Groups and Dynamics

Recent methods for proving pointwise ergodic theorems for probability-measure-preserving (pmp) actions of free groups rely crucially on weak mixing properties of Markov measures on the boundary of the free group, but no criterion for when such measures are weakly mixing was known. In joint work with Jenna Zomback, we give a complete characterization of these measures in terms of a new dynamical condition on actions, which we call k-chaining (for integers k≥1) and which is of independent interest. For every pmp action of a countable group, weak mixing is equivalent to 1-chaining. For nonsingular actions, k-chaining yields a family of invariants that are strictly stronger than weak mixing, forming a genuine hierarchy in which the condition becomes strictly weaker as k increases.