Entropy, Lyapunov exponents and the boundary deformation rate under the action of hyperbolic dynamical systems

We consider an Anosov diffeomorphism of a Riemannian manifold and characterize the deformation of the boundary of a small ball in under the action of in terms of the volume of a small neighbourhood of divided by the volume of . We prove that the logarithm of this ratio divided by tends to the sum of...

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Bibliographic Details
Published in:Journal of difference equations and applications 2016-01, Vol.22 (1), p.140-146
Main Authors: Gurevich, B.M., Komech, S.A.
Format: Article
Language:English
Subjects:
Anosov diffeomorphism
Boundaries
Chaos theory
Deformation
Dynamical systems
Entropy
Logarithms
Lyapunov exponents
Manifolds
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Summary:We consider an Anosov diffeomorphism of a Riemannian manifold and characterize the deformation of the boundary of a small ball in under the action of in terms of the volume of a small neighbourhood of divided by the volume of . We prove that the logarithm of this ratio divided by tends to the sum of the positive Lyapunov exponents of an arbitrary -invariant ergodic probability measure a.e. with respect to this measure, provided that increases not too fast. A statement concerning the measure-theoretic entropy of is stated as a corollary.
ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2015.1077816