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arXiv · 2103.16697

Germ-typicality of the coexistence of infinitely many sinks

Pierre Berger·Sylvain Crovisier·Enrique Pujals
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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.

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Pierre Berger, Sylvain Crovisier, Enrique Pujals. 2021-03-30. Germ-typicality of the coexistence of infinitely many sinks. https://arxiv.org/abs/2103.16697

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