Phénomène de Newhouse et bifurcations en dynamique holomorphe à plusieurs variables

Abstract : In this PhD thesis, we study Newhouse’s phenomenon and bifurcations in the context of dynamics in several complex variables. We prove three main Theorems. The first one is a complex Gap Lemma. In real dynamics, Newhouse’s Gap Lemma gives a criterion on the product of the thicknesses of two dynamical Cantor sets K and L to show that K ∩ L is not empty. We show a partial generalization of this result for dynamical Cantor sets in C. A relevant notion of thickness in this case is defined and we give some criterion on the product of two thicknesses to show that two dynamical Cantor sets in C must intersect. We also show that the thickness varies continuously, which generates persistent intersections of dynamical Cantor sets. In the second Theorem, 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. In contrary to the case of C^{2}, the degree is known. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz. In particular, we use a blender to produce robust tangencies. In the third and last result, 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. This gives a partial answer to a conjecture of Dujardin. 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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Sébastien Biebler. Phénomène de Newhouse et bifurcations en dynamique holomorphe à plusieurs variables. Mathématiques générales [math.GM]. Université Paris-Est, 2018. Français. ⟨NNT : 2018PESC1066⟩. ⟨tel-01940348⟩

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