Dynamical Systems Seminar

Given a measure-preserving group action $(\mu, X, G)$ one can associate to it a character (positive-definite conjugacy invariant function) on $G$ by \begin{align*} \chi(g)=\mu(\{x \in X: g x=x\}), g \in G . \end{align*} Anatoly Vershik suggested that for a "sufficiently rich" simple group $G$ every indecomposable character (extreme point in the space of characters) on $G$ can be obtained by formula (1) from some ergodic measure-preserving action of $G$. Given a Cantor minimal system one can associate to it two important groups of actions: the topological full group of the system and the approximately finite group of the related Bratteli diagram. In my talk I will explain the correspondence (1) for these groups and outline the proof of a generalization of Vershik's conjecture for them. The talk is based on an ongoing joint work with Konstantin Medynets.