April 2018, 38(4): 2125-2140. doi: 10.3934/dcds.2018087

Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting

1. 

College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei 050024, China

2. 

School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030031, China

3. 

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China

* The corresponding author

Received  May 2017 Published  January 2018

Fund Project: It is supported by NSFC(No:11371120,11771118). The third author is also supported by Fundamental Research Funds for the Central University, China(No:20720170004)

Metric entropies along a hierarchy of unstable foliations are investigated for $C^1 $ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.

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

F. AbdenurC. Bonatti and S. Crovisier, Nonuniform hyperbolicity for C1-generic diffeomorphisms, Israel J. of Math., 183 (2011), 1-60. doi: 10.1007/s11856-011-0041-5.

[2]

C. Bonatti, L. Díaz and M. Viana, Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspestive, vol. 102 of Encyclopedia Math. Sci., Springer-Verlag, Berlin, 2005.

[3]

E. CatsigerasM. Cerminara and H. Enrich, The Pesin entropy formula for C1 diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 35 (2015), 737-761. doi: 10.1017/etds.2013.93.

[4]

H.-Y. HuY.-X. Hua and W.-S. Wu, Unstable entropies and variational principle for partially hyperbolic diffeomorphisms, Advances in Mathematics, 321 (2017), 31-68. doi: 10.1016/j.aim.2017.09.039.

[5]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅰ: Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539. doi: 10.2307/1971328.

[6]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅱ: Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574. doi: 10.2307/1971329.

[7]

P. -D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, vol. 1606 of Lecture Notes in Math., Springer-Verlag, Berlin, 1995.

[8]

R. Mañé, A proof of Pesin's formula, Ergodic Theory Dynam. Systems, 1 (1981), 95-102.

[9]

V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obv'sv'c., 19 (1968), 179-210.

[10]

Y. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112.

[11]

Y. B. Pesin and Y. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.

[12]

E. R. Pujals, From hyperbolicity to dominated splitting, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow (eds. G. Forni, M. Lyubich, C. Pugh and M. Shub), vol. 51 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, (2007), 89-102.

[13]

V. A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Transl. , 1952 (1952), 55pp.

[14]

D. Ruelle, An inequality for the entropy of differentiable maps, Bull. Braz. Math. Soc., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[15]

M. Sambarino, A (short) survey on dominated splitting, Mathematical Congress of the Americas, 149-183, Contemp. Math., 656, Amer. Math. Soc., Providence, RI, 2016.

[16]

W.-X. Sun and X.-T. Tian, Dominated splitting and Pesin's entropy formula, Discrete Contin. Dyn. Syst., 32 (2010), 1421-1434.

[17]

X.-T. Tian, Pesin's entropy formula for systems between ${C}^1 $ and ${C}^{1+α} $, J. Stat. Phys., 156 (2014), 1184-1198. doi: 10.1007/s10955-014-1065-0.

show all references

References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Nonuniform hyperbolicity for C1-generic diffeomorphisms, Israel J. of Math., 183 (2011), 1-60. doi: 10.1007/s11856-011-0041-5.

[2]

C. Bonatti, L. Díaz and M. Viana, Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspestive, vol. 102 of Encyclopedia Math. Sci., Springer-Verlag, Berlin, 2005.

[3]

E. CatsigerasM. Cerminara and H. Enrich, The Pesin entropy formula for C1 diffeomorphisms with dominated splitting, Ergodic Theory Dynam. Systems, 35 (2015), 737-761. doi: 10.1017/etds.2013.93.

[4]

H.-Y. HuY.-X. Hua and W.-S. Wu, Unstable entropies and variational principle for partially hyperbolic diffeomorphisms, Advances in Mathematics, 321 (2017), 31-68. doi: 10.1016/j.aim.2017.09.039.

[5]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅰ: Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539. doi: 10.2307/1971328.

[6]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms: Part Ⅱ: Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574. doi: 10.2307/1971329.

[7]

P. -D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, vol. 1606 of Lecture Notes in Math., Springer-Verlag, Berlin, 1995.

[8]

R. Mañé, A proof of Pesin's formula, Ergodic Theory Dynam. Systems, 1 (1981), 95-102.

[9]

V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obv'sv'c., 19 (1968), 179-210.

[10]

Y. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112.

[11]

Y. B. Pesin and Y. G. Sinai, Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems, 2 (1982), 417-438.

[12]

E. R. Pujals, From hyperbolicity to dominated splitting, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow (eds. G. Forni, M. Lyubich, C. Pugh and M. Shub), vol. 51 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, (2007), 89-102.

[13]

V. A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Transl. , 1952 (1952), 55pp.

[14]

D. Ruelle, An inequality for the entropy of differentiable maps, Bull. Braz. Math. Soc., 9 (1978), 83-87. doi: 10.1007/BF02584795.

[15]

M. Sambarino, A (short) survey on dominated splitting, Mathematical Congress of the Americas, 149-183, Contemp. Math., 656, Amer. Math. Soc., Providence, RI, 2016.

[16]

W.-X. Sun and X.-T. Tian, Dominated splitting and Pesin's entropy formula, Discrete Contin. Dyn. Syst., 32 (2010), 1421-1434.

[17]

X.-T. Tian, Pesin's entropy formula for systems between ${C}^1 $ and ${C}^{1+α} $, J. Stat. Phys., 156 (2014), 1184-1198. doi: 10.1007/s10955-014-1065-0.

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