Characterization of hyperbolicity and generalized shadowing lemma

Davor Dragicevic; Sinisa Slijepcevic

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arXiv · 1107.3216

Characterization of hyperbolicity and generalized shadowing lemma

Davor Dragicevic·Sinisa Slijepcevic

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Abstract

J. Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization of nonuniform hyperbolicity and show that it is equivalent to invertibility of the same operator on a larger, Fr\'echet space. We apply it to obtain a condition for a diffeomorphism on the boundary of the set of Anosov diffeomorphisms to be nonuniformly hyperbolic. Finally we generalise the Shadowing lemma in the same context.

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Davor Dragicevic, Sinisa Slijepcevic. 2011-07-16. Characterization of hyperbolicity and generalized shadowing lemma. https://arxiv.org/abs/1107.3216

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