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Uniform hyperbolic approximations of measures with non-zero Lyapunov exponents

Authors: Stefano Luzzatto and Fernando J. Sánchez-Salas
Journal: Proc. Amer. Math. Soc. 141 (2013), 3157-3169
MSC (2010): Primary 37D25
Published electronically: May 24, 2013
MathSciNet review: 3068969
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Abstract: We show that for any $ C^{1+\alpha } $ diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence $ \Omega _{n} $ of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence $ \{\mu _{n}\} $ of $ f $-invariant ergodic probability measures with $ supp (\mu _{n}) \subseteq \Omega _{n} $ we have $ \mu _{n}\to \mu $ in the weak-$ * $ topology.

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Additional Information

Stefano Luzzatto
Affiliation: Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy

Fernando J. Sánchez-Salas
Affiliation: Departamento de Matemáticas, Facultad Experimental de Ciencias, Universidad del Zulia, Avenida Universidad, Edificio Grano de Oro, Maracaibo, Venezuela

Keywords: Non-uniformly hyperbolic systems, uniformly hyperbolic systems, approximation of hyperbolic measures
Received by editor(s): June 7, 2011
Received by editor(s) in revised form: July 14, 2011, and November 25, 2011
Published electronically: May 24, 2013
Additional Notes: Most of this work was carried out at the Abdus Salam International Centre for Theoretical Physics (ICTP). The second author was partially supported by the Associateship Programme of ICTP
Communicated by: Bryna Kra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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