Countable compacta admitting homeomorphisms with positive sequence entropy
In a topological dynamical system \((X,T)\) given by a compact metric space \(X\) and a homeomorphism \(T: X\to X\), let \(\Omega (T)\) be the set of non-wandering points and \(h_{top}^s(T)\) be the topological sequence entropy, i.e., \(h_{top}^s(T) = \sup_A h_{top}^A(T)\) where \(h_{top}^A(T)\) is the topological sequence entropy with respect to the sequence \(A\). While the topological entropy equals the topological entropy on the set of non-wandering points, the authors give an example of a countable compact metric space \(X\) and a homeomorphism \(T: X\to X\) such that \(h_{top}^s(T) \neq h_{top}^s(T|_{\Omega (T)})\). Further, for a countable compact metric space \(X\) let \(d(X)\) be the derived degree of \(X\), i.e., the countable ordinal number \(\alpha\) such that for the derived sets of \(X\) of orders \(\alpha\) and \(\alpha +1\) we have \(X^{(\alpha)} \neq 0\) and \(X^{(\alpha +1)} = 0\), respectively. In the paper under review it is shown that \(h_{top}^s(T)=0\) whenever \(d(X)\leq 1\). On the other hand, if \(d(X)\geq 2\) then there exists a homeomorphism \(T\) on \(X\) such that the set \(X^{(1)}\) of all limit points of \(X\) is the so-called sequence entropy set of \((X,T)\).
scientific article; zbMATH DE number 5375869
scientific article