Abstract
We study several properties of expansive group actions on metric spaces and obtain some relations between the expansivity for subgroup and group actions. Through counter examples, the necessity of hypothesis is justified. We also study the expansivity of covering maps. Further, we define orbit expansivity for group actions on topological spaces and use it to characterize expansive actions.
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References
Aoki, N., Hiraide, K.: Topological Theory of Dynamical Systems. North-Holland, Mathematical Library, vol. 52. North-Holland Publishing Co., Amsterdam (1994)
Achigar, M., Artigue, A., Monteverde, I.: Expansive homeomorphisms on non-Hausdorff spaces. Topol. Appl. 207, 109–122 (2016)
Barzanouni, A.: Shadowing property on finitely generated group actions. J. Dyn. Syst. Geom. Theor. 12(1), 69–79 (2014)
Bryant, B.F.: Expansive self-homeomorphisms of a compact metric space. Am. Math. Monthly 69(5), 386–391 (1962)
Chung, N.P., Lee, K.: Topological stability and pseudo-orbit tracing property of group actions. Proc. Am. Math. Soc. 146(3), 1047–1057 (2018)
Das, R.: Expansive self-homeomorphisms on G-spaces. Period. Math. Hungar. 31(2), 123–130 (1995)
Das, T., Lee, K., Richeson, D., Wiseman, J.: Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces. Topol. Appl. 160(1), 149–158 (2013)
Hurder, S.: Dynamics of expansive group actions on the circle. Preprint
Kato, H.: Continuum-wise expansive homeomorphisms. Can. J. Math. 45(3), 576–598 (1993)
Meddaugh, J., Raines, B.: The structure of limit sets for \(\mathbb Z^{d}\)-actions. Discrete Contin. Dyn. Syst. 34(11), 4765–4780 (2014)
Osipov, A.V., Tikhomirov, S.B.: Shadowing for actions of some finitely generated groups. Dyn. Syst. 29(3), 337–351 (2014)
Oprocha, P.: Chain recurrence in multidimensional time discrete dynamical systems. Discrete Contin. Dyn. Syst. 20(4), 1039–1056 (2008)
Pacificio, M.J., Vieitez, J.L.: On measure expansive diffeomorphisms. Proc. Am. Math. Soc. 143(2), 811–819 (2015)
Shah, S., Das, R.: On collective sensitivity for \(\mathbb Z^{d}\)-actions. Dyn. Syst. 31(2), 221–227 (2016)
Shi, E., Sun, B.: Fixed point properties of nilpotent group actions on 1-arcwise connected continua. Proc. Am. Math. Soc. 137(2), 771–775 (2009)
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Barzanouni, A., Divandar, M.S. & Shah, E. On Properties of Expansive Group Actions. Acta Math Vietnam 44, 923–934 (2019). https://doi.org/10.1007/s40306-019-00330-9
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DOI: https://doi.org/10.1007/s40306-019-00330-9