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. 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.  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

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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

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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

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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. 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.  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. 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.  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

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.  Google Scholar

[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.  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. 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.  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. 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.  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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