On the three-legged accessibility property

TL;DR

This paper investigates the three-legged accessibility property in partially hyperbolic diffeomorphisms, establishing conditions for minimal sets and transitivity, with applications to Anosov systems and skew products.

Contribution

It provides new criteria for robust three-legged accessibility and links this property to minimal sets and transitivity in partially hyperbolic systems.

Findings

Existence of a unique minimal set for one foliation under three-legged accessibility

Transitivity of the other foliation in such systems

Criteria for robust three-legged accessibility when center dimension is one

Abstract

We show that certain types of the three-legged accessibility property of a partially hyperbolic diffeomorphism imply the existence of a unique minimal set for one strong foliation and the transitivity of the other one. In case the center dimension is one, we also give a criteria to obtain three-legged accessibility in a robust way. We show some applications of our results to the time-one map of Anosov flows, skew products and certain Anosov diffeomorphisms with partially hyperbolic splitting.