On the non-robustness of intermingled basins

TL;DR

This paper investigates the robustness of intermingled basins in partially hyperbolic diffeomorphisms on the 3-torus, showing that such phenomena are not stable under perturbations.

Contribution

It demonstrates that on the 3-torus, the only partially hyperbolic examples with intermingled basins are non-robust, highlighting limitations in the stability of this phenomenon.

Findings

Intermingled basins are not robust on the 3-torus.

Partially hyperbolic diffeomorphisms with intermingled basins are structurally unstable.

The phenomenon is limited to specific, non-robust cases.

Abstract

It is well-known that it is possible to construct a partially hyperbolic diffeomorphism on the 3-torus in a similar way than in Kan's example. It has two hyperbolic physical measures with intermingled basins supported on two embedded tori with Anosov dynamics. A natural question is how robust is the intermingled basins phenomenon for diffeomorphisms defined on boundaryless manifolds? In this work we will show that on the 3-torus the only partially hyperbolic examples having hyperbolic physical measures with intermingled basins are not robust.