Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T04:26:17.658Z Has data issue: false hasContentIssue false
  • English
  • Français

On mostly expanding diffeomorphisms

Published online by Cambridge University Press:  02 May 2017

MARTIN ANDERSSON  and
CARLOS H. VÁSQUEZ
MARTIN ANDERSSON
Affiliation:
Departamento de Matemática Aplicada, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, 24020-140, Niterói – RJ, Brazil email [email protected]
CARLOS H. VÁSQUEZ
Affiliation:
Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work, we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such a class is $C^{r}$-open, $r>1$, among the partially hyperbolic diffeomorphisms and we prove that the mostly expanding condition guarantees the existence of physical measures and provides more information about the statistics of the system. Mañé’s classical derived-from-Anosov diffeomorphism on $\mathbb{T}^{3}$ belongs to this set.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 8 , December 2018 , pp. 2838 - 2859
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2) (2000), 351–398.Google Scholar
Alves, J. F., Dias, C. L., Luzzatto, S. and Pinheiro, V.. SRB measures for partially hyperbolic systems whose central direction is weakly expanding. Preprint, 2014, arXiv:1403.2937.Google Scholar
Andersson, M.. Robust ergodic properties in partially hyperbolic dynamics. Trans. Amer. Math. Soc. 362(4) (2010), 1831–1867.Google Scholar
Barreira, L. and Pesin, Ya.. Lectures on Lyapunov exponents and smooth ergodic theory. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69) . American Mathematical Society, Providence, RI, 2001, pp. 3–106; Appendix A by M. Brin and Appendix B by D. Dolgopyat, H. Hu and Pesin.Google Scholar
Bonatti, C., Díaz, L. J. and Ures, R.. Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms. J. Inst. Math. Jussieu 1(4) (2002), 513–541.Google Scholar
Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102) . Springer, Berlin, 2005.Google Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.Google Scholar
Bortolotti, R. T.. Physical measures for certain partially hyperbolic attractors on 3-Manifolds. Preprint, June 2015, arXiv: e-prints.Google Scholar
Burns, K., Dolgopyat, D. and Pesin, Ya.. Partial hyperbolicity, Lyapunov exponents and stable ergodicity. J. Stat. Phys. 108(5–6) (2002), 927–942, dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays.Google Scholar
Burns, K., Dolgopyat, D., Pesin, Y. and Pollicott, M.. Stable ergodicity for partially hyperbolic attractors with negative central exponents. J. Mod. Dyn. 2(1) (2008), 63–81.Google Scholar
Dolgopyat, D.. On dynamics of mostly contracting diffeomorphisms. Comm. Math. Phys. 213(1) (2000), 181–201.Google Scholar
Dolgopyat, D., Viana, M. and Yang, J.. Geometric and measure-theoretical structures of maps with mostly contracting center. Comm. Math. Phys. 341(3) (2016), 991–1014.Google Scholar
Hasselblatt, B. and Pesin, Y.. Partially hyperbolic dynamical systems. Handbook of Dynamical Systems. Vol. 1B Elsevier B. V., Amsterdam, 2006, pp. 1–55.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.Google Scholar
Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.Google Scholar
Nobili, F.. Minimality of invariant laminations for partially hyperbolic attractors. Nonlinearity 28(6) (2015), 1897–1918.Google Scholar
Pesin, Y.. On the work of Dolgopyat on partial and nonuniform hyperbolicity. J. Mod. Dyn. 4(2) (2010), 227–241.Google Scholar
Pesin, Y. and Climenhaga, V.. Open problems in the theory of non-uniform hyperbolicity. Discrete Contin. Dyn. Syst. 27(2) (2010), 589–607.Google Scholar
Pesin, Ya. B. and Sinaĭ, Ya. G.. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2(3–4) (1982), 417–438.Google Scholar
Pugh, C. and Shub, M.. Ergodic attractors. Trans. Amer. Math. Soc. 312(1) (1989), 1–54.Google Scholar
Pujals, E. R. and Sambarino, Martín. A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26(1) (2006), 281–289.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle. Invent. Math. 172(2) (2008), 353–381.Google Scholar
Tsujii, M.. Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1) (2005), 37–132.Google Scholar
Ures, R. and Vásquez, C. H.. On the non-robustness of intermingled basins. Ergod. Th. & Dynam. Sys. (2016), 1–17, doi:10.1017/etds.2016.33, published online 4 July.Google Scholar
Vásquez, C. H.. Statistical stability for diffeomorphisms with dominated splitting. Ergod. Th. & Dynam. Sys. 27(1) (2007), 253–283.Google Scholar
Vásquez, C. H.. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. J. Mod. Dyn. 3(2) (2009), 233–251.Google Scholar
Viana, M. and Yang, J.. Physical measures and absolute continuity for one-dimensional center direction. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(5) (2013), 845–877.Google Scholar
Young, L.-S.. What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108(5–6) (2002), 733–754, dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays.Google Scholar