Germ-typicality of the coexistence of infinitely many sinks

TL;DR

This paper demonstrates that the coexistence of infinitely many sinks, known as the Newhouse phenomenon, is a typical behavior in certain dynamical systems, using a new renormalization approach and exploring heterodimensional cycles.

Contribution

It introduces the concept of germ-typicality in dynamics and proves that the Newhouse phenomenon is locally $C^r$-germ-typical near dissipative bicycles, with a novel renormalization scheme for heterodimensional cycles.

Findings

Newhouse phenomenon is locally $C^r$-germ-typical near dissipative bicycles.

A new renormalization scheme stabilizes heterodimensional cycles across regularity classes.

Parablenders emerge through unfolding heterodimensional cycles.

Abstract

In the spirit of Kolmogorov typicality, we introduce the notion of germ-typicality: in a space of dynamics, it encompass all these phenomena that occur for a dense and open subset of parameters of any generic parametrized family of systems. For any $2\le r<\infty$, we prove that the Newhouse phenomenon (the coexistence of infinitely many sinks) is locally $C^r$-germ-typical, nearby a dissipative bicycle: a dissipative homoclinic tangency linked to a special heterodimensional cycle. During the proof we show a result of independent interest: the stabilization of some heterodimensional cycles for any regularity class $r\in \{1, \dots, \infty\}\cup \{\omega\}$ by introducing a new renormalization scheme. We also continue the study of the paradynamics done in [Be15,Be17,BCP16] and prove that parablenders appear by unfolding some heterodimensional cycles.