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Hyperbolic Lyapunov–Perron regular points and smooth invariant measures

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  17 June 2022

NAWAF ALANSARI
NAWAF ALANSARI*
Affiliation:
Department of Mathematics, Pennsylvania State University, State College, PA, USA
*
email: [email protected]
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Abstract

For a $C^{1+\alpha }$ diffeomorphism f of a compact smooth manifold, we give a necessary and sufficient condition that guarantees that if the set of hyperbolic Lyapunov–Perron regular points has positive volume, then f preserves a smooth measure. We use recent results on symbolic coding of $\chi $ -non-uniformly hyperbolic sets and results concerning the existence of SRB measures for them.

Keywords

Lyapunov–Perron regularitysmooth measuresSRBhyperbolicityentropy

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D05: Hyperbolic orbits and sets
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 7 , July 2023 , pp. 2177 - 2200
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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