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CONTINUOUS SHADOWING AND STABILITY FOR GROUP ACTIONS

Kim, Sang Jin

  • Received : 2018.01.17
  • Accepted : 2018.10.12
  • Published : 2019.01.01

Abstract

Recently, Chung and Lee [2] introduced the notion of topological stability for a finitely generated group action, and proved a group action version of the Walters's stability theorem. In this paper, we introduce the concepts of continuous shadowing and continuous inverse shadowing of a finitely generated group action on a compact metric space X with respect to various classes of admissible pseudo orbits and study the relationships between topological stability and continuous shadowing and continuous inverse shadowing property of group actions. Moreover, we introduce the notion of structural stability for a finitely generated group action, and we prove that an expansive action on a compact manifold is structurally stable if and only if it is continuous inverse shadowing.

Keywords

continuous shadowing;expansiveness;group action;inverse shadowing;structural stability;topological stability

References

  1. T. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  2. N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc. 146 (2018), no. 3, 1047-1057.
  3. R. M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl. 189 (1995), no. 2, 409-423. https://doi.org/10.1006/jmaa.1995.1027
  4. P. E. Kloeden and J. Ombach, Hyperbolic homeomorphisms and bishadowing, Ann. Polon. Math. 65 (1997), no. 2, 171-177. https://doi.org/10.4064/ap-65-2-171-177
  5. K. Lee, Continuous inverse shadowing and hyperbolicity, Bull. Austral. Math. Soc. 67 (2003), no. 1, 15-26. https://doi.org/10.1017/S0004972700033487
  6. K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 533-540. https://doi.org/10.3934/dcds.2005.13.533
  7. J. Lewowicz and M. Cerminara, Some open problems concerning expansive systems, Rend. Istit. Mat. Univ. Trieste 42 (2010), 129-141.
  8. A. V. Osipov and S. B. Tikhomirov, Shadowing for actions of some finitely generated groups, Dyn. Syst. 29 (2014), no. 3, 337-351. https://doi.org/10.1080/14689367.2014.902037
  9. S. Yu. Pilyugin, Inverse shadowing in group actions, Dyn. Syst. 32 (2017), no. 2, 198-210. https://doi.org/10.1080/14689367.2016.1173651
  10. C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), no. 3, 425-437. https://doi.org/10.1216/RMJ-1977-7-3-425
  11. P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 231-244, Lecture Notes in Math., 668, Springer, Berlin, 1978.
  12. K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J. 79 (1980), 145-149. https://doi.org/10.1017/S0027763000018997

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)