Pseudo-Orbit Tracing and Algebraic actions of countable amenable groups

Abstract

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li's algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

21 pages, some minor errors corrected

References in corpus (2)

• Symbolic dynamics on amenable groups: the entropy of generic shifts
• Topological stability and pseudo-orbit tracing property of group actions

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• The non-coexistence of distality and expansivity for group actions on infinite compacta
• Extracting invariant of conjugacy from independence entropy
• Scale pressure for amenable group actions