Topological stability and pseudo-orbit tracing property of group actions

Chung, Nhan-Phu; Lee, Keonhee

doi:10.1090/proc/13654

Topological stability and pseudo-orbit tracing property of group actions
HTML articles powered by AMS MathViewer

by Nhan-Phu Chung and Keonhee Lee PDF

Proc. Amer. Math. Soc. 146 (2018), 1047-1057 Request permission

Abstract:

In this paper we extend the concept of topological stability from homeomorphisms to group actions on compact metric spaces and prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property, then it is topologicaly stable. This represents a group action version of P. Walter’s stability theorem [Lecture Notes in Math., vol. 668, Springer, 1978, pp. 231–244]. Moreover we give a class of group actions with topological stability or pseudo-orbit tracing property. In particular, we establish a characterization of subshifts of finite type over finitely generated groups in terms of the pseudo-orbit tracing property.

References

N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland Mathematical Library, vol. 52, North-Holland Publishing Co., Amsterdam, 1994. Recent advances. MR 1289410
Miklós Abért and Nikolay Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 5, 1657–1677. MR 2966663, DOI 10.4171/JEMS/344
Joseph Auslander, Eli Glasner, and Benjamin Weiss, On recurrence in zero dimensional flows, Forum Math. 19 (2007), 107–114.
Tullio Ceccherini-Silberstein and Michel Coornaert, Cellular automata and groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. MR 2683112, DOI 10.1007/978-3-642-14034-1
Nhan-Phu Chung and Guohua Zhang, Weak expansiveness for actions of sofic groups, J. Funct. Anal. 268 (2015), no. 11, 3534–3565. MR 3336733, DOI 10.1016/j.jfa.2014.12.013
Michel Coornaert and Athanase Papadopoulos, Symbolic dynamics and hyperbolic groups, Lecture Notes in Mathematics, vol. 1539, Springer-Verlag, Berlin, 1993. MR 1222644
Hu Huyi, Shi Enhui, and Zhenqi Jenny Wang, Some ergodic and rigidity properties of discrete Heisenberg group actions, preprint, arXiv:1405.1120.
Adrian Ioana, Cocycle superrigidity for profinite actions of property (T) groups, Duke Math. J. 157 (2011), no. 2, 337–367. MR 2783933, DOI 10.1215/00127094-2011-008
Keonhee Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations 262 (2017), no. 6, 3467–3487. MR 3592647, DOI 10.1016/j.jde.2016.04.029
Zbigniew Nitecki, On semi-stability for diffeomorphisms, Invent. Math. 14 (1971), 83–122. MR 293671, DOI 10.1007/BF01405359
Piotr Oprocha, Shadowing in multi-dimensional shift spaces, Colloq. Math. 110 (2008), no. 2, 451–460. MR 2353915, DOI 10.4064/cm110-2-8
Alexey V. Osipov and Sergey B. Tikhomirov, Shadowing for actions of some finitely generated groups, Dyn. Syst. 29 (2014), no. 3, 337–351. MR 3227777, DOI 10.1080/14689367.2014.902037
Sergei Yu. Pilyugin, Inverse shadowing in group actions, to appear in Dyn. Syst, http://dx.doi.org/10.1080/14689367.2016.1173651.
Sergei Yu. Pilyugin and Sergei B. Tikhomirov, Shadowing in actions of some abelian groups, Fund. Math. 179 (2003), no. 1, 83–96. MR 2028929, DOI 10.4064/fm179-1-7
Peter Walters, Anosov diffeomorphisms are topologically stable, Topology 9 (1970), 71–78. MR 254862, DOI 10.1016/0040-9383(70)90051-0
Peter Walters, On the pseudo-orbit tracing property and its relationship to stability, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977) Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 231–244. MR 518563