Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets of the plane associated to a holomorphic IFS. Our main result is a complex version of Newhouse's Gap Lemma : we show that under some assumptions, if the product t(K)t(L) of the thicknesses of two Cantor sets K and L is larger than 1, then K and L have non empty intersection. Since in addition this thickness varies continuously, this gives a criterion to get a robust intersection between two Cantor sets in the plane.
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]. Contents Introduction: State of the art and main results 0.1. Wandering Fatou components 0.2. Statistical complexity of the dynamics: Emergence 0.3. Emergence of wandering stable components in the Newhouse domain 0.4. Outline of the proof and organization of the manuscript 1. The geometric model 1.1. System of type (A, C) 1.2. Unfolding of wild type 2. Examples of families displaying the geometric model 2.1. A simple example of unfolding of wild type (A, C) 2.2. Natural examples satisfying the geometric model 2.3. Proof of Theorem 2.10 3. Sufficient conditions for a wandering stable domain 3.1. Implicit representations and initial bounds 3.2. Normal form and definitions of the sets B j and Bj 3.3. Proof of Theorems 1.32 and 1.34 4. Parameter selection
We show that there exists a polynomial automorphism f of C 3 of degree 2 such that for every automorphism g sufficiently close to f , g admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each d ≥ 2, there exists an open set of polynomial automorphisms of degree at most d in which the automorphisms having infinitely many sinks are dense. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz. Preliminaries Choice of a quadratic polynomialIn the following, we will consider the euclidean norm on C n for n ∈ {1, 2, 3}.Notation 2.1.1. We denote by D ⊂ C the unit disk, and by D(0, r) the disk centered at 0 of radius r for any r > 0. In particular, D(0, 1) = D.Notation 2.1.2. We will denote by dist the distance induced by the euclidean norm on C n for n ∈ {1, 2, 3}.Notation 2.1.3. For every z ∈ C 3 and i ∈ {1, 2, 3}, we denote by pr i (z) the i thcoordinate of z.In the following proposition, we carefully choose a family of quadratic polynomials with special properties. Proposition 2.1.4. For every integer q > 1, there exists a disk C ⊂ C of center c0 ∈ C, a holomorphic family (pc)c∈C of quadratic polynomials, two integers m and r, a disk D ′ with D ⊂ D ′ such that we have : 1. for every c ∈ C, p −r c (D) (resp. p −r c (D ′ )) admits two components D1, D2 (resp. D ′ 1 , D ′ 2 ) included in D (resp. D ′ ) such that p r c is univalent on both D1 and D2 (resp. D ′ 1 and D ′ 2 ). Denote by αc = n≥0 (p r c ) −n (D1) and γc = n≥0 (p r c ) −n (D2) which are two fixed points of p r c . 2. for every c ∈ C, αc is a repulsive fixed point of pc, |p ′ c (αc)| > 2 and we have : A := rαc 0 = γc 0 + pc 0 (γc 0 ) + • • • + p r−1 c 0 (γc 0 ) := B and |A − B| > 1
We study the phenomenon of robust bifurcations in the space of holomorphic maps of P 2 (C). We prove that any Lattès example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattès map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in C 2 with a well-oriented complex curve. Then we show that any Lattès map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry.
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