American Institute of Mathematical Sciences

Advanced
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 & 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.  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

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. 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

[1]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159
[2]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993
[3]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[4]

Vladimír Špitalský. Local correlation entropy. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249
[5]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
[6]

Zhiyu Wang, Yan Guo, Zhiwu Lin, Pingwen Zhang. Unstable galaxy models. Kinetic & Related Models, 2013, 6 (4) : 701-714. doi: 10.3934/krm.2013.6.701
[7]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
[8]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[9]

Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291
[10]

Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911
[11]

Edmond W. H. Lee. Equational theories of unstable involution semigroups. Electronic Research Announcements, 2017, 24: 10-20. doi: 10.3934/era.2017.24.002
[12]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829
[13]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[14]

Marina Chugunova, Roman M. Taranets. New dissipated energy for the unstable thin film equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 613-624. doi: 10.3934/cpaa.2011.10.613
[15]

Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451
[16]

Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 797-808. doi: 10.3934/dcdss.2012.5.797
[17]

Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443
[18]

C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002
[19]

Giulia Cavagnari, Antonio Marigonda. Measure-theoretic Lie brackets for nonsmooth vector fields. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 845-864. doi: 10.3934/dcdss.2018052
[20]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (227)
  • HTML views (104)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences