Advanced Search
Article Contents
Article Contents

Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems

  • * Corresponding author: Weisheng Wu

    * Corresponding author: Weisheng Wu 

X. Wang and Y. Zhu are supported by NSFC (Nos: 11771118, 11801336), W. Wu is supported by NSFC (No: 11701559). The first author is also supported by the Innovation Fund Designated for Graduate Students of Hebei Province (No: CXZZBS2018101) and China Scholarship Council (CSC)

Abstract Full Text(HTML) Related Papers Cited by
  • Let $ \mathcal{F} $ be a random partially hyperbolic dynamical system generated by random compositions of a set of $ C^2 $-diffeomorphisms. For the unstable foliation, the corresponding local unstable measure-theoretic entropy, local unstable topological entropy and local unstable pressure via the dynamics of $ \mathcal{F} $ along the unstable foliation are introduced and investigated. And variational principles for local unstable entropy and local unstable pressure are obtained respectively.

    Mathematics Subject Classification: Primary: 37D30, 37D35; Secondary: 37H99.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] J. Bahnmüller and P.-D. Liu, Characterization of measures satisfying the Pesin entropy formula for random dynamical systems, Journal of Dynamics and Differential Equations, 10 (1998), 425-448.  doi: 10.1023/A:1022653229891.
    [2] H. HuY. Hua and W. Wu, Unstable entropies and variational principle for partially hyperbolic diffeomorphsims, Advances in Mathematics, 321 (2017), 31-68.  doi: 10.1016/j.aim.2017.09.039.
    [3] H. Hu and Y. Zhu, Quasi-stability of partially hyperbolic diffeomorphisms, Transaction of the American Mathematical Society, 366 (2014), 3787-3804.  doi: 10.1090/S0002-9947-2014-06037-6.
    [4] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: part Ⅰ: Characterization of measures satisfying Pesin's entropy formula, Annals of Mathematics, 122 (1985), 509-539.  doi: 10.2307/1971328.
    [5] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: part Ⅱ: Relations between entropy, exponents and dimension, Annals of Mathematics, 122 (1985), 540-574.  doi: 10.2307/1971329.
    [6] P.-D. Liu, Random perturbations of Axiom A basic sets, Journal of Statistical Physics, 90 (1998), 467-490.  doi: 10.1023/A:1023280407906.
    [7] P.-D. Liu, Survey: Dynamics of random transformations: Smooth ergodic theory, Ergodic Theory and Dynamical Systems, 21 (2001), 1279-1319.  doi: 10.1017/S0143385701001614.
    [8] P.-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, volume 1606 of Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg, 1995. doi: 10.1007/BFb0094308.
    [9] X. Ma and E. Chen, A local variational principle for random bundle transformations, Stochastics and Dynamics, 13 (2013), 1250023, 21pp. doi: 10.1142/S0219493712500232.
    [10] X. MaE. Chen and A. Zhang, A relative local variational principle for topological pressure, Science China Mathematics, 53 (2010), 1491-1506.  doi: 10.1007/s11425-010-3038-3.
    [11] V. A. Rokhlin, On the fundamental ideas of measure theory, American Mathematical Socitety Translations, 1952 (1952), 55 pp..
    [12] P. P. Romagnoli, A local variational principle for the topological entropy, Ergodic Theory and Dynamical Systems, 23 (2003), 1601-1610.  doi: 10.1017/S0143385703000105.
    [13] X. WangL. Wang and Y. Zhu, Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting, Discrete and Continuous Dynamical Systems, 38 (2018), 2125-2140.  doi: 10.3934/dcds.2018087.
    [14] X. Wang, W. Wu and Y. Zhu, Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems, preprint, arXiv: 1811.12674.
    [15] W. Wu, Local unstable entropies of partially hyperbolic diffeomorphisms, Ergodic Theory and Dynamical Systems, 2019. doi: 10.1017/etds.2019.3.
    [16] J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504.
  • 加载中

Article Metrics

HTML views(159) PDF downloads(354) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint