Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

Year 2021, , 934 - 948, 06.08.2021

Zahra Shabani Ali Barzanouni Xinxing Wu

https://doi.org/10.15672/hujms.784081

Abstract

We introduce various new type of recurrent sets for finitely generated semigroups on non-compact metric spaces that are conjugacy invariant, and obtain some basic properties of chain recurrent sets for semigroups via these new definitions. Moreover, we define the notion of weak shadowing property for finitely generated group actions on compact metric spaces, which is weaker than that of shadowing property, and prove the equivalence of the shadowing and weak shadowing properties for the finitely generated group actions on a generalized homogeneous space without isolated points.

Keywords

Group actions, Recurrent sets, shadowing property, weak shadowing property

Supporting Institution

National Natural Science Foundation of China

Project Number

11601449

References

• [1] S.A. Ahmadi, X. Wu, and G. Chen, Topological chain and shadowing properties of dynamical systems on uniform spaces, Topology Appl. 275, 107153, 2020.
• [2] J. Ahn, K. Lee, and S. Lee, Persistent actions on compact metric spaces, J. Chungcheong Math. Soc. 30 (1), 61–66, 2017.
• [3] D.V. Anosov, On a certain class of invariant sets of smooth dynamical systems, Proc. 5th Int. Conf. on Non-Linear Oscillations, 2, 39–45, 1970.
• [4] A. Artigue, Lipschitz perturbations of expansive systems, Discrete Contin. Dyn. Syst. 35 (5), 1829–1841, 2015.
• [5] A.Z. Bahabadi, Shadowing and average shadowing properties for iterated function systems, Georgian Math. J. 22 (2) 179–184, 2015.
• [6] R. Bowen, Equilibrium states and ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.
• [7] M. Carvakho, F.B. Rodrigues, and P. Varandas, A variational principle for free semigroup actions, Adv. Math. 334, 450–487, 2018.
• [8] S.K. Choi, C. Chu, and K. Lee, Recurrence in persistent dynamical systems, Bull. Aust. Math. Soc. 43 (3), 509–517, 1991.
• [9] N. Chung and K. Lee, Topological stability and pseudo orbit tracing property of group actions, Proc. Amer. Math. Soc. 146 (3), 1047–1057, 2018.
• [10] Z. Ding, J. Yin, and X. Luo, The multi-transitivity of free semigroup actions, Stoch. Dyn. 20 (5), 2050040, 2020.
• [11] A. Fakhari and F.H. Ghane, On shadowing: ordinary and ergodic, J. Math. Anal. Appl. 364 (1), 151–155, 2010.
• [12] Y. Ju, D. Ma and Y. Wang, Topological entropy of free semigroup actions for noncompact sets, Discrete Contin. Dyn. Syst. 39 (2), 995–1017, 2019.
• [13] K. Lee, N. Nguyen, and Y. Yang, Topological stability and spectral decomposition for homeomorphisms on noncompact spaces, Discrete Contin. Dyn. Syst. 38 (5), 2487– 2503, 2018.
• [14] J. Lewowics, Persistence in expansive systems, Ergodic Theory Dynam. Systems, 3 (4), 567-578, 1983.
• [15] Y. Liang and C. Zhao, Rates of recurrence for free semigroup actions, J. Dyn. Control Syst. 27, 417–425, 2021.
• [16] M. Mazure, Weak shadowing for discrete dynamical systems on nonsmooth manifolds, J. Math. Anal. Appl. 281 (2), 657–662, 2003.
• [17] P. Oprocha and X.Wu, On averaged tracing of periodic average pseudo orbits, Discrete Contin. Dyn. Syst. 37 (9), 4943–4957, 2017.
• [18] A.V. Osipov and S.B. Tikhomirov, Shadowing for actions of some finitely generated groups, Dyn. Syst. 29 (3), 337–351, 2014.
• [19] S.Y. Pilyugin, A.A. Rodionova, and K. Sakai, Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst. 9 (2), 287–308, 2003.
• [20] K. Sakai, Diffeomorphisms with the average-shadowing property on two-dimensional closed manifolds, Rocky Mountain J. Math. 30 (3), 1129–1137, 2000.
• [21] Z. Shabani, Ergodic shadowing of semigroup actions, Bull. Iranian Math. Soc. 46 (2), 303–321, 2020.
• [22] X. Wu, Chaos of transformations induced onto the space of probability measures, Internat. J. Bifur. Chaos 26 (13), 1650227, 2016.
• [23] X. Wu, S. Liang, Y. Luo, M. Xin, and X. Zhang, A remark on the limit shadowing property for iterated function systems, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (3), 107–114, 2019.
• [24] X.Wu, X. Ma, Z. Zhu, and T. Lu, Topological ergodic shadowing and chaos on uniform spaces, Internat. J. Bifur. Chaos 28 (3), 1850043, 2018.
• [25] X. Wu, P. Oprocha, and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (7), 1942–1972, 2016.
• [26] X. Wu, L. Wang, and J. Liang, The chain properties and average shadowing property of iterated function systems, Qual. Theory Dyn. Syst. 17 (1), 219–227, 2018.
• [27] X. Wu, X. Zhang, and X. Ma, Various shadowing in linear dynamical systems, Internat. J. Bifur. Chaos 29 (3), 1950042, 2019.
• [28] X. Wu, X. Zhang, and F. Sun, Volume-preserving diffeomorphisms with the $\mathscr{M}_0$- shadowing properties, Mediterr. J. Math. 18, 45, 2021.
• [29] X. Zhang and X. Wu, Diffeomorphisms with the $\mathscr{M}_0$-shadowing property, Acta Math. Sin. (Engl. Ser.) 35 (11), 1760–1770, 2019.
• [30] X. Zhang, X. Wu, Y. Luo, and X. Ma, A remark on limit shadowing for hyperbolic iterated function systems, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (3), 139–146, 2019.