Abstract
In this paper, we study the relationship between the multi-sensitivity and the topological maximal sequence entropy of dynamical systems for general group action. Furthermore, we also discuss the consistency of multi-sensitivity of a dynamical system (G ↷ X) and its hyperspace dynamical system G ↷ K(X). Moreover, we research the relationship between the multi-sensitivity of two dynamical systems and the multi-sensitivity of their product space dynamical system. Finally, we prove that if the topological sequence entropy of G ↷ X vanishes, then so does that of its induced system \(G\curvearrowright {\cal M}\left( X \right)\); if the topological sequence entropy of G ↷ X is positive, then that of its induced system \(G\curvearrowright {\cal M}\left( X \right)\) is infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslander, J., Yorke, J. A.: Interval maps, factors of maps and chaos. Tôhoku Math. J., 32, 177–188 (2002)
Beiglböck, M., Bergelson, V., Fish, A.: Sumset phenomenon in countable amenable groups. Adv. Math., 223, 416–432 (2010)
Coornaert, M.: Topological Dimension and Dynamical Systems, Springer, Cham, 2015
Glasner, E., Weiss, B.: Quasi-factors of zero entropy systems. J. Amer. Math. Soc., 8, 665–686 (1995)
Goodman, T. N. T.: Topological sequence entropy. Proc. London Math. Soc., 29, 331–350 (1974)
Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Comm. Math. Phys., 70, 133–160 (1979)
Liu, K., Qiao, Y., Xu, L.: Topological entropy of nonautonomous dynamical systems. J. Differential Equations, 268, 5353–5365 (2020)
Huang, W., Khilko, D., Kolyada, S., et al.: Dynamical compactness and sensitivity. J. Differential Equations, 260, 6800–6827 (2016)
Huang, W., Ye, X., Combinatorial lemmas and applications to dynamics. Adv. Math., 220, 1689–1716 (2009)
Huang, X., Liu, J., Zhu, C.: Bowen topological entropy of subsets for amenable group action. J. Math. Anal. Appl., 472, 1678–1715 (2019)
Kerr, D., Li, H.: Ergodic Theory. Independence and Dichotomies, Springer Monographs in Mathematics, Springer, Cham, 2016
Parthasarathy, K. R.: Introduction to Probability and Measure, Macmillan, London, 1977
Walter, P.: An introduction to ergodic theorem, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York, 1982
Wu, X.: A remark on topological sequence entropy. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27, 1750107, 7 pp. (2017)
Wu, X., Wang, J., Chen, G.: \({\cal F}\) and multi-sensitivity of hyperspatial dynamical. J. Math. Anal. Appl., 429, 16–26 (2015)
Acknowledgements
We thank the referees for their time and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSF of China (Grant No. 11671057) and NSF of Chongqing (Grant No. cstc2020jcyj-msxmX0694)
Rights and permissions
About this article
Cite this article
Huang, X.J., Zhu, B. The Multi-sensitivity and Topological Sequence Entropy of Dynamical System with Group Action. Acta. Math. Sin.-English Ser. 39, 663–684 (2023). https://doi.org/10.1007/s10114-022-0542-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-022-0542-7