The Kechris-Pestov-Todorcevic correspondence connects extreme amenability of non-Archimedean Polish groups with Ramsey properties of classes of finite structures. The purpose of the present paper is to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts.
Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications
Fixed points of local actions of nilpotent Lie groups on surfaces
Fixed points of Koch's maps
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