On sufficient conditions for the transitivity of homeomorphisms

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.