TY - JOUR

T1 - Pseudo-orbit tracing and algebraic actions of countable amenable groups

AU - Meyerovitch, T. O.M.

N1 - Funding Information:
Let us show that X has no off-diagonal asymptotic pairs. Suppose (x, y) ∈ X ×⋂X is an asymptotic pair. If follows that (x, y) satisfy (63) for some finite F ⊂ Γ. Because n Γn = {1} it follows that there exists n so that the map g ⤇→ gΓn is injective of F. Because (x, y) ∈ X × X ⊂ Xn+1 × Xn+1 it follows from Lemma 5.7 that x = y. □ Acknowledgements. I thank Nishant Chandgotia, Nhan-Phu Chung and Hanfeng Li for valuable comments on an early version of this paper, and the anonymous referee for valuable comments and, in particular, for detecting a minor error in an earlier version of the proof of Lemma 4.1. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement no. 333598 and from the Israel Science Foundation (grant no. 626/14).
Publisher Copyright:
© Cambridge University Press, 2018.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85070258582&partnerID=8YFLogxK

U2 - 10.1017/etds.2017.126

DO - 10.1017/etds.2017.126

M3 - Article

AN - SCOPUS:85070258582

VL - 39

SP - 2570

EP - 2591

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 9

ER -