On minimal actions of countable groups

Abstract

Our purpose here is to review some recent developments in the theory of dynamical systems whose common theme is a link between minimal dynamical systems, certain Ramsey type combinatorial properties, and the Lovasz local lemma (LLL). For a general countable group G the two classes of minimal systems we will deal with are (I) the minimal subsystems of the {\em subgroup system} (Sub(G), G), called URS's (uniformly recurrent subgroups), and (II) minimal {\em subshifts}; i.e. subsystems of the binary Bernoulli G-shift ({0, 1}^G, {\sig_g}_{g \in G}).

arXiv commentAn invited review

References in corpus (6)

• Kesten's theorem for Invariant Random Subgroups
• Uniformly recurrent subgroups and the ideal structure of reduced crossed products
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• Subgroup dynamics and $C^{\ast }$-simplicity of groups of homeomorphisms
• Free Subshifts with Invariant Measures from the Lovász Local Lemma
• On uniformly recurrent subgroups of finitely generated groups