Emergence of wandering stable components

Abstract:

We prove the existence of a locally dense set of real polynomial automorphisms of $\mathbb C^2$ 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 historic 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 historic, 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 $2\le r\le \infty$ and $r=\omega$. This implies a complement of the work of Kiriki and Soma [Adv. Math. 306 (2017), pp. 524–588], a proof of the last Taken’s problem in the $C^{\infty }$ and $C^\omega$-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 by Berger [Zoology in the Hénon family: twin babies and Milnor’s swallows, 2018].