Sequence Entropy Tuples, Independence, and Mean Sensitivity for Invariant Measures
Date:
Abstract. In this talk, I will discuss recent joint work with Leiye Xu and Shuhao Zhang on local notions of complexity for measure-preserving systems under actions of countably infinite discrete groups. The talk will focus on three related notions: sequence entropy tuples, IT tuples, and mean sensitive tuples.
It is known from the independence approach to dynamics that IT tuples are topological sequence entropy tuples, while measure-theoretic sequence entropy tuples are also known to be topological sequence entropy tuples. This naturally leads to the question of whether measure-theoretic sequence entropy tuples are directly related to IT tuples. Our first result answers this question affirmatively: every measure-theoretic sequence entropy tuple is an IT tuple.
I will also discuss two further results. The first gives an upper bound on the maximal sequence entropy in terms of the absence of essential sequence entropy K-tuples. The second concerns the amenable ergodic case, where sequence entropy tuples coincide with mean sensitive tuples along tempered Følner sequences.