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Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains
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 |
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.
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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