On the non-robustness of intermingled basins

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

On mostly expanding diffeomorphisms

“…We point out, however, that there is no possibility of having the phenomenon of intermingled basins of attraction in the mostly expanding case (see Lemma 4.5 and also [25]).…”

A robustly transitive diffeomorphism of Kan's type

“…It is well known that Kan's examples can be extended to boundaryless manifolds such as T 3 , following the same arguments. In this case, however, Ures and Vasquez proved recently in [29] that Kan's examples can not be robust. In fact, they proved for C r (r ≥ 2)-partially hyperbolic, dynamically coherent diffeomorphisms of T 3 with compact center leaves, if two physical measures are intermingled, then they must both support in periodic tori which are s, u-saturated.…”

“…Noting that Kan's aim is to show the existence of intermingle property, we may generalize Kan's examples as in [29] to the following Kan-like diffeomorphisms. We have mentioned that a beautiful description of Kan-like diffeomorphisms on T 3 has been given in [29]. They proved that every partially hyperbolic Kan-like diffeomorphism on T 3 must admit two s, u-saturated periodic tori as supports of physical measures.…”

On the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center leaves

“…Observe that the two torus {t = 0}, {t = 1/2} are invariant and support the SRB measures with intermingled basins on the 2−torus. We mention that Ures and Vasquez [24] proved the existence of su−torus for any partially hyperbolic C r , r > 1 diffeomorphism in T 3 with intermingled physical measures. In particular such diffeomorphisms are not accessible and so the set of these diffeomorphisms has empty interior in C r topology.…”

“…To get a diffeomorphism on a boundaryless manifold, we consider two such examples and glue them to find a partially hyperbolic diffeomorphism of T 3 admitting two SRB measures (See also [7] and [24]).…”