# American Institute of Mathematical Sciences

January  2020, 40(1): 81-105. doi: 10.3934/dcds.2020004

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

 1 College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China 2 Department of Applied Mathematics, College of Science, China Agricultural University, Beijing 100083, China 3 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

* Corresponding author: Weisheng Wu

Received  October 2018 Revised  June 2019 Published  October 2019

Fund Project: 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).

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.

Citation: Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004
##### References:
 [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.  Google Scholar [2] H. Hu, Y. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] P.-D. Liu, Random perturbations of Axiom A basic sets, Journal of Statistical Physics, 90 (1998), 467-490.  doi: 10.1023/A:1023280407906.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] X. Ma, E. 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.  Google Scholar [11] V. A. Rokhlin, On the fundamental ideas of measure theory, American Mathematical Socitety Translations, 1952 (1952), 55 pp..  Google Scholar [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.  Google Scholar [13] X. Wang, L. 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.  Google Scholar [14] X. Wang, W. Wu and Y. Zhu, Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems, preprint, arXiv: 1811.12674. Google Scholar [15] W. Wu, Local unstable entropies of partially hyperbolic diffeomorphisms, Ergodic Theory and Dynamical Systems, 2019. doi: 10.1017/etds.2019.3.  Google Scholar [16] J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504. Google Scholar

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##### References:
 [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.  Google Scholar [2] H. Hu, Y. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] P.-D. Liu, Random perturbations of Axiom A basic sets, Journal of Statistical Physics, 90 (1998), 467-490.  doi: 10.1023/A:1023280407906.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] X. Ma, E. 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.  Google Scholar [11] V. A. Rokhlin, On the fundamental ideas of measure theory, American Mathematical Socitety Translations, 1952 (1952), 55 pp..  Google Scholar [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.  Google Scholar [13] X. Wang, L. 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.  Google Scholar [14] X. Wang, W. Wu and Y. Zhu, Unstable entropy and unstable pressure for random partially hyperbolic dynamical systems, preprint, arXiv: 1811.12674. Google Scholar [15] W. Wu, Local unstable entropies of partially hyperbolic diffeomorphisms, Ergodic Theory and Dynamical Systems, 2019. doi: 10.1017/etds.2019.3.  Google Scholar [16] J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504. Google Scholar
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