Advanced Search
Article Contents
Article Contents

Center specification property and entropy for partially hyperbolic diffeomorphisms

Abstract Related Papers Cited by
  • Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $\mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, \mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $\Lambda$ is center topologically mixing then $f|_{\Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $ h(f)\leq h(f,\mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $\mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.
    Mathematics Subject Classification: Primary: 37D30, 37B40; Secondary: 37C50.


    \begin{equation} \\ \end{equation}
  • [1]

    L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Cambridge University Press, Cambridge, 2007.doi: 10.1017/CBO9781107326026.


    D. Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation, Journal of Modern Dynamics, 7 (2013), 565-604.doi: 10.3934/jmd.2013.7.565.


    C. Bonatti and D. Bohnet, Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics, to appear in Ergodic Theory Dynam. Systems, arXiv:1210.2835.


    C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia Math. Sci., 102, Springer, Berlin, 2005.


    R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math., 154 (1971), 377-397.


    M. Brin and J. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.doi: 10.1070/IM1974v008n01ABEH002101.


    P. D. Carrasco, Compact Dynamical Foliations, Ph.D thesis, University of Toronto, 2011.


    P. D. Carrasco, Compact dynamical foliations, to appear in Ergodic Theory Dynam. Systems, arXiv:1105.0052.


    F. Hertz, J. Hertz and R. Ures, Partially Hyperbolic Dynamics, $28^0$ Coloquio Brasileiro de Matematica, IMPA Mathematical Publications, IMPA-Rio de Janeiro, 2011.


    M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes in Math. 583, Springer, New York, 1977.


    H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing for partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 35 (2015), 412-430.doi: 10.1017/etds.2014.126.


    H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing and quasi-stability for dynamically coherent partially hyperbolic diffeomorphisms, arXiv:1405.0081.


    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511809187.


    S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing, Discrete Continuous Dynam. Systems, 33 (2013), 2901-2909.doi: 10.3934/dcds.2013.33.2901.


    Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics. European Math. Soc., Zurich, 2004.doi: 10.4171/003.


    C. Pugh, M. Shub and A. Wilkinson, Holder foliations, revisited, J. Modern Dyn., 6 (2012), 79-120.doi: 10.3934/jmd.2012.6.79.


    P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.

  • 加载中

Article Metrics

HTML views() PDF downloads(53) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint