Hyperbolicity and Types of Shadowing for C1 Generic Vector Fields

TL;DR

This paper investigates shadowing properties in C1 generic vector fields, establishing conditions under which isolated sets are hyperbolic and transitive, with applications to divergence-free fields.

Contribution

It demonstrates that various shadowing properties imply hyperbolicity and transitivity for generic vector fields, extending understanding of their dynamical structure.

Findings

Chain transitivity is implied by the limit shadowing property.

Shadowing properties lead to hyperbolic and topologically transitive sets.

Results apply to divergence-free vector fields.

Abstract

We study various types of shadowing properties and their implication for C1 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.