Rescaling of Markov shifts. (English) Zbl 0866.22006

For any compact metric space \(A\) and any positive integer \(d\) let \(A^{\mathbb Z^d}\) denote as usual the space of sequences \((a_{\mathbf n})_{\mathbf n\in \mathbb Z^d}\) of elements from \(A\) with indices in \(\mathbb Z^d\) (endowed with the product topology). Let \(\sigma:\mathbb Z^d\times A^{\mathbb Z^d}\rightarrow A^{\mathbb Z^d}\) denote the shift action of \(\mathbb Z^d\) on \(A^{\mathbb Z^d}\), that is, \(\sigma \left({\mathbf m}, (a_{\mathbf n})_{\mathbf n\in\mathbb Z^d}\right)=(a_{\mathbf n+\mathbf m})_{\mathbf n\in\mathbb Z^d}\). A closed subset \(\Sigma_{(F,P)}\subset A^{\mathbb Z^d}\) is called a Markov shift if there are a finite set \(F\subset\mathbb Z^d\) and a set \(P\subset A^F\) of finite sequences indexed by \(F\) such that \((a_{\mathbf n})_{\mathbf n\in\mathbb Z^d}\in \Sigma_{(F,P)}\) if and only if \((a_{\mathbf n+\mathbf m})_{\mathbf n\in F}\in P\) for any \(\mathbf m\in\mathbb Z^d\). The restriction of \(\sigma\), \(\sigma^{(F,P)}:\mathbb Z^d\times\Sigma_{(F,P)} \rightarrow \Sigma_{(F,P)}\), is then well defined and induces a discrete dynamical system \((\Sigma_{(F,P)},\sigma^{(F,P)})\) (which we simply denote again by \(\Sigma_{(F,P)}\)).
The present paper deals with so-called rescalings of Markov shifts. Namely, let \(M\) be a \(d\times d\) integer matrix with \(\det M\neq 0\). With the above notation, write \(M(F)=\{{\mathbf n}M: \mathbf n\in F\}\) and define \(M(P)\subset A^{M(F)}\) as follows: \((a_{\mathbf m})_{\mathbf m\in M(F)}\in M(P)\) if and only if \((a_{{\mathbf m}M^{-1}})_{\mathbf m\in M(F)}\in P\). Then \(\Sigma_{(M(F),M(P))}\) is called the \(M\)-rescaling of the Markov shift \(\Sigma_{(F,P)}\).
While \(\Sigma_{(F,P)}\) and \(\Sigma_{(M(F),M(P))}\) need not be topologically conjugate, it is shown in the paper that they always have the same topological entropy. Some examples (particularly from the theory of group automorphisms) are brought into consideration.
Markov shifts which are invariant under rescaling (that is, which are topologically conjugated to their \(M\)-rescaling for any matrix \(M\)) are also briefly studied. For example it is shown that if a Markov shift has \(s\) fixed points and is invariant under rescaling then its entropy is greater than or equal to \(\log s\).


37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B40 Topological entropy
22D40 Ergodic theory on groups
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37E99 Low-dimensional dynamical systems
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