Dominated splitting and Pesin's entropy formula

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

Dominated splitting and Pesin's entropy formula

Wenxiang Sun ^1, and Xueting Tian ^2,

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

Abstract
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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.

Keywords: Metric entropy, dominated splitting., Lyapunov exponents, Pesin's entropy formula.

Mathematics Subject Classification: 37A05, 37A35, 37D25, 37D3.

Citation: Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and 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.
[2]	L. Barreira and Y. B. Pesin, "Nonuniform Hyperbolicity," Cambridge Univ. Press, Cambridge, 2007.
[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.
[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.
[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.
[6]	P. Liu, Pesin's entropy formula for endomorphism, Nagoya Math. J., 150 (1998), 197-209.
[7]	P. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems, Math. Z., 230 (1999), 201-239. doi: 10.1007/PL00004694.
[8]	R. Mañé, A proof of Pesin's formula, Ergod. Th. and Dynam. Sys., 1 (1981), 95-102.
[9]	R. Mañé, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, London, 1987.
[10]	V. I. Oseledec, Multiplicative ergodic theorem, Liapunov characteristic numbers for dynamical systems, translated from Russian, Trans. Moscow Math. Soc., 19 (1968), 197-221.
[11]	Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.
[12]	D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Sox. Bras. Mat, 9 (1978), 83-87. doi: 10.1007/BF02584795.
[13]	A. Tahzibi, $C^1$-generic Pesin's entropy formula, C. R. Acad. Sci. Paris, 335 (2002), 1057-1062.
[14]	P. Walters, "An Introduction to Ergodic Theory," Springer-Verlag, 2001.
[15]	J. Yang, "$C^1$ Dynamics Far from Tangencies," Ph.D thesis, IMPA, Rio de Janeiro. Available from: http://www.preprint.impa.br.

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.
[2]	L. Barreira and Y. B. Pesin, "Nonuniform Hyperbolicity," Cambridge Univ. Press, Cambridge, 2007.
[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.
[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.
[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.
[6]	P. Liu, Pesin's entropy formula for endomorphism, Nagoya Math. J., 150 (1998), 197-209.
[7]	P. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems, Math. Z., 230 (1999), 201-239. doi: 10.1007/PL00004694.
[8]	R. Mañé, A proof of Pesin's formula, Ergod. Th. and Dynam. Sys., 1 (1981), 95-102.
[9]	R. Mañé, "Ergodic Theory and Differentiable Dynamics," Springer-Verlag, Berlin, London, 1987.
[10]	V. I. Oseledec, Multiplicative ergodic theorem, Liapunov characteristic numbers for dynamical systems, translated from Russian, Trans. Moscow Math. Soc., 19 (1968), 197-221.
[11]	Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.
[12]	D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Sox. Bras. Mat, 9 (1978), 83-87. doi: 10.1007/BF02584795.
[13]	A. Tahzibi, $C^1$-generic Pesin's entropy formula, C. R. Acad. Sci. Paris, 335 (2002), 1057-1062.
[14]	P. Walters, "An Introduction to Ergodic Theory," Springer-Verlag, 2001.
[15]	J. Yang, "$C^1$ Dynamics Far from Tangencies," Ph.D thesis, IMPA, Rio de Janeiro. Available from: http://www.preprint.impa.br.

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