On sufficient conditions for the transitivity of homeomorphisms

TL;DR

This paper establishes a precise condition involving invariant subsets and the barycenter property that determines when a homeomorphism with shadowing is topologically transitive, comparing it with known conditions for Anosov systems.

Contribution

It introduces a necessary and sufficient condition for transitivity in shadowing homeomorphisms and explores its implications and applications in dynamical systems.

Findings

Derived a condition linking invariant sets and the barycenter property for transitivity.

Compared this condition with other known transitivity conditions for Anosov diffeomorphisms.

Described the $C^1$ interior of diffeomorphisms satisfying the condition.

Abstract

We derive a necessary and sufficient condition for a homeomorphism with the shadowing property to be topologically transitive: to have an invariant subset $A$, dense in the non-wandering set, where the barycenter property holds. To elucidate its dynamical nature, we compare this condition with other properties known to be sufficient for an Anosov diffeomorphism to be topologically transitive. We also describe the $C^1$ interior of the set of diffeomorphisms which comply with this condition, discuss examples with a variety of dynamics and present some applications of interest.