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Theses

Dynamique d'applications non polynomiales et courants laminaires

Abstract : This thesis is devoted to holomorphic dynamics in two complex variables, and the theory of laminar currents which is closely related to it. We study the dynamics of a class of holomorphic mappings, introduced by Hubbard and Oberste Vorth, that are defined in some neighborhood of the unit bidisk and need not be rational. They have the same relationship with complex Hénon mappings as polynomial-like maps do with polynomials in one variable. These maps are proven to display several dynamical properties that parallel those of polynomial diffeomorphisms, as established by Bedford, Lyubich, Smillie, Fornæ ss and Sibony: existence of attracting closed positive currents, as well as a unique measure of maximal entropy, which describes the asymptotic distribution of saddle orbits. Laminar currents --a generalization of Sullivan's ``foliation cycles''-- were introduced by Bedford, Lyubich, and Smillie in the setting of two-dimensional holomorphic dynamics. We develop a general theory of such currents. We first give a geometric criterion on a sequence of plane algebraic curves, with degree growing to infinity, ensuring that the cluster values (in the sense of currents) are laminar; as a consequence laminarity of the dynamical ``Green'' current is derived for a class of rational selfmaps of the projective plane, including birational ones. For currents obtained in this way, we give a geometric interpretation of the wedge product, assuming a potential theoretic condition; we also prove such currents satisfy an ``analytic continuation'' property. This enables us to realize these currents as foliation cycles on an abstract lamination.
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Contributor : Romain Dujardin <>
Submitted on : Friday, December 19, 2003 - 10:04:13 AM
Last modification on : Friday, March 27, 2020 - 3:10:56 AM
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Romain Dujardin. Dynamique d'applications non polynomiales et courants laminaires. Mathématiques [math]. Université Paris Sud - Paris XI, 2002. Français. ⟨tel-00004028⟩

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