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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

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 & Continuous Dynamical Systems - A, 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, (2005).   Google Scholar

[2]

L. Barreira and Y. B. Pesin, "Nonuniform Hyperbolicity,", Cambridge Univ. Press, (2007).   Google Scholar

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J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic systems,, Ann. of Math., 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar

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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.   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.  doi: 10.2307/1971328.  Google Scholar

[6]

P. Liu, Pesin's entropy formula for endomorphism,, Nagoya Math. J., 150 (1998), 197.   Google Scholar

[7]

P. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems,, Math. Z., 230 (1999), 201.  doi: 10.1007/PL00004694.  Google Scholar

[8]

R. Mañé, A proof of Pesin's formula,, Ergod. Th. and Dynam. Sys., 1 (1981), 95.   Google Scholar

[9]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,", Springer-Verlag, (1987).   Google Scholar

[10]

V. I. Oseledec, Multiplicative ergodic theorem, Liapunov characteristic numbers for dynamical systems,, translated from Russian, 19 (1968), 197.   Google Scholar

[11]

Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55.  doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

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D. Ruelle, An inequality for the entropy of differentiable maps,, Bol. Sox. Bras. Mat, 9 (1978), 83.  doi: 10.1007/BF02584795.  Google Scholar

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A. Tahzibi, $C^1$-generic Pesin's entropy formula,, C. R. Acad. Sci. Paris, 335 (2002), 1057.   Google Scholar

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P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (2001).   Google Scholar

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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, (2005).   Google Scholar

[2]

L. Barreira and Y. B. Pesin, "Nonuniform Hyperbolicity,", Cambridge Univ. Press, (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.  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.   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.  doi: 10.2307/1971328.  Google Scholar

[6]

P. Liu, Pesin's entropy formula for endomorphism,, Nagoya Math. J., 150 (1998), 197.   Google Scholar

[7]

P. Liu, Entropy formula of Pesin type for noninvertible random dynamical systems,, Math. Z., 230 (1999), 201.  doi: 10.1007/PL00004694.  Google Scholar

[8]

R. Mañé, A proof of Pesin's formula,, Ergod. Th. and Dynam. Sys., 1 (1981), 95.   Google Scholar

[9]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,", Springer-Verlag, (1987).   Google Scholar

[10]

V. I. Oseledec, Multiplicative ergodic theorem, Liapunov characteristic numbers for dynamical systems,, translated from Russian, 19 (1968), 197.   Google Scholar

[11]

Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55.  doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar

[12]

D. Ruelle, An inequality for the entropy of differentiable maps,, Bol. Sox. Bras. Mat, 9 (1978), 83.  doi: 10.1007/BF02584795.  Google Scholar

[13]

A. Tahzibi, $C^1$-generic Pesin's entropy formula,, C. R. Acad. Sci. Paris, 335 (2002), 1057.   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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