Comptes Rendus
C 1 -generic Pesin's entropy formula
Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1057-1062.

The metric entropy of a C2-diffeomorphism with respect to an invariant smooth measure μ is equal to the average of the sum of the positive Lyapunov exponents of μ. This is the celebrated Pesin's entropy formula, hμ(f)=∫Mλi>0λi. The C2 regularity (or C1+α) of diffeomorphism is essential to the proof of this equality. We show that at least in the two dimensional case this equality is satisfied for a C1-generic diffeomorphism and in particular we obtain a set of volume preserving diffeomorphisms strictly larger than those which are C1+α where Pesin's formula holds.

L'entropie métrique d'un difféomorphisme C2, par rapport à une mesure invariante est égale à la moyenne de la somme des exposants de Lyapunov positifs. Ceci est la célèbre formule d'entropie de Pesin. La régularité du difféomorphisme est essentielle pour la preuve de cette égalité. Nous montrons que en dimension deux, cette égalité est satisfaite pour un difféomorphisme C1-générique et montrons qu'en particulier nous obtenons un ensemble de difféomorphismes conservatifs contenant strictement ceux qui sont C1+α , où la formule de Pesin est satisfaite.

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Published online:
DOI: 10.1016/S1631-073X(02)02609-2

Ali Tahzibi 1

1 Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
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Ali Tahzibi. $ \mathrm{C}^{\mathrm{1}}$-generic Pesin's entropy formula. Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1057-1062. doi : 10.1016/S1631-073X(02)02609-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02609-2/

[1] J. Bochi, Genericity of zero lyapunov exponents, Preprint, IMPA, 2001

[2] K. Kuratowski Topology, Vol. 1, Academic Press, 1966

[3] F. Ledrappier; L.-S. Young The metric entropy of diffeomorphisms. I. characterization of measures satisfying Pesin's entropy formula, Ann. of Math., Volume 122 (1985), pp. 509-539

[4] E. Zehnder A note on smoothing symplectic and volume preserving diffeomorphims, Lecture Notes in Math., 597, Springer-Verlag, 1977, pp. 828-854

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