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A hierarchy of topological systems with completely positive entropy. (English) Zbl 1480.37028

The authors introduce a hierarchy of topological dynamical systems that lie between uniformly positive entropy (UPE) and topological completely positive entropy (CPE). The hierarchy basically consists of three levels. The authors present a local version of a result by T. Meyerovitch [Ergodic Theory Dyn. Syst. 39, No. 9, 2570–2591 (2019; Zbl 1431.37030)] by using the formalism of entropy pairs. Then, they prove that a sort of converse also holds.
Let \((X, T)\) be a \(G\)-topological dynamical system (\(G\)-TDS), where \(X\) is a compact metric space and \(T\) is a left \(G\)-action on \(X\) by homeomorphisms. Denote the set of entropy pairs by \(\mathrm{E}(X, T)\) and by \(\mathrm{A}^{\varepsilon}(X, T)\) the set of pairs \((x, y)\) for which \(d(gx, gy) > \varepsilon\) for finitely many \(g\in G\) and by \(\mathrm{A}(X,T)=\bigcap\limits_{\varepsilon>0}\mathrm{A}^{\varepsilon}(X, T)\) the set of asymptotic pairs. Let \(\Delta\) be the diagonal of \(X^2\). The authors prove the following claim.
Theorem. Let \((X, T)\) be a \(G\)-TDS with the pseudo-orbit tracing property.
(1) If \((x, y)\in \mathrm{E}(X, T)\), then \((x, y)\in\overline{\mathrm{A}^{\varepsilon}(X, T)}\setminus \Delta\) for every \(\varepsilon>0\) and there exists an invariant measure \(\mu\) such that \(x, y \in \mathrm{supp}(\mu)\).
(2) If \((x, y)\in \mathrm{A}(X, T)\) and there exists an invariant measure \(\mu\) such that \(x, y \in \mathrm{supp}(\mu)\), then \((x, y)\in \mathrm{E}(X, T)\cup \Delta\).
A new proof of a result by R. Pavlov [Ergodic Theory Dyn. Syst. 38, No. 5, 1894–1922 (2016; Zbl 1396.37010)] is given and this is extended to countable amenable groups. Further, the authors prove that there is a topologically weakly mixing \(\mathbb{Z}^{3}\)-SFT with topological CPE which does not have BCE. Some examples are given. Finally, they prove that for every countable ordinal \(\alpha\) the CPE class \(\alpha\) is non-empty. The asymptotic hierarchy of subshifts is studied. In particular, it is proved that there exists a \(\mathbb{Z}^{3}\)-SFT in the asymptotic class 3.

MSC:

37B40 Topological entropy
37B65 Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B02 Dynamics in general topological spaces
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References:

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