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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 [email protected]
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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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References

Bowen, R.. On Axiom A Diffeomorphisms (Regional Conference Series in Mathematics, 35) . American Mathematical Society, Providence, RI, 1978.Google Scholar
Chung, N.-P. and Lee, K.. Topological stability and pseudo-orbit tracing property of group actions. Proc. Amer. Math. Soc., to appear, http://www.ams.org/journals/proc/0000-000-00/S0002-9939-2017-13654-5/S0002-9939-2017-13654-5.pdf.Google Scholar
Chung, N.-P. and Li, H.. Homoclinic groups, IE groups, and expansive algebraic actions. Invent. Math. 199(3) (2015), 805–858.Google Scholar
Danilenko, A. I.. Entropy theory from the orbital point of view. Monatsh. Math. 134(2) (2001), 121–141.Google Scholar
Frisch, J. and Tamuz, O.. Symbolic dynamics on amenable groups: the entropy of generic shifts. Ergod. Th. & Dynam. Sys. 37(4) (2017), 1187–1210.Google Scholar
Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9(4) (1989), 691–735.Google Scholar
Lind, D. and Schmidt, K.. Homoclinic points of algebraic Z d -actions. J. Amer. Math. Soc. 12(4) (1999), 953–980.Google Scholar
Oprocha, P.. Shadowing in multi-dimensional shift spaces. Colloq. Math. 110(2) (2008), 451–460.Google Scholar
Ornstein, D. S. and Weiss, B.. Every transformation is bilaterally deterministic. Israel J. Math. 21(2–3) (1975), 154–158.Google Scholar
Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128) . Birkhäuser, Basel, 1995.Google Scholar
Schmidt, K.. The cohomology of higher-dimensional shifts of finite type. Pacific J. Math. 170(1) (1995), 237–269.Google Scholar
Schmidt, K.. Representations of toral automorphisms. Topology Appl. 205 (2016), 88–116.Google Scholar
Walters, P.. On the pseudo-orbit tracing property and its relationship to stability. The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State University, Fargo, ND, 1977) (Lecture Notes in Mathematics, 668) . Springer, Berlin, 1978, pp. 231–244.Google Scholar