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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Conditions for the Existence of SBR Measures
for ``Almost Anosov'' Diffeomorphisms


Author: Huyi Hu
Journal: Trans. Amer. Math. Soc. 352 (2000), 2331-2367
MSC (1991): Primary 58F11, 58F15; Secondary 28D05
Published electronically: December 10, 1999
MathSciNet review: 1661238
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Abstract: A diffeomorphism $f$ of a compact manifold $M$ is called ``almost Anosov'' if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure $\mu $ that has absolutely continuous conditional measures on unstable manifolds. The measure $\mu $ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, $\frac{1}{n} \sum _{i=0}^{n-1}\delta _{f^{i}x}$ tends to either an SBR measure or $\delta _{p}$ for almost every $x$ with respect to Lebesgue measure. ($\delta _{x}$ is the Dirac measure at $x$.) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of $f$ at $p$.


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Additional Information

Huyi Hu
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: Department of Mathematics, Penn State University, University Park, Pennsylvania 16801
Email: hu@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02477-0
PII: S 0002-9947(99)02477-0
Keywords: Almost Anosov diffeomorphism, SBR measure, infinite SBR measure, local H\"{o}lder condition
Received by editor(s): November 23, 1997
Published electronically: December 10, 1999
Additional Notes: The author of this work was supported by NSF under grants DMS-8802593 and DMS-9116391, and by DOE (Office of Scientific Computing).
Article copyright: © Copyright 2000 American Mathematical Society