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Pseudo-orbit tracing and algebraic actions of countable amenable groups

Published online by Cambridge University Press:  24 January 2018

TOM MEYEROVITCH
TOM MEYEROVITCH*
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva 8410501, Israel email mtom@math.bgu.ac.il
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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 that they do. 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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 9 , September 2019 , pp. 2570 - 2591
Copyright
© Cambridge University Press, 2018 

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