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April  2012, 32(4): 1421-1434. doi: 10.3934/dcds.2012.32.1421

## Dominated splitting and Pesin's entropy formula

 1 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China 2 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received  May 2010 Revised  September 2011 Published  October 2011

Let $M$ be a compact manifold and $f:\,M\rightarrow M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting $T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$, then we give an estimation through Lyapunov characteristic exponents from below in Pesin's entropy formula, i.e., the metric entropy $h_\mu(f)$ satisfies $$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ where $\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and $\lambda_1(x)\geq\lambda_2(x)\geq\cdots\geq\lambda_{dim\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $\mu.$
Consequently, we obtain that Pesin's entropy formula always holds for (1) volume-preserving Anosov diffeomorphisms, (2) volume-preserving partially hyperbolic diffeomorphisms with one-dimensional center bundle, (3) volume-preserving diffeomorphisms far away from homoclinic tangency, and (4) generic volume-preserving diffeomorphisms.
Citation: Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
##### References:
 [1] Ch. Bonatti, L. Diaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective," Springer-Verlag, Berlin, Heidelberg, 2005. Google Scholar [2] L. Barreira and Y. B. Pesin, "Nonuniform Hyperbolicity," Cambridge Univ. Press, Cambridge, 2007. Google Scholar [3] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic systems, Ann. of Math., 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.  Google Scholar [4] F. Ledrappier and J. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergod. Th. and Dynam. Sys., 2 (1982), 203-219.  Google Scholar [5] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539. doi: 10.2307/1971328.  Google Scholar [6] P. Liu, Pesin's entropy formula for endomorphism, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar [7] P. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems, Math. Z., 230 (1999), 201-239. doi: 10.1007/PL00004694.  Google Scholar [8] R. Mañé, A proof of Pesin's formula, Ergod. Th. and Dynam. Sys., 1 (1981), 95-102.  Google Scholar [9] R. Mañé, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, London, 1987. Google Scholar [10] V. I. Oseledec, Multiplicative ergodic theorem, Liapunov characteristic numbers for dynamical systems, translated from Russian, Trans. Moscow Math. Soc., 19 (1968), 197-221.  Google Scholar [11] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar [12] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Sox. Bras. Mat, 9 (1978), 83-87. doi: 10.1007/BF02584795.  Google Scholar [13] A. Tahzibi, $C^1$-generic Pesin's entropy formula, C. R. Acad. Sci. Paris, 335 (2002), 1057-1062.  Google Scholar [14] P. Walters, "An Introduction to Ergodic Theory," Springer-Verlag, 2001. Google Scholar [15] J. Yang, "$C^1$ Dynamics Far from Tangencies,", Ph.D thesis, ().   Google Scholar

show all references

##### References:
 [1] Ch. Bonatti, L. Diaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective," Springer-Verlag, Berlin, Heidelberg, 2005. Google Scholar [2] L. Barreira and Y. B. Pesin, "Nonuniform Hyperbolicity," Cambridge Univ. Press, Cambridge, 2007. Google Scholar [3] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic systems, Ann. of Math., 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.  Google Scholar [4] F. Ledrappier and J. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergod. Th. and Dynam. Sys., 2 (1982), 203-219.  Google Scholar [5] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539. doi: 10.2307/1971328.  Google Scholar [6] P. Liu, Pesin's entropy formula for endomorphism, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar [7] P. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems, Math. Z., 230 (1999), 201-239. doi: 10.1007/PL00004694.  Google Scholar [8] R. Mañé, A proof of Pesin's formula, Ergod. Th. and Dynam. Sys., 1 (1981), 95-102.  Google Scholar [9] R. Mañé, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, London, 1987. Google Scholar [10] V. I. Oseledec, Multiplicative ergodic theorem, Liapunov characteristic numbers for dynamical systems, translated from Russian, Trans. Moscow Math. Soc., 19 (1968), 197-221.  Google Scholar [11] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar [12] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Sox. Bras. Mat, 9 (1978), 83-87. doi: 10.1007/BF02584795.  Google Scholar [13] A. Tahzibi, $C^1$-generic Pesin's entropy formula, C. R. Acad. Sci. Paris, 335 (2002), 1057-1062.  Google Scholar [14] P. Walters, "An Introduction to Ergodic Theory," Springer-Verlag, 2001. Google Scholar [15] J. Yang, "$C^1$ Dynamics Far from Tangencies,", Ph.D thesis, ().   Google Scholar
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