Groups & Dynamics Seminar

Let Γ be a countably infinite discrete group. A Γ-flow X (i.e., a nonempty compact Hausdorff space equipped with a continuous action of Γ) is called S-minimal for a subset S⊆Γ if the partial orbit S⋅x is dense for every point x∈X. (When S=Γ, we recover the usual notion of minimality.) Despite the simplicity of the definition, given a group Γ, finding an S-minimal dynamical system is typically quite difficult (in particular even when Γ is the free group and S is a subgroup it was not previously known).   In this talk, I will discuss a very recent result on how to construct S-minimal systems for any countable collection of infinite subsets simultaneously. Although the problem is purely dynamical, the techniques make heavy use of recent ideas from descriptive set theory. This is joint work with Anton Bernshteyn