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 H{\'e}non 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 (2017), 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 in [Ber18].