A robustly transitive diffeomorphism of Kan's type

TL;DR

This paper constructs a family of partially hyperbolic skew-product diffeomorphisms on the 3-torus that are robustly transitive, have two physical measures with intermingled basins, and exhibit a dichotomy under perturbation.

Contribution

It introduces a new class of robustly transitive diffeomorphisms of Kan's type with intermingled basins and analyzes their stability properties.

Findings

All examples are not topologically mixing.

Perturbations lead to either a unique physical measure with mixing or two measures with intermingled basins.

The constructed diffeomorphisms are robustly transitive on $ ext{T}^3$.

Abstract

We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.