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 | References | Similar Articles | Additional Information

Abstract: A diffeomorphism of a compact manifold 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 that has absolutely continuous conditional measures on unstable manifolds. The measure is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, tends to either an SBR measure or for almost every with respect to Lebesgue measure. ( is the Dirac measure at .) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of at .

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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

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