Non existence of attractors and dynamics around some wild homoclinic classes

TL;DR

This paper constructs examples of generic diffeomorphisms lacking attractors and analyzes their complex accumulation properties, revealing new insights into wild homoclinic classes and their dynamics.

Contribution

It introduces new examples of diffeomorphisms without attractors and studies their accumulation by other classes using DA constructions and semiconjugacy.

Findings

Existence of generic diffeomorphisms without attractors.

Wild homoclinic classes are accumulated by infinitely many classes.

Examples have a unique Milnor attractor robustly.

Abstract

We present new examples of generic diffeomorphisms without attractors. Also, we study how these wild classes are accumulated by infinitely many other classes (obtaining that the chain recurrence classes different from the only quasi-attractor are contained in center stable manifolds). The construction relies on some derived from Anosov (DA) constructions and uses strongly the semiconjugacy obtained by these diffeomorphisms. An interesting feature of this examples is that we can show that robustly, they present a unique attractor in the sense of Milnor.