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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 |
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.
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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