# American Institute of Mathematical Sciences

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. Abdenur, C. 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. Catsigeras, M. 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. Hu, Y.-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. Abdenur, C. 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. Catsigeras, M. 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. Hu, Y.-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.
 [1] Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421 [2] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [3] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006 [4] Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367 [5] Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433 [6] Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119 [7] Xiaomin Zhou. A formula of conditional entropy and some applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4063-4075. doi: 10.3934/dcds.2016.36.4063 [8] Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205 [9] Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003 [10] Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 [11] Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215 [12] Jairo Bochi, Michal Rams. The entropy of Lyapunov-optimizing measures of some matrix cocycles. Journal of Modern Dynamics, 2016, 10: 255-286. doi: 10.3934/jmd.2016.10.255 [13] Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555 [14] Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic & Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479 [15] Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91 [16] Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957 [17] Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107 [18] Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433 [19] Miroslav Bulíček, Annegret Glitzky, Matthias Liero. Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 697-713. doi: 10.3934/dcdss.2017035 [20] Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237

2016 Impact Factor: 1.099