Bounded chaining in measurable dynamics

Abstract

We introduce a one-parameter family of notions between double ergodicity and metric ergodicity for measure-class preserving (i.e., nonsingular) actions of countable groups on standard probability spaces, providing infinitely many new invariants distinguishing weakly mixing actions. We call these properties (essentially) $k$-chaining, for $k\in \mathbb{N}$. We apply this framework to study boundary actions of free groups of finite rank $r\ge 1$, where the boundary is equipped with a stationary Markov measure. We prove that in this context, weak mixing is equivalent to $(2r-1)$-chaining, as well as to strict irreducibility of the transition matrix of the Markov measure.

24 pages, 3 figures. Cosmetic changes and added discussion of how Theorem 1.2 relates to the existing results in the literature