Universal flows of closed subgroups of S∞S_{\infty} and relative extreme amenability

This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of $S_{\infty}$. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings, and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows to provide a partial answer to a question posed by Kechris, Pestov and Todorcevic.Comment: 13 pages. Updated references, some statements made more precise. arXiv admin note: text overlap with arXiv:1201.127

The oscillation stability problem for the Urysohn sphere: A combinatorial approach

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for $\ell_2$ in the context of the Urysohn space \Ur. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.Comment: 19 page

Ramsey precompact expansions of homogeneous directed graphs

In 2005, Kechris, Pestov and Todorcevic provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow, immediately leading to an explicit representation of this invariant in many concrete cases. More recently, the framework was generalized allowing for further applications, and the purpose of this paper is to apply these new methods in the context of homogeneous directed graphs. In this paper, we show that the age of any homogeneous directed graph allows a Ramsey precompact expansion. Moreover, we verify the relative expansion properties and consequently describe the respective universal minimal flows

Polish groups with metrizable universal minimal flows

International audienceWe prove that if the universalminimal flow of a Polish group G is metrizable and contains a comeagre orbit O, then it is isomorphic to the completion of the homogeneous space G/O and show how this result translates naturally in terms of structural Ramsey theory. We also investigate universal minimal proximal flows and describe concrete representations of them in a number of examples

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

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.En 2003, Kechris, Pestov et Todorcevic démontrèrent que la structure de certains espaces métriques - dits ultrahomogènes - est intimement liée au comportement combinatoire de la classe de leurs sous-espaces métriques finis. La présente thèse a pour but d'explorer les différents aspects de cette connexion. Dans la première partie, la notion d'ultrahomogénéité métrique et les espaces ultrahomogènes complets séparables les plus remarquables, à savoir la sphère unité S_H de l'espace de Hilbert, l'espace de Baire et la sphère d'Urysohn S_U (à isométrie près, le seul espace complet séparable ultrahomogène et universel pour la classe des espaces métriques séparables de diamètre inférieur à 1) sont présentés. Dans la seconde partie, la notion de classe de Ramsey d'espaces métriques finis ordonnés est introduite et mise en lien avec les propriétés dynamiques des groupes d'isométries des espaces ultrahomogènes. Une importance particulière est attachée au théorème de Nesetril et à sa conséquence (originalement due à Pestov) selon laquelle toute action continue du groupe des autoisométries de S_U sur un compact admet un point fixe. Des résultats analogues sont ensuite obtenus dans d'autres cas, en particulier les espaces ultramétriques et l'espace de Baire. La troisième partie est quant à elle axée sur la notion de stabilité par oscillations. Pour la sphere de l'espace de Hilbert, la stabilité par oscillations n'est pas satisfaite ; il sagit d'un résultat essentiel en analyse fonctionnelle dû à Odell et Schlumprecht et équivalent à l'existence d'une application uniformément continue f de S_H dans [0,1] qui ne stabilise (ne devient presque constante) sur aucune copie isométrique de S_H dans S_H. En revanche, pour la majorité des autres espaces séparables ultrahomogènes, rien ne permet de démontrer ou de réfuter la stabilité par oscillations. C'est à ce problème qu'est consacré l'essentiel de la dernière partie. Cela conduit à la caractérisation complète des espaces ultramétriques séparables ultrahomogènes stables par oscillations et à une solution partielle dans le cas de la sphère d'Urysohn S_U