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Théorie de Ramsey structurale des espaces métriques et dynamique topologique des groupes d'isométries

Abstract : In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present thesis is to explore the different aspects of this connection. In Part 1, the notion of metric ultrahomogeneity is presented as well as the most remarkable complete separable ultrahomogeneous metric spaces, that is, the unit sphere S_H of the Hilbert space, the Baire space, and the Urysohn sphere S_U (up to isometry, the unique complete separable ultrahomogeneous metric space universal for the class of all separable metric spaces with diameter less or equal to 1). In Part 2, the notion of Ramsey class of finite ordered metric space is introduced and related to the dynamical properties of the isometry groups attached to ultrahomogeneous spaces. A particular attention is paid to Nesetril theorem and its consequence (originally due to Pestov) according to which every continuous action of the autoisometry group of S_U on a compact Hausdorff space has a fixed point. Analogous results are then obtained in other similar situations, such as the ultrametric spaces and the Baire space. As for Part 3, it focuses on the notion of oscillation stability. For S_H, oscillation stability does not hold. This is a deep result in functional analysis due to Odell and Schlumprecht and equivalent to the existence of a uniformly continuous f from S_H to [0,1] that does not stabilize (does not become almost constant) on any isometric copy of S_H in S_H. However, for most of the other ultrahomogeneous spaces, no result is presently known concerning oscillation stability. The last part of the thesis is essentially devoted to that problem. This leads to a complete characterization of the complete separable ultrahomogeneous ultrametric spaces, as well well as to a partial solution in the case of the Urysohn sphere S_U.
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https://tel.archives-ouvertes.fr/tel-00139239
Contributor : Lionel Nguyen van Thé <>
Submitted on : Friday, March 30, 2007 - 2:42:33 AM
Last modification on : Saturday, March 28, 2020 - 2:23:47 AM
Long-term archiving on: : Wednesday, April 7, 2010 - 1:47:50 AM

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Lionel Nguyen van Thé. Théorie de Ramsey structurale des espaces métriques et dynamique topologique des groupes d'isométries. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2006. Français. ⟨tel-00139239⟩

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