Topological entropy in continuous orbit equivalence of one-sided topological Markov shifts
January 01, 1970
In this paper, we prove that the continuous orbit equivalence class of a one-sided topological Markov shift contains a one-sided topological Markov shift whose topological entropy is greater than an arbitrary prescribed positive real number, and also contains a one-sided topological Markov shift whose topological entropy is less than an arbitrary prescribed positive real number.
Mathematics Subject Classification: Primary 37B10, 37B40; Secondary 37A20, 46L80.
Keywords and phrases: topological entropy, topological Markov shift, continuous orbit equivalence, Cuntz–Krieger algebra
F. Sugisaki in [1] and [2] proved that any Cantor minimal system is strongly orbit equivalent to Cantor minimal systems of all entropies ([1] for finite entropies, [2] for infinite entropy). His results gave a complete answer for the conjecture raised by Boyle–Handelman [3] (cf. [4]). Inspired by Sugisaki’s results, in this paper, we study a relationship between topological entropy and continuous orbit equivalence of one-sided topological Markov shifts. Cantor minimal systems are minimal homeomorphisms of Cantor sets. On the contrary, one-sided topological Markov shifts are surjective continuous maps having many periodic points. Let \(A=[A(i,j)]_{i,j=1}^N\) be an irreducible non-permutation matrix with entries in \(\{0,1\}\). The one-sided topological Markov shift \((X_A,\sigma_A)\) is defined to be a shift transformation \(\sigma_A\) on the shift space \(X_A\) consisting of all right one-sided sequences \((x_n)_{n\in \mathbb{N}}\) of \(x_n \in \{1,\dots,N\}\) satisfying \(A(x_n, x_{n+1}) =1\) for all \(n \in \mathbb{N}\). The shift \(\sigma_A\) on \(X_A\) is defined by \(\sigma_A((x_n)_{n \in \mathbb{N}}) = (x_{n+1})_{n \in \mathbb{N}}\). By equipping \(X_A\) with the relative topology of the infinite product topology on \(\{1,\dots, N\}^\mathbb{N}\) of discrete set \(\{1,\dots,N\}\), the shift space \(X_A\) is homeomorphic to a Cantor discontinuum. In [5], the notion of continuous orbit equivalence in one-sided topological Markov shifts was introduced in the following way. Two one-sided topological Markov shifts \((X_A,\sigma_A)\) and \((X_B,\sigma_B)\) are continuously orbit equivalent, written as \((X_A, \sigma_A) \underset{{{\operatorname{COE}}}}{\sim}(X_B,\sigma_B)\), if there exist a homeomorphism \(h: X_A \to X_B\) and continuous maps \(k_1, l_1: X_A \to {\mathbb{Z}}_+=\{ 0,1,\dots,\}\) and \(k_2, l_2: X_B \to {\mathbb{Z}}_+\) satisfying \[\begin{align} \sigma_B^{k_1(x)}(h(\sigma_A(x))) = & \sigma_B^{l_1(x)}(h(x))\quad \text{ for } x \in X_A, \\ \sigma_A^{k_2(y)}(h^{-1}(\sigma_B(y))) = & \sigma_A^{l_2(y)}(h^{-1}(y))\quad \text{ for } y \in X_B. \end{align}\] In [6], the classification of continuous orbit equivalence was completed so that we know \((X_A, \sigma_A) \underset{{{\operatorname{COE}}}}{\sim}(X_B,\sigma_B)\) if and only if the associated Cuntz–Krieger algebras \({{\mathcal{O}}_A}, {{\mathcal{O}}_B}\) are isomorphic and \({{\operatorname{det}}}(I_N - A) = {{\operatorname{det}}}(I_M-B)\), where the matrix size of \(A\) is \(N\) and that of \(B\) is \(M\), and \(I_N, I_M\) denote the identity matrices, respectively. Let us denote by \(\operatorname{BF}(A^t)\) the quotient group \(\mathbb{Z}^N/(I_N -A^t)\mathbb{Z}^N\), which is well-known as the Bowen–Franks group for the matrix \(A^t\) ([7], cf. [8], [9], etc.). Since the isomorphism class of \({{\mathcal{O}}_A}\) is completely determined by its \(\operatorname{K}\)-group together with the position \([1_{{\mathcal{O}}_A}]\) of the unit \(1_{{{\mathcal{O}}_A}}\) of \({{\mathcal{O}}_A}\) in \(\operatorname{K}_0({{\mathcal{O}}_A})\) by [10], and \((\operatorname{K}_0({{\mathcal{O}}_A}), [1_{{\mathcal{O}}_A}])\cong (\operatorname{BF}(A^t), [1_N])\) by [11], we see that the triplet \((\operatorname{BF}(A^t), [1_N], {{\operatorname{det}}}(I_N-A))\) is a complete set of invariants of the continuous orbit equivalence of \((X_A, \sigma_A)\), where \([1_N]\) denotes the class of the vector \((1,\dots,1)\) in the quotient group \(\operatorname{BF}(A^t)\).
On the other hand, it is well-known as Parry’s theorem [12] that the topological entropy \(h_{\operatorname{top}}(X_A, \sigma_A)\) of a topological Markov shift \((X_A,\sigma_A)\) is computed to be \(\log \lambda_A\), where \(\lambda_A\) is the Perron–Frobenius eigenvalue of the underlying matrix \(A\) (cf. [9], [8])).
In this paper, we prove that the continuous orbit equivalence class of a one-sided topological Markov shift contains a one-sided topological Markov shift whose topological entropy is greater than an arbitrary prescribed positive real number (Theorem 2), and also contains a one-sided topological Markov shift whose topological entropy is less than an arbitrary prescribed positive real number (Theorem 5).
In what follows, we denote by \({\mathbb{Z}}_+\) and \(\mathbb{N}\) the set of nonnegative integers and the set of positive integers, respectively.
In this section, we study upper entropies in continuous orbit equivalence class. We have to provide a couple of notions and lemmas to show Theorem 2. An essential matrix means a matrix for which none of its rows or columns is zero.
Let \(A=[A(i,j)]_{i,j=1}^N\) be an \(N\times N\) essential matrix with entries in nonnegative integers. A directed graph \({\mathcal{G}}_A=({\mathcal{V}}_A, \mathcal{E}_A)\) naturally associates to the matrix \(A\) in the following way. The vertex set \({\mathcal{V}}_A\) is defined as \(\{1,\dots,N\}\). For two vertices \(i,j \in {\mathcal{V}}_A\) with \(A(i,j) \ne 0\), define \(A(i,j)\) multiple directed edges from \(i\) to \(j\). The set of such edges is the edge set \(\mathcal{E}_A\). If \(\alpha \in \mathcal{E}_A\) is a directed edge from \(i\) to \(j\), we write \(s(\alpha) = i\) and \(t(\alpha) =j\). We then have a matrix \(A^{[2]}=[A^{[2]}(\alpha,\beta)]_{\alpha,\beta \in \mathcal{E}_A}\) with entries in \(\{0,1\}\) such that \(A^{[2]}(\alpha,\beta) =1\) if \(t(\alpha) = s(\beta)\), otherwise \(0\). The matrix \(A^{[2]}\) is called the second higher block matrix for \(A\) (cf. [9]). Define \({\mathcal{V}}_A\times \mathcal{E}_A\)-matrix \(D=[D(i,\alpha)]_{i \in {\mathcal{V}}_A, \alpha\in \mathcal{E}_A}\) and \(\mathcal{E}_A\times {\mathcal{V}}_A\)-matrix \(E=[E(\beta,j)]_{\beta\in \mathcal{E}_A, j \in {\mathcal{V}}_A}\) by setting for \(i,j\in {\mathcal{V}}_A\) and \(\alpha,\beta \in \mathcal{E}_A\) \[\label{eq:DE} D(i,\alpha)= \begin{cases} 1 & \text{ if } i = s(\alpha), \\ 0 & \text{ otherwise} \end{cases} \quad \text{ and } \quad E(\beta,j)= \begin{cases} 1 & \text{ if } j = t(\beta), \\ 0 & \text{ otherwise,} \end{cases}\tag{1}\] so that \(A= DE, \, A^{[2]} = ED\). The matrix \(D\) is called a division matrix which is a rectangular matrix with entries in \(\{0,1\}\) and with exactly one \(1\) in each column and at least one \(1\) in each row, and \(E\) is called an amalgamation matrix which is a rectangular matrix with entries in \(\{0,1\}\) and with exactly one \(1\) in each row and at least one \(1\) in each column. Let \(M\) be the cardinal number \(|\mathcal{E}_A|\) of the edge set \(\mathcal{E}_A\). The following lemma is well-known (cf. [9]).
Lemma 1.
The map \(x \in \mathbb{Z}^N \to D^t x \in \mathbb{Z}^M\) induces an isomorphism \(\Phi_{A}:[x] \in \operatorname{BF}(A^t) \to [D^t x] \in \operatorname{BF}({A^{[2]}}^t)\) such that \(\Phi_{A}([1_N]) = [1_M]\).
\({{\operatorname{det}}}(I_N - A) = {{\operatorname{det}}}(I_M - A^{[2]})\) and \(\lambda_A = \lambda_A^{[2]}\).
Hence the Cuntz–Krieger algebras \({{\mathcal{O}}_A}\) and \({\mathcal{O}}_{A^{[2]}}\) are isomorphic and \((X_A, \sigma_A) \underset{{{\operatorname{COE}}}}{\sim}(X_{A^{[2]}},\sigma_{A^{[2]}})\).
We shall use a slight variant of the construction in [13]. Although the result needed below follows directly from [13], the following explicit form of the matrix \(\Bar{A}\) is useful because it keeps the original matrix \(A\) as the upper-left block. This makes the later reduction to the case where the matrix has a non-zero diagonal entry more transparent.
Lemma 2 (cf. [13]). For an irreducible non-permutation matrix \(A=[A(i,j)]_{i,j=1}^N\) with entries in \(\{0,1\}\), let \(\Bar{A}\) be the \((N+3)\times (N+3)\) matrix defined by \[\Bar{A} = \setcounter{MaxMatrixCols}{15} {\small \begin{bmatrix} A(1,1) &\cdots&A(1,N-1) &A(1,N) &0 &0 &0 \\ \vdots & &\vdots &\vdots &\vdots &\vdots &\vdots \\ A(N-1,1)&\cdots&A(N-1,N-1)&A(N-1,N) &0 &0 & 0 \\ A(N,1) &\cdots &A(N,N-1) &A(N,N) &0 &1 &0 \\ 0 &\cdots &0 &1 &0 &0 &0 \\ 0 &\cdots &0 &0 &1 &0 &1 \\ 0 &\cdots &0 &0 &0 &1 &1 \end{bmatrix} } = \setcounter{MaxMatrixCols}{15} {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{A}}& & &0 &0 & 0 \\ & & & &0 &1 &0 \\ 0 &\cdots &0 &1 &0 &0 &0 \\ 0 &\cdots &0 &0 &1 &0 &1 \\ 0 &\cdots &0 &0 &0 &1 &1 \end{bmatrix}. }\] Then there exists an isomorphism \(\Psi: \operatorname{BF}({\Bar{A}}^t) \to \operatorname{BF}({A}^t)\) such that \[\label{eq:Psi} \Psi([1_{N+3}]) = [1_N] \quad \text{ and } \quad {{\operatorname{det}}}(I_{N+3}-\Bar{A}) = - {{\operatorname{det}}}(I_N - A).\tag{2}\]
Proof. The pair \((I_{N+3}-\Bar{A}^t, [1_{N+3}])\) of the matrix \(I_{N+3}-\Bar{A}^t\) and the vector \([1_{N+3}]\) is transformed by elementary operations of matrices in the following way: \[\begin{align} &(I_{N+3}-\Bar{A}^t, [1_{N+3}]) \\ = & \left( {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{I}}_{N}\text{\huge{-}}\text{\huge{A}}^{\text{\large{t}}} & & &0 &0 & 0 \\ & & & &-1 &0 &0 \\ 0 &\cdots &0 &0 &1 &-1 &0 \\ 0 &\cdots &0 &-1 &0 &1 &-1 \\ 0 &\cdots &0 &0 &0 &-1 &0 \end{bmatrix}, \begin{bmatrix} 1 \\ \vdots \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} } \right) (\text{add } (N+3)\text{th column to } (N+2)\text{th column}) \\ \to & \left( {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{I}}_{N}\text{\huge{-}}\text{\huge{A}}^{\text{\large{t}}} & & &0 &0 & 0 \\ & & & &-1 &0 &0 \\ 0 &\cdots &0 &0 &1 &-1 &0 \\ 0 &\cdots &0 &-1 &0 &0 &-1 \\ 0 &\cdots &0 &0 &0 &-1 &0 \end{bmatrix}, \begin{bmatrix} 1 \\ \vdots \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} } \right) (\text{subtract } (N+3)\text{th column from } N\text{th column}) \\ \to & \left( {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{I}}_{N}\text{\huge{-}}\text{\huge{A}}^{\text{\large{t}}} & & &0 &0 & 0 \\ & & & &-1 &0 &0 \\ 0 &\cdots &0 &0 &1 &-1 &0 \\ 0 &\cdots &0 &0 &0 &0 &-1 \\ 0 &\cdots &0 &0 &0 &-1 &0 \end{bmatrix}, \begin{bmatrix} 1 \\ \vdots \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} } \right) (\text{add } (N+1)\text{th row to } N\text{th row}) \\ \to & \left( {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{I}}_{N}\text{\huge{-}}\text{\huge{A}}^{\text{\large{t}}} & & &0 &0 & 0 \\ & & & &0 &-1 &0 \\ 0 &\cdots &0 &0 &1 &-1 &0 \\ 0 &\cdots &0 &0 &0 &0 &-1 \\ 0 &\cdots &0 &0 &0 &-1 &0 \end{bmatrix}, \begin{bmatrix} 1 \\ \vdots \\ 1 \\ 2 \\ 1 \\ 1 \\ 1 \end{bmatrix} } \right) (\text{subtract } (N+3)\text{th row from } N\text{th row}) \\ \to & \left( {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{I}}_{N}\text{\huge{-}}\text{\huge{A}}^{\text{\large{t}}} & & &0 &0 & 0 \\ & & & &0 &0 &0 \\ 0 &\cdots &0 &0 &1 &-1 &0 \\ 0 &\cdots &0 &0 &0 &0 &-1 \\ 0 &\cdots &0 &0 &0 &-1 &0 \end{bmatrix}, \begin{bmatrix} 1 \\ \vdots \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} } \right)(\text{add } (N+1)\text{th column to } (N+2)\text{th column}) \\ \to & \left( {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{I}}_{N}\text{\huge{-}}\text{\huge{A}}^{\text{\large{t}}} & & &0 &0 & 0 \\ & & & &0 &0 &0 \\ 0 &\cdots &0 &0 &1 &0 &0 \\ 0 &\cdots &0 &0 &0 &0 &-1 \\ 0 &\cdots &0 &0 &0 &-1 &0 \end{bmatrix}, \begin{bmatrix} 1 \\ \vdots \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} } \right)(\text{exchange } (N+2)\text{th row and } (N+3)\text{th row}) \\ \to & \left( {\small \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{I}}_{N}\text{\huge{-}}\text{\huge{A}}^{\text{\large{t}}} & & &0 &0 & 0 \\ & & & &0 &0 &0 \\ 0 &\cdots &0 &0 &1 &0 &0 \\ 0 &\cdots &0 &0 &0 &-1 &0 \\ 0 &\cdots &0 &0 &0 &0 &-1 \end{bmatrix}, \begin{bmatrix} 1 \\ \vdots \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} } \right). \end{align}\] By the above operations of matrices, we see that there exists an isomorphism \(\Psi: \operatorname{BF}({\Bar{A}}^t) \to \operatorname{BF}({A}^t)\) satisfying 2 . Since all the operations except the final exchange of two rows preserve the determinant, and the final exchange changes its sign, the determinant formula in 2 follows. ◻
Therefore we have the following lemma.
Lemma 3. For an irreducible non-permutation matrix \(A=[A(i,j)]_{i,j=1}^N\) with entries in \(\{0,1\}\), let \(\Bar{\Bar{A}}\) be the \((N+6)\times (N+6)\) matrix \(({\Bar{A}})^{\bar{}}\) defined by \[\begin{align} \Bar{\Bar{A}} = & {\small \setcounter{MaxMatrixCols}{15} \begin{bmatrix} A(1,1) &\cdots &A(1,N-1) &A(1,N) &0 &0 &0 &0 &0 &0\\ \vdots & &\vdots &\vdots &\vdots &\vdots&\vdots&\vdots &\vdots&\vdots \\ A(N-1,1) &\cdots &A(N-1,N-1)&A(N-1,N) &0 &0 & 0 &0 &0 &0\\ A(N,1) &\cdots &A(N,N-1) &A(N,N) &0 &1 &0 &0 &0 &0 \\ 0 &\cdots &0 &1 &0 &0 &0 &0 &0 &0 \\ 0 &\cdots &0 &0 &1 &0 &1 &0 &0 &0 \\ 0 &\cdots &0 &0 &0 &1 &1 &0 &1 &0 \\ 0 &\cdots &0 &0 &0 &0 &1 &0 &0 &0 \\ 0 &\cdots &0 &0 &0 &0 &0 &1 &0 &1 \\ 0 &\cdots &0 &0 &0 &0 &0 &0 &1 &1 \end{bmatrix} }\\ = & {\small \setcounter{MaxMatrixCols}{15} \begin{bmatrix} & & & &0 &0 &0 \\ & & & &\vdots &\vdots&\vdots \\ &\text{\huge{\Bar{A}}}& & &0 &0 &0 \\ & & & &0 &1 &0 \\ 0 &\cdots &0 &1 &0 &0 &0 \\ 0 &\cdots &0 &0 &1 &0 &1 \\ 0 &\cdots &0 &0 &0 &1 &1 \end{bmatrix} } \end{align}\] Then there exists an isomorphism \(\Phi: \operatorname{BF}({\Bar{\Bar{A}}}^t) \to \operatorname{BF}({A}^t)\) such that \[\Phi([1_{N+6}]) = [1_N] \quad \text{ and } \quad {{\operatorname{det}}}(I_{N+6}- {\Bar{\Bar{A}}}) = {{\operatorname{det}}}(I_N - A).\] Hence we have \((X_{\Bar{\Bar{A}}}, \sigma_{\Bar{\Bar{A}}}) \underset{{{\operatorname{COE}}}}{\sim} (X_A,\sigma_A).\)
We note that there exist nonzero diagonal entries of the above matrix \({\Bar{\Bar{A}}}\).
Proposition 1. For an irreducible non-permutation matrix \(A\) with entries in \(\{0,1\}\) and a positive integer \(n \in \mathbb{N}\), there exists an irreducible non-permutation matrix \(B\) with entries in \(\{0,1\}\) such that \[(X_A, \sigma_A) \underset{{{\operatorname{COE}}}}{\sim} (X_B,\sigma_B) \quad \text{ and }\quad \lambda_B \geq n\] where \(\lambda_B\) is the Perron–Frobenius eigenvalue of \(B\).
Proof. Let \(A=[A(i,j)]_{i,j=1}^N\) be an irreducible non-permutation matrix with entries in \(\{0,1\}\). By considering \(\Bar{\Bar{A}}\) in Lemma 3 instead of \(A\), we may assume that \(A(p,p) =1\) for some \(p =1,\dots,N\). Put the matrix \(\widetilde{A}:= A -I_N\), so that \(\widetilde{A}(p,p) =0.\) Take an arbitrary fixed positive integer \(n\). Since \(A\) is irreducible, there exists \(q\ne p\) such that \(A(p,q) =1\) and hence \(\widetilde{A}(p,q) =1\). Let \(\widetilde{A}^{\prime}\) be the \(N \times N\) matrix obtained from \(\widetilde{A}\) by adding \(n\) times \(p\)th row to the \(q\)th row. The matrix \(\widetilde{A}^{\prime}\) is of the form \[\widetilde{A}^{\prime} = P\cdot \widetilde{A}\] for the \(N\times N\) invertible matrix \(P=[P(i,j)]_{i,j=1}^N\in GL(N,\mathbb{Z})\) defined by \[P(i,j) = \begin{cases} 1 & \text{ if } i=j, \\ n & \text{ if } (i, j) = (q,p),\\ 0 & \text{ otherwise.} \end{cases}\] The matrix \(\widetilde{A}^{\prime} + I_N\) is an irreducible non-permutation matrix having its entries in nonnegative integers, because \((\widetilde{A}^{\prime} + I_N)(i,j) \ge A(i,j)\) for all \(i,j=1,\dots,N\). Define the matrix \(B\) to be the second higher block matrix \((\widetilde{A}^{\prime}+I_N)^{[2]}\) of \(\widetilde{A}^{\prime} +I_N\). Let \(M\) denote the size of the matrix \(B\), so that \(B\) is an \(M\times M\) irreducible non-permutation matrix with entries in \(\{0,1\}\). By using Lemma 1, we have an isomorphism \(\Phi: \operatorname{BF}(A^t) \to \operatorname{BF}(B^t)\) such that \(\Phi([1_N]) = [1_M]\). Since \(B\) is the second higher block matrix of \(\widetilde{A}^\prime + I_N\), we have \({{\operatorname{det}}}(I_M -B) ={{\operatorname{det}}}(I_N -(\widetilde{A}^\prime + I_N)).\) By noting \({{\operatorname{det}}}(P) = 1\), the equalities \[{{\operatorname{det}}}(I_N-A) ={{\operatorname{det}}}(-\widetilde{A}) = {{\operatorname{det}}}(-\widetilde{A}^\prime) = {{\operatorname{det}}}(I_N -(\widetilde{A}^\prime + I_N)) ={{\operatorname{det}}}(I_M - B)\] hold. We thus conclude that \((X_B, \sigma_B) \underset{{{\operatorname{COE}}}}{\sim} (X_A,\sigma_A).\) Let us denote by \(\lambda_B\) and \(\lambda_{\widetilde{A}^\prime + I_N}\) the Perron–Frobenius eigenvalues of the matrices \(B\) and \(\widetilde{A}^\prime + I_N\), respectively, so that \(\lambda_B = \lambda_{\widetilde{A}^\prime + I_N}\). As \((\widetilde{A}^{\prime} + I_N)(q,q) \ge n\), the subgraph with the single vertex \(q\) has at least \(n\) multiple loops, so that \(\lambda_{\widetilde{A}^{\prime} + I_N} \ge n\) and hence \(\lambda_B \ge n.\) We therefore end the proof of Proposition 1. ◻
It is well-known as Parry’s theorem [12] that the topological entropy \(h_{\operatorname{top}}(X_A, \sigma_A)\) of a topological Markov shift \((X_A,\sigma_A)\) is computed to be \(\log \lambda_A\) (cf. [9]). We thus obtain the following theorem.
Theorem 2. For an irreducible non-permutation matrix \(A\) with entries in \(\{0,1\}\), and \(R>0\), there exists an irreducible non-permutation matrix \(B\) with entries in \(\{0,1\}\) such that \[(X_A, \sigma_A) \underset{{{\operatorname{COE}}}}{\sim} (X_B,\sigma_B) \quad \text{ and }\quad h_{\operatorname{top}}(X_B, \sigma_B) >R.\]
In this section, we study lower entropies in a continuous orbit equivalence class. In what follows, we prove that a one-sided topological Markov shift \((X_A, \sigma_A)\) is continuously orbit equivalent to a topological Markov shift \((X_B, \sigma_B)\) whose topological entropy \(h_{{{\operatorname{top}}}}(X_B, \sigma_B)\) is less than an arbitrary prescribed positive real number. Throughout the section, let \(A=[A(i,j)]_{i,j=1}^N\) be an irreducible non-permutation matrix with entries in \(\{0,1\}\).
Lemma 4. For \(m \in \mathbb{N}\), define the \(Nm \times Nm\)-matrix \(A^{<m>}\) as the block matrix: \[\label{eq:matrixA9412360m62125} A^{<m>}: = \begin{bmatrix} 0 & I_N & 0 &\cdots& 0 \\ 0 & 0 & I_N &\ddots& \vdots \\ \vdots&\ddots&\ddots&\ddots& 0 \\ 0 &\cdots&0 &0 &I_N \\ A &0 &\cdots&0 & 0 \end{bmatrix},\tag{3}\] where \(0\)s in the above matrix denote the \(N\times N\) zero matrices. Then we have
\((\operatorname{BF}(A^{<m> t}), [1_{Nm}]) \cong (\operatorname{BF}(A^t), m [1_N]).\)
\({{\operatorname{det}}}(I_{Nm} - A^{<m>}) = {{\operatorname{det}}}(I_N - A).\)
\(\lambda_{A^{<m>}} = \lambda_A^{\frac{1}{m}}.\)
Proof. (i), (ii) We have the following sequence of elementary operations of matrices: \[\begin{align} & (I_{Nm}- A^{<m> t}, [1_{Nm}]) \\ = & \left( \begin{bmatrix} I_N & 0 &\cdots& 0& -A^t\\ -I_N & I_N & 0 &\cdots& 0 \\ 0 &\ddots&\ddots&\ddots& \vdots \\ \vdots&\ddots&-I_N &I_N &0 \\ 0 &\cdots& 0 &-I_N & I_N \end{bmatrix}, \, \begin{bmatrix} 1_N \\ 1_N \\ \vdots \\ \vdots \\ 1_N \\ \end{bmatrix} \right) \text{ (add mth column to (m-1)th column)} \\ \to & \left( \begin{bmatrix} I_N & 0 &\cdots&-A^t & -A^t\\ -I_N & I_N & 0 &\cdots& 0 \\ 0 &\ddots&\ddots&\ddots& \vdots \\ \vdots&\ddots&-I_N &I_N &0 \\ 0 &\cdots & 0 &0 & I_N \end{bmatrix}, \, \begin{bmatrix} 1_N \\ 1_N \\ \vdots \\ \vdots \\ 1_N \\ \end{bmatrix} \right) \text{ (add (m-1)th column to (m-2)th column)} \\ \to & \cdots \\ \to & \left( \begin{bmatrix} I_N-A^t&-A^t &\cdots&-A^t & -A^t\\ 0 & I_N & 0 &\cdots& 0 \\ 0 &\ddots&\ddots&\ddots& \vdots \\ \vdots&\ddots& 0 &I_N & 0 \\ 0 &\cdots& 0 &0 & I_N \end{bmatrix}, \, \begin{bmatrix} 1_N \\ 1_N \\ \vdots \\ \vdots \\ 1_N \\ \end{bmatrix} \right) \text{ (add A^t\times mth row to 1th row)}\\ \to & \left( \begin{bmatrix} I_N-A^t&-A^t &\cdots&-A^t & 0\\ 0 & I_N & 0 &\cdots& 0 \\ 0 &\ddots&\ddots&\ddots& \vdots \\ \vdots&\ddots& 0 &I_N & 0 \\ 0 &\cdots& 0 &0 & I_N \end{bmatrix}, \, \begin{bmatrix} 1_N + A^t1_N\\ 1_N \\ \vdots \\ \vdots \\ 1_N \\ \end{bmatrix} \right) \text{ (add A^t\times (m-1)th row to 1th row)}\\ \to & \cdots \\ \to & \left( \begin{bmatrix} I_N-A^t&0 &\cdots&0 & 0\\ 0 & I_N & 0 &\cdots& 0 \\ 0 &\ddots&\ddots&\ddots& \vdots \\ \vdots&\ddots& 0 &I_N & 0 \\ 0 &\cdots& 0 &0 & I_N \end{bmatrix}, \, \begin{bmatrix} 1_N +(m-1)A^t 1_N\\ 1_N \\ \vdots \\ \vdots \\ 1_N \\ \end{bmatrix} \right). \end{align}\] Hence we have \[\begin{align} (\operatorname{BF}(A^{<m> t}), [1_{Nm}]) \cong (\operatorname{BF}(A^t), [1_N + (m-1)A^t 1_N]) \cong (\operatorname{BF}(A^t), m [1_N]). \end{align}\] The above matrix operations also shows that \({{\operatorname{det}}}(I_{Nm} - A^{<m>}) = {{\operatorname{det}}}(I_N - A).\)
(iii) Let \(\lambda_{A^{<m>}}\) be the Perron–Frobenius eigenvalue of \(A^{<m>}\). One may find a sequence of positive vectors \(v_i\in {\mathbb{R}}^N,\, i=1,\dots,m\) such that \[A^{<m>} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{bmatrix} =\lambda_{A^{<m>}} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{bmatrix} \quad \text{ so that } \quad \begin{bmatrix} v_2 \\ v_3 \\ \vdots \\ v_m\\ Av_1 \end{bmatrix} = \begin{bmatrix} \lambda_{A^{<m>}} v_1 \\ \lambda_{A^{<m>}} v_2 \\ \vdots \\ \lambda_{A^{<m>}} v_{m-1} \\ \lambda_{A^{<m>}} v_m \end{bmatrix}.\] This shows that \[Av_1 = (\lambda_{A^{<m>}})^m v_1.\] Hence \(v_1\) is a positive eigenvector of \(A\) for the positive eigenvalue \((\lambda_{A^{<m>}})^m\) which is the Perron-Frobenius eigenvalue \(\lambda_A\) for the matrix \(A\), because of the uniqueness of the Perron-Frobenius eigenvalue. ◻
By using the above lemma, we have the following proposition.
Proposition 3. Let \(A=[A(i,j)]_{i,j=1}^N\) be an irreducible non-permutation matrix with entries in \(\{0,1\}\). Assume that \([1_N]\) is a torsion element in the quotient group \(\operatorname{BF}(A^t)\). Then there exists a sequence \(A_n, \, n \in \mathbb{N}\) of irreducible non-permutation matrices with entries in \(\{0,1\}\) such that \[\label{eq:theorem2} \begin{cases} \bullet & (X_A,\sigma_A) \underset{{{\operatorname{COE}}}}{\sim} (X_{A_n}, \sigma_{A_n}), \quad n \in \mathbb{N}, \\ \bullet & \lambda_{A_n} \downarrow 1. \end{cases}\qquad{(1)}\]
Proof. Assume that \([1_N]\) is a torsion element in the quotient group \(\operatorname{BF}(A^t)\). Hence one may find \(\ell \in \mathbb{N}\) such that \(\ell [1_N] =0\) in \(\mathbb{Z}^N/( I_N - A^t) \mathbb{Z}^N\). For any \(n \in \mathbb{N}\), put \(m_n = 1 + n \ell\), so that we have \[m_n[1_N] = [1_N] + n \ell[1_N] = [1_N].\] Lemma 4 tells us that \[(\operatorname{BF}(A^{<m_n> t}), [1_{Nm_n}]) \cong (\operatorname{BF}(A^t), m_n [1_N]) \cong (\operatorname{BF}(A^t), [1_N]).\] Since \({{\operatorname{det}}}(I_{Nm_n} - A^{<m_n>}) = {{\operatorname{det}}}(I_N - A),\) by putting \(A_n := A^{<m_n>}\), we have \[(X_A,\sigma_A) \underset{{{\operatorname{COE}}}}{\sim} (X_{A_n}, \sigma_{A_n}), \quad n \in \mathbb{N}.\] Since \(\lambda_{A^{<m_n>}} = \lambda_A^{\frac{1}{m_n}},\) we get \(\lambda_{A_n} \downarrow 1.\) ◻
The above proof does not work in case that \([1_N]\) is non-torsion in \(\operatorname{BF}(A^t)\), so we have to provide several lemmas to treat general cases. We use the operation in matrices called the Parry-Sullivan move ([14]). Let \(A=[A(i,j)]_{i,j=1}^N\) be an irreducible non-permutation matrix with entries in \(\{0,1\}\). Take an arbitrary fixed \(p \in \{1,\dots,N\}\), and put the new vertex set \(\Sigma_{[p]} = \{1,\dots, N \} \cup \{p'\}\). Define the matrix \(A_{[p]} =[A_{[p]}(i,j)]_{i,j \in \Sigma_{[p]}}\) over \(\Sigma_{[p]}\) by setting \[A_{[p]}(i,j) = \begin{cases} A(i,j) & \text{ for } i\ne p, p' \text{ and } j\ne p',\\ A(i,j) & \text{ for } i=p' \text{ and } j\ne p', \\ 1 & \text{ for } i=p \text{ and } j=p', \\ 0 & \text{ for } i=p \text{ and } j\ne p', \\ 0 & \text{ for } i \ne p \text{ and } j =p', \end{cases}\] so that \[A_{[p]} = \begin{bmatrix} A(1,1) &\cdots & A(1,p) & 0 & A(1,p+1) &\cdots & A(1,N) \\ \vdots & & \vdots &\vdots&\vdots & & \vdots \\ A(p-1,1)&\cdots &A(p-1,p)& 0 &A(p-1,p+1)&\cdots &A(p-1,N) \\ 0 &\cdots &0 & 1 &0 &\cdots &0 \\ A(p,1) &\cdots &A(p,p) & 0 &A(p,p+1) &\cdots &A(p,N) \\ \vdots & & \vdots &\vdots&\vdots & & \vdots \\ A(N,1) &\cdots & A(N,p) & 0 & A(N,p+1) &\cdots & A(N,N) \end{bmatrix}.\]
Lemma 5. \(\lambda_{A_{[p]}} < \lambda_A\).
Proof. Let \(v=[v_i]_{i=1}^N\) be a positive eigenvector for the Perron-Frobenius eigenvalue \(\lambda_A\) so that \(Av =\lambda_Av\). We then have the equality: \[{ \begin{bmatrix} A(1,1) &\cdots & A(1,p) & 0 & A(1,p+1) &\cdots & A(1,N) \\ \vdots & & \vdots &\vdots &\vdots & & \vdots \\ A(p-1,1)&\cdots &A(p-1,p)& 0 &A(p-1,p+1)&\cdots &A(p-1,N) \\ 0 &\cdots &0 & 1 &0 &\cdots &0 \\ A(p,1) &\cdots &A(p,p) &\lambda_A-1&A(p,p+1) &\cdots &A(p,N) \\ A(p+1,1)&\cdots &A(p+1,p)&0 &A(p+1,p+1)&\cdots &A(p+1,N)\\ \vdots & & \vdots &\vdots &\vdots & & \vdots \\ A(N,1) &\cdots & A(N,p) & 0 & A(N,p+1) &\cdots & A(N,N) \end{bmatrix} \begin{bmatrix} v_1 \\ \vdots \\ v_{p-1} \\ v_p \\ \lambda_A v_{p} \\ v_{p+1}\\ \vdots \\ v_{N} \end{bmatrix} = \lambda_A \begin{bmatrix} v_1 \\ \vdots \\ v_{p-1} \\ v_p \\ \lambda_Av_{p} \\ v_{p+1}\\ \vdots \\ v_{N} \end{bmatrix}. }\] Let us denote by \(A'_{[p]}\) the above \((N+1)\times (N+1)\) matrix in the left hand side. It differs from \(A_{[p]}\) only at \((p+1,p+1)\)-component. We note that both of the matrices \(A_{[p]}\) and \(A'_{[p]}\) are irreducible, because so is the matrix \(A\). By the above equality, we see that \(\lambda_A\) is the Perron-Frobenius eigenvalue of \(A'_{[p]}\), because the vector in the right-hand is a positive vector. As \(0< \lambda_A -1\) and hence \(A_{[p]}< A'_{[p]}\), we know that \(\lambda_{A_{[p]}} < \lambda_{A'_{[p]}} =\lambda_A\) by [15]. ◻
It is clear that \({{\operatorname{det}}}(I_{N+1}-A_{[p]})={{\operatorname{det}}}(I_N-A)\).
The proof of the following lemma is essentially seen in the proof of [7], so we omit its proof.
Lemma 6. The map \(\eta: \mathbb{Z}^{N+1}\to \mathbb{Z}^N\) defined by \[\eta([x_1,\dots, x_{p-1}, x_p, x_{p'}, x_{p+1},\dots, x_N]) =[x_1,\dots, x_{p-1}, x_{p}+x_{p'}, x_{p+1}, x_{p+2}, \dots, x_N])\] induces an isomorphism \(\bar{\eta}: \operatorname{BF}(A_{[p]}^t) \to \operatorname{BF}(A^t)\) so that \[\eta([1_{N+1}]) =[(\overbrace{1,\dots,1}^{\text{p-1 times}}, 2,\overbrace{1,\dots,1}^{\text{N-p times}})]\]
By using the above lemma repeatedly together with Lemma 4 and Lemma 5, we have the following lemma.
::: {#lem:A_{[p],k} .lemma} Lemma 7. For any \(p \in \{1,\dots,N\}\) and \(k\in {\mathbb{Z}}_+\), there exists an \((N+k)\times (N+k)\) matrix \(A_{[p],k}\) with entries in \(\{0,1\}\) such that there exists an isomorphism \(\bar{\eta}: \operatorname{BF}(A_{[p],k}^t) \to \operatorname{BF}(A^t)\) such that \[\begin{gather} \bar{\eta}([1_{N+k}]) =[(\overbrace{1,\dots,1}^{\text{p-1 times}}, k+1,\overbrace{1,\dots,1}^{\text{N-p times}})], \\ {{\operatorname{det}}}(I_{N+k} - A_{[p],k}) = {{\operatorname{det}}}(I_N - A), \qquad 0< \lambda_{A_{[p],k}} < \lambda_A. \end{gather}\] :::
We provide one more lemma.
Lemma 8. Let \(e_i \in \mathbb{Z}^N\) be the vector whose \(i\)th component is one, and the other components are zeros. Let \([e_i]\) be the class of \(e_i\) in \(\operatorname{BF}(A^t)\). Then for any element \(u \in \operatorname{BF}(A^t)\), there exist positive integers \(m_1, \dots, m_N \in \mathbb{N}\) such that \[\label{eq:um951m95n} u = m_1[e_1] + \cdots +m_N [e_N].\tag{4}\]
Proof. Let \(p\) be the period of the matrix \(A\). Then \(A^p\) is permutation-similar to a direct sum of \(p\) primitive matrices, say \(B_1\oplus B_2 \oplus \cdots \oplus B_p\). There exists \(n\in \mathbb{N}\) such that every entry of \(B_i^n\) is not less than \(2\) for all \(i = 1,2,\dots,p\). For \(e := e_1 + e_2 +\cdots+ e_N,\) we have \[0 =[({(A^t)}^{pn}- I_N)e] =[{(A^t)}^{pn} e -e ] \quad \text{ in } \operatorname{BF}(A^t),\] and the vector \({(A^t)}^{pn} e -e\) is a linear combination of \({e_i}^\prime\) s with positive coefficients. Hence for any \(a_1 e_1 + a_2 e_2 +\cdots+ a_N e_N \in \mathbb{Z}^N\), by taking large enough \(k \in \mathbb{N}\) and adding \(k({(A^t)}^{pn}e -e)\), we may find a vector in \(\mathbb{N}^N\) within the same equivalence class in \(\operatorname{BF}(A^t)\). ◻
We thus have the following proposition.
Proposition 4. Let \(A=[A(i,j)]_{i,j=1}^N\) be an irreducible non-permutation matrix with entries in \(\{0,1\}\). For any positive real number \(\epsilon\), there exists an irreducible non-permutation matrix \(B\) with entries in \(\{0,1\}\) such that \[(X_A,\sigma_A) \underset{{{\operatorname{COE}}}}{\sim} (X_B, \sigma_B) \quad \text{ and } \quad 1 < \lambda_B <1+\epsilon.\]
Proof. For the matrix \(A=[A(i,j)]_{i,j=1}^N\), there exists an irreducible non-permutation matrix \(A_\circ=[A_\circ(i,j)]_{i,j=1}^{N_\circ}\) with entries in \(\{0,1\}\) such that \[\operatorname{BF}(A_\circ^t) \cong \operatorname{BF}(A^t), \quad {{\operatorname{det}}}(I_{N_\circ}-A_{\circ}) = {{\operatorname{det}}}(I_N - A), \quad [1_{N_\circ}] =0 \text{ in } \operatorname{BF}(A_\circ^t)\] (cf. [6]). For any \(\epsilon>0\), by Lemma 4, one may find \(m \in \mathbb{N}\) such that \[(X_{A_\circ^{<m>}}, \sigma_{A_\circ^{<m>}}) \underset{{{\operatorname{COE}}}}{\sim} (X_{A_\circ}, \sigma_{A_\circ}), \qquad 1< \lambda_{A_\circ^{<m>}} <1+\epsilon,\] because \(m [1_{N_\circ}] = 0.\) Put \(N_1 = m N_\circ\). As \(\operatorname{BF}(A_\circ^{<m> t}) \cong \operatorname{BF}(A^t)\) Lemma 8 tells us that there exist \(m_1,\dots, m_{N_1} \in \mathbb{N}\) such that \[(\operatorname{BF}(A_\circ^{<m> t}), m_1[e_1] + \cdots + m_{N_1} [e_{N_1}]) \cong (\operatorname{BF}(A^t), [1_N]).\] By using Lemma 7 at each \(i=1,\dots, N_1\), one may find an irreducible non-permutation matrix \(B=[B(i,j)]_{i,j=1}^M\) with entries in \(\{0,1\}\) such that \[(\operatorname{BF}(B^t), [1_M]) \cong (\operatorname{BF}(A^t), [1_N]), \quad {{\operatorname{det}}}( I_M - B) = {{\operatorname{det}}}(I_N - A), \quad 1<\lambda_B < \lambda_{A_\circ^{<m>}},\] so that \((X_B, \sigma_B) \underset{{{\operatorname{COE}}}}{\sim} (X_A, \sigma_A)\) and \(1< \lambda_B <1+\epsilon.\) ◻
We thus reach the following theorem.
Theorem 5. Let \(A\) be an irreducible non-permutation matrix with entries in \(\{0,1\}\). For any positive real number \(r\), there exists an irreducible non-permutation matrix \(B\) with entries in \(\{0,1\}\) such that \[(X_A,\sigma_A) \underset{{{\operatorname{COE}}}}{\sim} (X_B, \sigma_B) \quad \text{ and } \quad h_{{{\operatorname{top}}}}(X_B,\sigma_B) <r.\]
Let \(A =[A(i,j)]_{i,j=1}^N\) be the full \(N\)-shift with \(N >1\) whose entries \(A(i,j), \, i,j=1,\dots,N\) are all \(1\)s.
1. Upper entropy for full shifts.
Following the proof of Proposition 1 in Section 2, we concretely construct a one-sided topological Markov shift \((X_{B},\sigma_{B})\) such that \((X_A,\sigma_A) \underset{{{\operatorname{COE}}}}{\sim} (X_{B}, \sigma_{B})\) and \(h_{{{\operatorname{top}}}}(X_B,\sigma_B)\) is greater than a prescribed integer. Let \(m\) be an arbitrary fixed positive integer. Let \(p =N\) and \(q = N-1\), Put the matrix \(\widetilde{A}= A -I_N\). Let \(\widetilde{A}^{\prime}\) be the \(N \times N\) matrix obtained from \(\widetilde{A}\) by adding \(m\) times \(p\)th row to the \(q\)th row. The matrix \(x I_N - (\widetilde{A}^{\prime}+I_N)\) is of the form \[x I_N - (\widetilde{A}^{\prime} + I_N) = {\small \begin{bmatrix} x-1 &-1 &-1 & \dots & -1 &-1 \\ -1 & x-1 &-1 & \dots & -1 &-1 \\ \vdots&\ddots& \ddots&\ddots &\vdots&\vdots \\ -1 &\cdots&-1 &x-1 &-1 &-1 \\ -(m+1)&\cdots&\cdots&-(m+1)&x-(m+1)&-1 \\ -1 &\cdots&\cdots &-1 &-1 &x-1 \end{bmatrix}. }\] It is a direct computation to show that \[{{\operatorname{det}}}(xI_N - (\widetilde{A}^{\prime} + I_N)) = x^{N-2}\{x^2 -(m+N)x + m\}\] and hence \[\lambda_{\widetilde{A}^{\prime} + I_N} = \frac{1}{2}\{ m+N + \sqrt{(m+N)^2 -4m} \}.\] Let \(B\) be the second higher block matrix \((\widetilde{A}^{\prime}+I_N)^{[2]}\) of \(\widetilde{A}^{\prime} +I_N\). As \(\lambda_B = \lambda_{\widetilde{A}^{\prime} + I_N}\) the one-sided topological Markov shift \((X_{B},\sigma_{B})\) is continuously orbit equivalent to the full \(N\)-shift \((X_A,\sigma_A)\) such that \(\lambda_B> m + N-1\).
2. Lower entropy for full shifts.
Let \(n\) be an arbitrary fixed positive integer. Put \(m_n = 1 + n(N-1).\) Let \(A^{<m_n>}\) be the matrix defined by 3 for the full \(N\)-shift \(A\) and the positive integer \(m_n\). It is a direct computation to show that \[{{\operatorname{det}}}(xI_{Nm_n} - A^{<m_n>}) =x^{Nm_n -m_n}(x^{m_n} - N)\] and hence \[\lambda_{A^{<m_n>}} = N^{\frac{1}{m_n}}\] Put \(B = A^{<m_n>}\). The one-sided topological Markov shift \((X_{B},\sigma_{B})\) is continuously orbit equivalent to the full \(N\)-shift \((X_A,\sigma_A)\) such that \(\lambda_B =N^{\frac{1}{m_n}}\).
Acknowledgments: K. Matsumoto is supported by JSPS KAKENHI Grant Number 24K06775. H. Matui is supported by JSPS KAKENHI Grant Number 23K22397.