Hyperbolicity and types of shadowing for $C^1$ generic vector fields

Raquel Ribeiro

doi:10.3934/dcds.2014.34.2963

Article Contents

2014, Volume 34, Issue 7: 2963-2982. Doi: 10.3934/dcds.2014.34.2963

This issue Previous Article Discrete admissibility and exponential trichotomy of dynamical systems Next Article The Aubry-Mather theorem for driven generalized elastic chains

Hyperbolicity and types of shadowing for $C^1$ generic vector fields

Raquel Ribeiro^1,

1.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro

Received: June 30, 2013

Revised: August 31, 2013

Published: June 2014

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Abstract

We study various types of shadowing properties and their implication for $C^1$ generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i) the set is chain transitive and satisfies the (classical) shadowing property, (ii) the set satisfies the limit shadowing property, or (iii) the set satisfies the (asymptotic) shadowing property with the additional hypothesis that stable and unstable manifolds of any pair of critical orbits intersect each other. In our proof we essentially rely on the property of chain transitivity and, in particular, show that it is implied by the limit shadowing property. We also apply our results to divergence-free vector fields.

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Mathematics Subject Classification: Primary: 37C20, 37D20; Secondary: 37C10, 37C50.

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