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Vanishing transverse entropy in smooth ergodic theory

Published online by Cambridge University Press:  27 September 2010

FRANÇOIS LEDRAPPIER  and
JIAN-SHENG XIE
FRANÇOIS LEDRAPPIER
Affiliation:
LPMA, UMR 7599, Boǐte courrier 188, Université Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France (email: francois.ledrappier@upmc.fr)
JIAN-SHENG XIE
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China (email: jiansheng.xie@gmail.com)
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Abstract

For a measure-preserving transformation, the entropy being zero means that there is no increasing σ-algebra. In this note, we prove that a similar phenomenon occurs for C2 diffeomorphisms when considering the increment between the partial entropies associated with different exponents.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1229 - 1235
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Barreira, L. and Pesin, Ya.. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents (Encyclopedia of Mathematics and its Applications, 115). Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
[2]Barreira, L., Pesin, Ya. and Schmeling, J.. Dimension and product structure of hyperbolic measures. Ann. of Math. (2) 149 (1999), 755–783.Google Scholar
[3]Einsiedler, M. and Lindenstrauss, E.. On measures invariant under diagonalizable actions: the rank-one case and the general low-entropy method. J. Mod. Dyn. 2 (2008), 83–128.CrossRefGoogle Scholar
[4]Kalinin, B. and Katok, A.. Invariant measures for actions of higher rank abelian groups. Smooth Ergodic Theory and its Applications (Proceedings of Symposia in Pure Mathematics, 69). American Mathematical Society, Providence, RI, 2001, pp. 593–637.CrossRefGoogle Scholar
[5]Kalinin, B., Katok, A. and Rodriguez Hertz, F.. Nonuniform measure rigidity. Preprint, 2008.Google Scholar
[6]Katok, A. and Rodriguez Hertz, F.. Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups. Preprint, 2010.CrossRefGoogle Scholar
[7]Katok, A. and Spatzier, R.. Invariant measures for higher-rank hyperbolic abelian actions. Ergod. Th. & Dynam. Sys. 16 (1996), 751–778.CrossRefGoogle Scholar
[8]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Part I: characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122 (1985), 509–539.CrossRefGoogle Scholar
[9]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Part II: relations between entropy, exponents and dimension. Ann. of Math. (2) 122 (1985), 540–574.CrossRefGoogle Scholar
[10]Liu, P.-D. and Qian, M.. Smooth Ergodic Theory of Random Dynamical Systems (Lecture Notes in Mathematics, 1606). Springer, New York, 1995.CrossRefGoogle Scholar
[11]Peres, Y., Schlag, W. and Solomyak, B.. Sixty years of Bernoulli convolution. Fractal Geometry and Stochastics, II (Greisswald/Koserow, 1988) (Progress in Probablity, 46). Birkhaüser, Basel, 2000,pp. 39–65.CrossRefGoogle Scholar
[12]Qian, M., Xie, J.-S. and Zhu, S.. Smooth Ergodic Theory for Endomorphisms (Lecture Notes in Mathematics, 1978). Springer, New York, 2009.CrossRefGoogle Scholar
[13]Rokhlin, V. A.. Lectures on the entropy theory of transformations with invariant measures. Russian Math. Surveys 22(5) (1967), 1–52.CrossRefGoogle Scholar