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Nonexistence of SBR measures for some diffeomorphisms that are ‘Almost Anosov’

Published online by Cambridge University Press:  19 September 2008

Huyi Hu  and
Lai-Sang Young
Huyi Hu
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
Lai-Sang Young
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90024, USA
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Abstract

The purpose of this paper is to present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Each example has a fixed point at which the larger eigenvalue is equal to one and the smaller eigenvalue is less than one.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 1 , February 1995 , pp. 67 - 76
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[A]Anosov, D.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. of Math. 90 (1970), 1–235.Google Scholar
[B]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Springer Lecture Notes in Mathematics 470. Springer: Berlin, 1975.Google Scholar
[BC]Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. Math. 133 (1991), 73–169.CrossRefGoogle Scholar
[BY]Benedicks, M. and Young, L.-S.. Sinai–Bowen–Ruelle measures for certain Hénon maps. Inventiones. To appear.Google Scholar
[HP]Hirsch, M. and Pugh, C.. Stable manifolds and hyperbolic sets. Amer. Math. Soc. Proc. Symp. Pure Math. 14 (1970), 133–164.CrossRefGoogle Scholar
[HPS]Hirsch, M., Pugh, C. and Shub, M.. Invariant manifolds. Lecture Notes in Mathematics 583. Springer: Berlin, 1977.Google Scholar
[L]Ledrappier, F.. Propriétés ergodiques des measure de Sinai. Publ. Math. IHES 59 (1984), 163–188.CrossRefGoogle Scholar
[LS]Ledrappier, F. and Strelcyn, J.-M.. Estimation from below in Pesin's entropy formula. Ergod. Th. & Dynam. Sys. 2 (1982), 203–219.CrossRefGoogle Scholar
[LY]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Ann. Math. 122 (1985), 509–574.CrossRefGoogle Scholar
[P]Pesin, Ya. B.. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR Izvestija 10 (1978), 1261–1305.CrossRefGoogle Scholar
[Pi]Pianigiani, G.. First return map and invariant measures. Israel J. Math. 35 (1980), 32–48.CrossRefGoogle Scholar
[R]Rohlin, V. A.. Lectures on the theory of entropy of transformations with invariant measures. Russ. Math. Surveys 22:5 (1967), 1–54.CrossRefGoogle Scholar
[S]Sinai, Ya. G.. Gibbs measures in ergodic theory. Russ. Math. Surveys 27:4 (1972), 21–69.CrossRefGoogle Scholar