No Access

Random entropy expansiveness for diffeomorphisms with dominated splittings

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China

E-mail Address: mizeya@163.com

Search for more papers by this author

https://doi.org/10.1142/S0219493720500148Cited by:3 (Source: Crossref)

Abstract

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for $C^{1}$ partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider $C^{1}$ diffeomorphism $f$ with dominated splitting $T M = E^{c u} \oplus E_{1}^{c} \oplus \dots \oplus E_{k}^{c} \oplus E^{c s}$ such that $dim E_{i}^{c} = 1$ for every $1 \leq i \leq k$ , and all the Lyapunov exponents are non-negative along $E^{c u}$ and non-positive along $E^{c s}$ , we prove the asymptotically random entropy expansiveness for $f$ .

Keywords:

AMSC: 37A35, 37D30

References

1. J. Alves, Statistical analysis of non-uniformly expanding dynamical systems, Lect. Notes 24th Braz. Math. Colloq. IMPA (Rio De Janeiro, 2003), pp. 85–88. Google Scholar
2. R. Bowen, Entropy expansive maps, Trans. Amer. Math. Soc. 164 (1972) 323–331. Web of Science, Google Scholar
3. M. Boyle, D. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math. 14 (2002) 713–757. Web of Science, Google Scholar
4. M. Brin and Y. Pesin, Partially hyperbolic dynamical systems, Izv. Akad, Nauk SSSR Ser. Mat. 38 (1974) 170–212. Google Scholar
5. Y. Cao and D. Yang, On pesin’s entropy formula for dominated splittings without mixed behavior, J. Differential Equations 261 (2016) 3964–3986. Web of Science, Google Scholar
6. W. Cowieson and L.-S. Young, SRB measures as zero-noise limits, Ergodic Theory Dynam. Systems 25 (2005) 1115–1138. Web of Science, Google Scholar
7. S. Crovisier and R. Potrie, Introduction to partially hyperbolic dynamics (2018), https://www.math.u-psud.fr/crovisie/00-CP-Trieste-Version1.pdf. Google Scholar
8. S. Crovisier, M. Sambarino and D. Yang, Partial hyperbolicity and homoclinic tangencies, J. Eur. Math. Soc. 17 (2015) 1–49. Web of Science, Google Scholar
9. L. J. Diaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Entropy expansiveness for partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst. 32 (2012) 4195–4207. Web of Science, Google Scholar
10. M. W. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathmatics, Vol. 583 (Springer-Verlag, 1977). Google Scholar
11. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, 1995). Google Scholar
12. Yu. Kifer, Ergodic Theory of Random Transformations (Birkhauser, 1986). Google Scholar
13. G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc. 15 (2010) 2043–2060. Web of Science, Google Scholar
14. P. Liu and K. Lu, A note on partially hyperbolic attractors: Entropy conjecture and SRB measures, Discrete Contin. Dyn. Syst. 35 (2015) 341–352. Web of Science, Google Scholar
15. P. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, Vol. 1606 (Springer-Verlag, 1995). Google Scholar
16. Z. Mi, Y. Cao and D. Yang, SRB measures for attractors with continuous invariant splittings, Math. Z. 288 (2018) 135–165. Web of Science, Google Scholar
17. M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut. 21 (2008) 293–317. Web of Science, Google Scholar
18. M. J. Pacifico and J. L. Vieitez, Robust entropy expansiveness implies generic domination, Nonlinearity 23 (2009) 1971–1990. Web of Science, Google Scholar
19. C. Pugh and M. Shub, Ergodicity of Anosov actions, Invent. Math. 15 (1972) 1–23. Web of Science, Google Scholar
20. E. R. Pujals and M. Sambarino, Topics on homoclinic bifurcation, dominated splitting, robust transitivity and related results, Handb. Dyn. Syst. 1 (2004) 327–378. Google Scholar
21. M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc. 80 (1974) 27–41. Web of Science, Google Scholar
22. Y. Sinai, Kolmogorovs work on ergodic theory, Ann. Prob. 17 (1989) 833–839. Web of Science, Google Scholar
23. M. Viana, Stochastic dynamics of deterministic systems, Lect. Notes XXI Braz. Math. Colloq. IMPA (Rio de Janeiro, 1997). Google Scholar
24. P. Walters, An Introduction to Ergodic Theory (Springer-Verlag, 1982). Google Scholar
25. L.-S. Young, What are SRB measures, and which dynamical systems have them? J. Statist. Phys. 108 (2002) 733–754. Web of Science, Google Scholar
26. Y. Zang, D. Yang and Y. Cao, The entropy conjecture for dominated splitting with multi 1D centers via upper semi-continuity of the metric entropy, Nonlinearity 30 (2017) 3076–3087. Web of Science, Google Scholar