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

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

Xinsheng Wang 1, , Weisheng Wu 2,, and  Yujun Zhu 3,

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.

Keywords: Random dynamical system, local unstable measure-theoretic entropy, local unstable topological entropy, local unstable pressure, variational principle.
Mathematics Subject Classification: Primary: 37D30, 37D35; Secondary: 37H99.
Citation: Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 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.

[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.

[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. 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.

[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. 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.

[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.

show all references

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.

[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.

[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. 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.

[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. 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.

[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.

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