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Zero-entropy invariant measures for skew product diffeomorphisms

Published online by Cambridge University Press:  17 July 2009

PENG SUN
PENG SUN*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA (email: [email protected])
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Abstract

In this paper, we study some skew product diffeomorphisms with non-uniformly hyperbolic structure along fibers and show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. For such diffeomorphisms, our result gives an affirmative answer to the question posed by Herman as to whether a smooth diffeomorphism of positive topological entropy would fail to be uniquely ergodic. The proof is based on some techniques that are analogous to those developed by Pesin and Katok, together with an investigation of certain combinatorial properties of the projected return map on the base.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 923 - 930
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Avila, A. and Krikorian, R.. Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2) 164 (2006), 911–940.CrossRefGoogle Scholar
[2]Barreira, L. and Pesin, Y.. Smooth ergodic theory and nonuniformly hyperbolic dynamics. Handbook of Dynamical Systems, Vol. 1B. Elsevier, Amsterdam, 2006, pp. 57–263.CrossRefGoogle Scholar
[3]Barreira, L. and Pesin, Y.. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
[4]Béguin, F., Crovisier, S. and Le Roux, F.. Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy–Rees technique. Ann. Sci. École Norm. Sup. (4) 40 (2007), 251–308.CrossRefGoogle Scholar
[5]Herman, M.. Construction d’un difféomorphisme minimal d’entropie topologique non nulle. Ergod. Th. & Dynam. Sys. 1 (1981), 65–76.CrossRefGoogle Scholar
[6]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes. Études. Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
[7]Katok, A.. Nonuniform hyperbolicity and structure of smooth dynamical systems. Proceedings of the International Congress of Mathematicians (Warsaw, 1983). PWN, Warsaw, 1984, pp. 1245–1253.Google Scholar
[8]Katok, A. and Mendoza, L.. Dynamical systems with nonuniformly hyperbolic behavior. Introduction to the Modern Theory of Dynamical Systems by A. Katok and B. Hasselblatt. Cambridge University Press, Cambridge, 1995, supplement.CrossRefGoogle Scholar
[9]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Ann. of Math. (2) 122(2) (1985), 509–574.CrossRefGoogle Scholar
[10]Pesin, Y.. Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. USSR-Izv. 40 (1976), 1261–1305.CrossRefGoogle Scholar
[11]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat. 9 (1978), 83–87.CrossRefGoogle Scholar
[12]Ruelle, D. and Wilkinson, A.. Absolutely singular dynamical foliations. Comm. Math. Phys. 219 (2001), 481–487.CrossRefGoogle Scholar
[13]Sun, P.. Measures of intermediate entropies for skew product diffeomorphisms, in preparation.Google Scholar