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Emergence of wandering stable components

Abstract : We prove the existence of a locally dense set of real polynomial automorphisms of C 2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historical, high emergent, stable domain. We show that this model can be embedded into families of Hénon maps of explicit degree and also in an open and dense set of 5-parameter C r-families of surface diffeomorphisms in the Newhouse domain, for every 2 ≤ r ≤ ∞ and r = ω. This implies a complement of the work of Kiriki and Soma (2017), a proof of the last Taken's problem in the C ∞ and C ω-case. The main difficulty is that here perturbations are done only along finite-dimensional parameter families. The proof is based on the multi-renormalization introduced in [Ber18].
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https://hal.archives-ouvertes.fr/hal-02449496
Contributor : Pierre Berger <>
Submitted on : Wednesday, January 22, 2020 - 5:41:31 PM
Last modification on : Saturday, March 28, 2020 - 1:55:14 AM

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  • HAL Id : hal-02449496, version 1
  • ARXIV : 2001.08649

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Pierre Berger, Sebastien Biebler. Emergence of wandering stable components. 2020. ⟨hal-02449496⟩

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