Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T13:24:09.516Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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