On the Continuity of the Topological Entropy of Non-autonomous Dynamical Systems

Article

Abstract

Let M be a compact Riemannian manifold. The set \(\text {F}^{r}(M)\) consisting of sequences \((f_{i})_{i\in {\mathbb {Z}}}\) of \(C^{r}\)-diffeomorphisms on M can be endowed with the compact topology or with the strong topology. A notion of topological entropy is given for these sequences. I will prove this entropy is discontinuous at each sequence if we consider the compact topology on \(\text {F}^{r}(M)\). On the other hand, if \( r\ge 1\) and we consider the strong topology on \(\text {F}^{r}(M)\), this entropy is a continuous map.

Keywords

Topological entropy Strong topology Non-autonomous dynamical systems Non-stationary dynamical systems 

Mathematics Subject Classification

37A35 37B40 37B55 

References

  1. Arnoux, P., Fisher, A.M.: Anosov families, renormalization and nonstationary subshifts. Erg. Th. Dym. Sys. 25, 661–709 (2005)CrossRefMATHGoogle Scholar
  2. Block, L.: Noncontinuity of topological entropy of maps of the Cantor set and of the interval. Procedings of the American mathematical society, 50 (1975)Google Scholar
  3. Dai, X., Zhou, Z., Geng, X.: Some relations between Hausdorff-dimensions and entropies. J. Sci. China Ser. A 41(10), 1068–1075 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. Hirsch, M.W.: Differential Topology, Graduate Texts in Mathematics, vol. 33. Springer-Verlag, New York-Heidelberg-Berlin (1976)Google Scholar
  5. Jin-lian, Zhang, Lan-xin, Chen: Lower bounds of the topological entropy for nonautonomous dynamical systems. Appl. Math. J. Chin. Univ. 24(1), 76–82 (2009)MathSciNetCrossRefGoogle Scholar
  6. Kloeden, P., Rasmussen, M.: Nonautonomous dynamical systems. Am. Math. Soc. 176 (2011)Google Scholar
  7. Kolyada, S., Snoha, L.: Topologial entropy of nonautonomous dynamical systems. Random Comput. Dyn. 4(2–3), 205–233 (1996)MATHGoogle Scholar
  8. Muentes, J.: Structural Stability of the Anosov families (2017) (Preprint)Google Scholar
  9. Muentes, J.: Another classification of dynamical systems on the circle (2017) (Preprint)Google Scholar
  10. Newhouse, S.: Continuity properties of entropy. Ann. Math 129, 215–235 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. Saghin, R., Yang, J.: Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows. Israel J. Math. 215 (2016)Google Scholar
  12. Shao, H., Shi, Y., Zhu, H.: Estimations of topological entropy for nonautonomous discrete systems. J. Differ. Equ. Appl. 22(3), 474–484 (2016)CrossRefMATHGoogle Scholar
  13. Walters, P.: An Introduction to ergodic Theory. Springer, Berlin (1982)CrossRefMATHGoogle Scholar
  14. Yano, K.: A remark on the topological entropy of homeomorphisms. Invent. Math 59, 215–220 (1980)MathSciNetCrossRefMATHGoogle Scholar
  15. Zhu, Y., Zhang, J., He, L.: Topological entropy of a sequence of monotone maps on circles. J. Korean Math. Soc. 43(2), 373–382 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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