$ be an action of $\mathbb{R}$ on ${{\mathcal{O}}_{A}}$. Denote by ${{\mathcal{O}}^{a}_{A}}$ the set of analytic elements of the action $\alpha$ ([22, 8.12]). Following after [3], for a real number $1<\beta\in\mathbb{R}$, a state $\psi\in{\mathfrak{S}}({{\mathcal{O}}_{A}})$ is called a $\log{\beta}$-KMS state for the $\alpha$-action if $\psi$ satisfies the condition (1.1). The equality (1.1) is called the KMS condition.

The following proposition has been seen in Exel’s paper [9] in a slightly different form. We will give a direct proof which is different from those of [9] and [8].

Proposition 2.2** (Exel [9]).**

1. (i)

For $f\in F_{\theta}(X_{A})$ and $1<\beta\in\mathbb{R}$, if $\psi\in{\mathfrak{S}}({{\mathcal{O}}_{A}})$ is a $\log\beta$-KMS state for the $\gamma^{f}$-action, by putting $\phi=(1-f){\log\beta}$, the restriction $\varphi=\psi|_{{{\mathcal{D}}_{A}}}\in{\mathfrak{S}}({{\mathcal{D}}_{A}})$ of $\psi$ to ${{\mathcal{D}}_{A}}$ satisfies $\varphi\circ\lambda_{\phi}=\beta\varphi$ on ${{\mathcal{D}}_{A}}$. 2. (ii)

For $\phi\in F_{\theta}(X_{A})$ and $1<\beta\in\mathbb{R}$, if $\varphi\in{\mathfrak{S}}({{\mathcal{D}}_{A}})$ is a state satisfying $\varphi\circ\lambda_{\phi}=\beta\varphi$ on ${{\mathcal{D}}_{A}}$, then by putting $f=1-\frac{1}{\log\beta}\phi$, the state $\psi=\varphi\circ E_{A}\in{\mathfrak{S}}({{\mathcal{O}}_{A}})$ is a $\log\beta$-KMS state for the $\gamma^{f}$-action.

Proof.

(i) Since $e^{\phi}=\beta^{1-f}\in{{\mathcal{D}}_{A}}$ and $\beta^{-f}S_{j}=\gamma_{i\log{\beta}}^{f}(S_{j})$, we have

[TABLE]

Let us note that the set of finite linear combinations of elements of the form $S_{\mu}S_{\nu}^{*},\,\mu,\nu\in B_{*}(X_{A})$ is dense in the analytic elements of ${{\mathcal{O}}_{A}}$ for the action $\gamma^{f}$. By the KMS condition (1.1), we have

[TABLE]

(ii) Suppose that $\varphi\in{\mathfrak{S}}({{\mathcal{D}}_{A}})$ satisfies $\varphi\circ\lambda_{\phi}=\beta\varphi$. Put $f=1-\frac{1}{\log\beta}\phi$ and the state $\psi=\varphi\circ E_{A}\in{\mathfrak{S}}({{\mathcal{O}}_{A}})$. Since the set of finite linear combinations of elements of the form $S_{\mu}S_{\nu}^{*},\,\mu,\nu\in B_{*}(X_{A})$ is dense in ${{\mathcal{O}}^{a}_{A}}$ and $\psi$ vanishes outside of diagonal elements of ${{\mathcal{F}}_{A}}$, it suffices to show the KMS condition (1.1) only for the following two cases and two subcases for each.

Case (1) $x=aS_{\nu}^{*},y=S_{\nu}b$.

Subcase (1-1) $a=S_{\xi}S_{\eta}^{*}$ and $b=S_{\eta}S_{\zeta}S_{\zeta}^{*}S_{\xi}^{*}$ where $|\xi|=|\eta|$.

Subcase (1-2) $a=S_{\eta}S_{\zeta}S_{\zeta}^{*}S_{\xi}^{*}$ and $b=S_{\xi}S_{\eta}^{*}$ where $|\xi|=|\eta|$.

Case (2) $x=S_{\nu}b,y=aS_{\nu}^{*}$.

Subcase (2-1) $a=S_{\xi}S_{\eta}^{*}$ and $b=S_{\eta}S_{\zeta}S_{\zeta}^{*}S_{\xi}^{*}$ where $|\xi|=|\eta|$.

Subcase (2-2) $a=S_{\eta}S_{\zeta}S_{\zeta}^{*}S_{\xi}^{*}$ and $b=S_{\xi}S_{\eta}^{*}$ where $|\xi|=|\eta|$.

Throughout the above four subcases, we put $n=|\nu|,\,k=|\xi|=|\eta|,\,m=|\zeta|.$ We note the identity $\phi^{\ell}+f^{\ell}\log\beta=\ell\log\beta$ for $\ell\in\mathbb{N}$ holds. To show the equality $\psi(y\gamma_{i\log\beta}^{f}(x))=\psi(xy)$ for each of the four subcases, we use the equality $\varphi\circ\lambda_{\phi}=\beta\varphi$ of the assumption.

For the subcase (1-1), we have

[TABLE]

For the subcase (1-2), we have

[TABLE]

For the subcase (2-1), we have

[TABLE]

For the subcase (2-2), we have

[TABLE]

Therefore $\psi$ is a $\log\beta$-KMS state for the $\gamma^{f}$-action on ${{\mathcal{O}}_{A}}$. ∎

R. Exel proved the following fact that will be useful in our further discussions.

Lemma 2.3** (Exel [9]).**

For $f\in F_{\theta}(X_{A}),$ there exists a unique positive real number $\beta_{f}\in\mathbb{R}$ such that there exists a $\log\beta_{f}$-KMS state for the $\gamma^{f}$-action on ${{\mathcal{O}}_{A}}$. The admitted KMS state is faithful and unique.

3 Continuous orbit equivalence and KMS conditions

Throughout the section, two irreducible square matrices $A,B$ with entries in $\{0,1\}$ are fixed. The Ruelle operator $\lambda_{\phi}$ on ${{\mathcal{D}}_{A}}$ is denote by $\lambda^{A}_{\phi}$, and the action $\gamma^{f}$ on ${{\mathcal{O}}_{A}}$ is denoted by $\gamma^{A,f}$. We use similar notation $\lambda_{\phi}^{B}$ and $\gamma^{B,f}$ for the matrix $B$.

Let $h:X_{A}\rightarrow X_{B}$ be a homeomorphism which gives rise to a continuous orbit equivalence between $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$. Take $k_{1},l_{1}\in C(X_{A},{\mathbb{Z}}_{+})$ and $k_{2},l_{2}\in C(X_{B},{\mathbb{Z}}_{+})$ satisfying (1.4) and (1.5), respectively. Recall that $c_{1}\in C(X_{A},{\mathbb{Z}}_{+})$ and $c_{2}\in C(X_{B},{\mathbb{Z}}_{+})$ are defined by $c_{1}(x)=l_{1}(x)-k_{1}(x),x\in X_{A}$ and $c_{2}(y)=l_{2}(y)-k_{2}(y),y\in X_{B}$, respectively.

Proposition 3.1** ([15, Corollary 3.4]).**

Let $h:X_{A}\rightarrow X_{B}$ be a homeomorphism which gives rise to a continuous orbit equivalence between $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$. Then there exists an isomorphism $\Phi:{{\mathcal{O}}_{A}}\rightarrow{{\mathcal{O}}_{B}}$ such that $\Phi({{\mathcal{D}}_{A}})={{\mathcal{D}}_{B}}$ and

[TABLE]

Recall that $r_{A},r_{B}$ denote the Perron-Frobenius eigenvalues of $A,B$, respectively.

Lemma 3.2**.**

Suppose that one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are continuously orbit equivalent via a homeomorphism $h:X_{A}\longrightarrow X_{B}$.

1. (i)

There exists a $\log r_{A}$-KMS state for the $\gamma^{B,c_{2}}$-action on ${{\mathcal{O}}_{B}}$, and similarly there exists a $\log r_{B}$-KMS state for the $\gamma^{A,c_{1}}$-action on ${{\mathcal{O}}_{A}}$. 2. (ii)

Put $\phi_{c_{2}}=(1-c_{2})\log r_{A}\in F_{\theta}(X_{B})$ and $\phi_{c_{1}}=(1-c_{1})\log r_{B}\in F_{\theta}(X_{A})$. Then we have $r_{\phi_{c_{2}}}=r_{A}$ and $r_{\phi_{c_{1}}}=r_{B}$, where $r_{\phi_{c_{2}}}$ and $r_{\phi_{c_{1}}}$ are positive eigenvalues of the operators $\lambda^{B}_{\phi_{c_{2}}}$ and $\lambda^{A}_{\phi_{c_{1}}}$ satisfying (2.2) on ${{\mathcal{D}}_{B}}$ and on ${{\mathcal{D}}_{A}}$, respectively.

Proof.

(i) By [7], there exists a unique $\log r_{A}$-KMS state written $\varphi_{A}$ for the gauge action on ${{\mathcal{O}}_{A}}$. Let $\varPhi:{{\mathcal{O}}_{A}}\longrightarrow{{\mathcal{O}}_{B}}$ be the isomorphism satisfying (3.1). We then have

[TABLE]

Since $\Phi(\gamma^{A}_{i\log r_{A}}(x))=\gamma^{B,c_{2}}_{i\log r_{A}}(\varPhi(x)),x\in{{\mathcal{O}}^{a}_{A}}$, the state $\varphi_{A}\circ\varPhi^{-1}$ is a $\log r_{A}$-KMS state for the $\gamma^{B,c_{2}}$-action on ${{\mathcal{O}}_{B}}$. We similarly know that there exists a $\log r_{B}$-KMS state for the $\gamma^{A,c_{1}}$-action on ${{\mathcal{O}}_{A}}$.

(ii) Take a strictly positive function $g_{\phi_{c_{2}}}\in{{\mathcal{D}}_{B}}$ and a faithful state $\varphi_{\phi_{c_{2}}}$ on ${{\mathcal{D}}_{B}}$ satisfying (2.2) for the Ruelle operator $\lambda^{B}_{\phi_{c_{2}}}$. By Lemma 2.3, the state $\varphi_{A}\circ\varPhi^{-1}$ in (i) is a unique $\log r_{A}$-KMS state for the $\gamma^{B,c_{2}}$-action on ${{\mathcal{O}}_{B}}$. Put $\varphi^{\prime}=\varphi_{A}\circ\varPhi^{-1}|_{{{\mathcal{D}}_{B}}}$. By Proposition 2.2, KMS-states and normalized positive eigenvectors of the Ruelle operator bijectively correspond so that we have

[TABLE]

As in Lemma 2.3, a KMS state is unique, so that $\varphi_{\phi_{c_{2}}}=\varphi^{\prime}$. Hence we have

[TABLE]

By (2.2), we have

[TABLE]

∎

We thus have the following proposition that states an asymptotic relation between two Perron-Frobenius eigenvalues $r_{A}$ and $r_{B}$.

Proposition 3.3**.**

Suppose that one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are continuously orbit equivalent. Let $\varphi_{A}$ (resp. $\varphi_{B}$) be the unique KMS state for the gauge action of ${{\mathcal{O}}_{A}}$ (resp. ${{\mathcal{O}}_{B}}$). Let $r_{A}$ (resp. $r_{B}$) be the Perron-Frobenius eigenvalue of the matrix $A$ (resp. $B$). Then there exist positive constants $C_{A},C_{B}$ such that

[TABLE]

where

[TABLE]

Proof.

Put $\phi_{c_{2}}=(1-c_{2})\log r_{A}$. For the Ruelle operator $\lambda^{B}_{\phi_{c_{2}}}$, take a strictly positive function $g_{\phi_{c_{2}}}\in{{\mathcal{D}}_{B}}$ and a faithful state $\varphi_{\phi_{c_{2}}}$ on ${{\mathcal{D}}_{B}}$ satisfying (2.2). By Lemma 2.1 (ii), we have for $b\in{{\mathcal{D}}_{B}}$

[TABLE]

so that

[TABLE]

By (2.1), we see

[TABLE]

By (3.2) with $r_{\phi_{c_{2}}}=r_{A}$ from Lemma 3.2, we have

[TABLE]

Since $\varphi_{B}\circ\lambda_{B}^{n}=r_{B}^{n}\varphi_{B}$, we thus have

[TABLE]

By letting $b=1$, we have

[TABLE]

Put $C_{B}=\varphi_{B}(g_{\phi_{c_{2}}})$. Since $\varphi_{B}$ is faithful, the number $C_{B}$ is positive. We thus have a desired equality. By putting $C_{A}=\varphi_{A}(g_{\phi_{c_{1}}})$ for a stictly positive function $g_{\phi_{c_{1}}}\in{{\mathcal{D}}_{A}}$ with $\phi_{c_{1}}=(1-c_{1})\log r_{A}$, we similarly have the other equality

[TABLE]

∎

We now obtain the main result of this paper.

Theorem 3.4**.**

Let $A$ and $B$ be irreducible, non-permutation matrices with entries in $\{0,1\}$. Suppose that one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are continuously orbit equivalent. Let $\varphi_{A}$ and $\varphi_{B}$ be the unique KMS states for the gauge actions of ${{\mathcal{O}}_{A}}$ and of ${{\mathcal{O}}_{B}}$, respectively. Let $r_{A}$ and $r_{B}$ be the Perron–Frobenius eigenvalues of the matrix $A$ and of the matrix $B$, respectively. Denote by $h_{top}(\sigma_{A})$ and $h_{top}(\sigma_{B})$ the topological entropy of $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$, respectivly. Then we have

[TABLE]

Proof.

Since $h_{top}(\sigma_{A})=\log r_{A}$ and $h_{top}(\sigma_{B})=\log r_{B}$ ([11], [20]), we get the desired equalities by Proposition 3.3. ∎

In [13], a notion of strongly continuously orbit equivalence between one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ was introduced. Suppose that $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are continuously orbit equivalent via a homeomorphism $h:X_{A}\longrightarrow X_{B}.$ Let $c_{1}:X_{A}\longrightarrow\mathbb{Z}$ be the cocycle function for $h:X_{A}\longrightarrow X_{B}.$ If there exists a continuous function $b_{1}:X_{A}\longrightarrow\mathbb{Z}$ such that

[TABLE]

then $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are said to be strongly continuously orbit equivalent. Although we have already known that strongly continuous orbit equivalence of one-sided topological Markov shifts implies topological conjugacy of their two-sided topological Markov shifts ([13, Theorem 5.5]) so that $h_{top}(\sigma_{A})=h_{top}(\sigma_{B})$, we may prove it as an immediate corollary of the above theorem in the following way.

Corollary 3.5**.**

Let $A$ and $B$ be irreducible, non-permutation matrices with entries in $\{0,1\}$. If one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are strongly continuously orbit equivalent, then we have $h_{top}(\sigma_{A})=h_{top}(\sigma_{B})$.

Proof.

The equality (3.3) implies

[TABLE]

As the function $b_{1}(x)-b_{1}(\sigma_{A}^{n}(x)),x\in X_{A}$ is bounded, the equality $h_{top}(\sigma_{A})=h_{top}(\sigma_{B})$ follows from Theorem 3.4. ∎

4 Example

Let $A$ and $B$ be the matrices:

[TABLE]

so that $r_{A}=2,\,r_{B}=\frac{1+\sqrt{5}}{2}.$ They are both irreducible and not permutations. The subshift $X_{A}$ is the on-sided full shift over $\{1,2\}$, whereas $X_{B}$ is the one-sided subshift over $\{1,2\}$ forbidden the word $(2,2)$. Let $h:X_{A}\rightarrow X_{B}$ be a homeomorphism defined by substituting the word $(2,1)$ for $2$ in $X_{A}$ such as

[TABLE]

Then the one-sided topological Markov shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are continuously orbit equivalent as in [12] via the homeomorphism $h:X_{A}\rightarrow X_{B}$. For $x=(x_{n})_{n\in\mathbb{N}},y=(y_{n})_{n\in\mathbb{N}}$, we put

[TABLE]

and

[TABLE]

The functions $k_{1},l_{1}\in C(X_{A},{\mathbb{Z}}_{+}),k_{2},l_{2}\in C(X_{B},{\mathbb{Z}}_{+})$ satisfy the equalities (1.4), (1.5). Hence we have the cocycle functions

[TABLE]

We will ensure the equalities (1.6) and (1.7) in the following concrete way.

1. Equality (1.6).

As $c_{1}(\sigma_{A}^{k}(x))=x_{k+1}$ for $x=(x_{n})_{n\in\mathbb{N}}\in X_{A}$, we have

[TABLE]

It then follows that

[TABLE]

Sine the restriction of the KMS state $\varphi_{A}$ to the subalgebra ${{\mathcal{D}}_{A}}$ is the Bernoulli measure, we see that $\varphi_{A}(\chi_{U_{\mu}})=\frac{1}{2^{n}}$ for $\mu\in B_{n}(X_{A})$. Therefore we have

[TABLE]

Now we have

[TABLE]

so that we have $\varphi_{A}(r_{B}^{-c_{1}^{n}})=\frac{1}{2^{n}}$ and hence

[TABLE]

2. Equality (1.7).

As $c_{2}(\sigma_{B}^{k}(y))=2-y_{k+1}$ for $y=(y_{n})_{n\in\mathbb{N}}\in X_{B}$, we have

[TABLE]

It then follows that by $r_{A}=2$

[TABLE]

Let $S_{1},S_{2}$ be the canonical generating partial isometries of the Cuntz–Krieger algebra ${{\mathcal{O}}_{B}}$ which satisfies

[TABLE]

By the identification of the function $\chi_{U_{\nu}}$ with the projection $S_{\nu}S_{\nu}^{*}$, the equality (4.1) tells us that

[TABLE]

As the value $\varphi_{B}(S_{\nu}S_{\nu}^{*})$ for $\nu\in B_{n}(X_{B})$ is not equal to $\frac{1}{2^{n}}$, we can not use an analogous method to the computation of $\varphi_{A}(S_{\mu}S_{\mu}^{*})$ above. We then use an operator $(S_{1}^{*}+\sqrt{2}S_{2}^{*})^{n}(S_{1}+\sqrt{2}S_{2})^{n}$ in the following way. By the identities $\varphi_{B}(S_{\nu}S_{\nu}^{*})=\frac{1}{r_{B}^{n}}\varphi_{B}(S_{\nu}^{*}S_{\nu})$ for $\nu\in B_{n}(X_{B})$ and

[TABLE]

we have

[TABLE]

Put $H_{n}=(S_{1}^{*}+\sqrt{2}S_{2}^{*})^{n}(S_{1}+\sqrt{2}S_{2})^{n}.$ It is easy to see that $H_{n}$ is written as

[TABLE]

As $\varphi_{B}$ is a KMS state on ${{\mathcal{O}}_{B}}$, the equality $1=\varphi_{B}(S_{1}^{*}S_{1})=r_{B}\varphi_{B}(S_{1}S_{1}^{*})$ holds so that $\varphi_{B}(S_{1}S_{1}^{*})=\frac{1}{r_{B}}$. It follows that

[TABLE]

We thus have

[TABLE]

Acknowledgments: The author would like to deeply thank the referee for careful reading and lots of helpful advices in the presentation of the paper. This work was supported by JSPS KAKENHI Grant Numbers 15K04896, 19K03537.