Conditions for the Existence of SBR Measures for “Almost Anosov” Diffeomorphisms
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- by Huyi Hu
- Trans. Amer. Math. Soc. 352 (2000), 2331-2367
- DOI: https://doi.org/10.1090/S0002-9947-99-02477-0
- Published electronically: December 10, 1999
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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 $p$.References
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Bibliographic 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
- 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).
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 2331-2367
- MSC (1991): Primary 58F11, 58F15; Secondary 28D05
- DOI: https://doi.org/10.1090/S0002-9947-99-02477-0
- MathSciNet review: 1661238