The topological entropy of stable sets for bi-orderable amenable groups
The authors deal with the topological entropy of stable sets for countable discrete infinite bi-orderable amenable group actions. Let \((X,d)\) be a compact metric space and bi-orderable group \(G\) continuously act on \(X\). Then \((X, G)\) is called a \(G\)-system. Let \(S\) be an infinite subset of \(G\). A pair \((x, y)\in X\times X\) is called an \(S\)-asymptotic pair if for each \(\varepsilon > 0\), there are only finitely many elements \(s\in S\) with \(d(sx, sy) >\varepsilon\). Let \((X, G)\) be a \(G\)-system, \(\mu\) be a \(G\)-invariant ergodic measure with \(h_{\mu}(G) > 0\) and \(\Phi\) the algebraic past of \(G\). The authors prove that if \(S\) is a \(\Phi\)-admissible subsemigroup of \(G\) and \(\cup_{n=1}^{\infty}F_n \subset (\Phi\cap\{e_G\})\), then for \(\mu\)-a.e. \(x\in X\), there exists a closed subset \(A(x)\subseteq W_S(x, G)\) such that \(h_{\text{top}}(A(x), {F_n})\geq h_{\mu}(G)\), where \(h_{\mu}(G)\) and \(h_{\text{top}}(A(x),{F_n})\) denote the measure-theoretic entropy of group \(G\) and the Bowen entropy of \(A(x)\) with respect to the two-sided Folner sequence \(\{Fn\}\) respectively. Then they prove the following: Theorem. Let \((X, G)\) be a \(G\)-system and \(\mu\) be a \(G\)-invariant ergodic measure with \(h_{\mu}(G) > 0\). If \(S\) is an infinite subset of \(G\) with \(|\{s\in S : s <_{\Phi} g\}| <\infty\) for each \(g\in G\), then for \(\mu\)-a.e. \(x\in X\), there exists a closed subset \(E(x)\subseteq \overline{W_S(x, G)}\) such that (1) \(E(x)\) is a weakly mixing set; (2) \(h_{\text{top}}(A(x), {F_n})\geq h_{\mu}(G)\). Let \(G\) be a finitely generated torsion-free nilpotent group. It is proved that there exist algebraic \(\Phi\) and \(\Phi\)-admissible subsemigroup \(S\) of \(G\) such that \(|\{s\in S : s <_{\Phi} g\}| <\infty\) for each \(g\in G\).
scientific article; zbMATH DE number 7348196
scientific article