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Propriétés extrémales et caractéristiques des exemples de Lattès

Abstract : In the first part of the thesis, we characterize the Lattès examples among the holomorphic endomorphisms of CP(k) by the absolute continuity of the measure of maximal entropy. This implies a characterization of the Lattès examples in terms of the Lyapounoff exponents of this measure. These results show that, generically, the maximal entropy measure of a holomorphic endomorphismof CP(k) is not absolutely continuous with respect to the Lebesgue measure, and at least one of its exponents is strictly larger than log d /2. This solves a question asked by Fornaess and Sibony. The characterization of the Lattès examples by their maximal entropy measure is based on a renormalization principle. The proof use th pluripotentialist interpretation of this measure as a Monge-Ampère mass. The second part is devoted to the study of the attracting basin of the origin for the polynomial lifts of Lattès examples. We show that the boundary of these domains is a quotient of a compact spherical hypersurface. These domains are surprising, because they are very close to the euclidian ball, and admit non injective self proper holomorphic maps. We get the desingularization of the boundary of the attracting basin in a line bundle over a torus, thanks to theta functions. We describe the singularities that appear by using some elements of invariant theory.
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Contributor : Christophe Dupont <>
Submitted on : Wednesday, March 26, 2003 - 2:22:56 PM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Tuesday, September 7, 2010 - 4:31:27 PM


  • HAL Id : tel-00002634, version 1



Christophe Dupont. Propriétés extrémales et caractéristiques des exemples de Lattès. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2002. Français. ⟨tel-00002634⟩



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