Berger domains and Kolmogorov typicality of infinitely many invariant circles

TL;DR

This paper introduces Berger domains for diffeomorphism families and demonstrates that having infinitely many attracting invariant circles is a typical phenomenon in these domains, extending previous results on attractor coexistence.

Contribution

It defines Berger domains inspired by Newhouse domains and proves the Kolmogorov typicality of infinitely many invariant circles within these domains.

Findings

Infinitely many attracting invariant circles are Kolmogorov typical in certain Berger domains.

Berger domains generalize Newhouse domains for families of diffeomorphisms.

The coexistence of infinitely many attractors is prevalent in higher-dimensional non-dissipative settings.

Abstract

Using the novel notion of parablender, P. Berger proved that the existence of finitely many attractors is not Kolmogorov typical in parametric families of diffeomorphisms. Here, motivated by the concept of Newhouse domains we define Berger domains for families of diffeomorphisms. As an application, we show that the coexistence of infinitely many attracting invariant smooth circles is Kolmogorov typical in certain non-sectionally dissipative Berger domains of parametric families in dimension three or greater.