Independence and mean sensitivity in minimal systems under group actions

Abstract: In this paper, we mainly study the relation between regularity, independence and mean sensitivity for minimal systems. In the first part, we show that if a minimal system is incontractible, or local Bronstein with an invariant Borel probability measure, then the regularity is strictly bounded by the infinite independence. In particular, the following two types of minimal systems are applicable to our result: (1) The acting group of the minimal system is a virtually nilpotent group. (2) The minimal system is a proximal extension of its maximal equicontinuous factor and admits an invariant Borel probability measure. Items (1) and (2) correspond to Conjectures 1 and 2 from Huang, Lian, Shao, and Ye (J. Funct. Anal., 2021); item (1) verifies Conjecture 1 in the virtually nilpotent case, and item (2) gives an affirmative answer to Conjecture 2. In the second part, for a minimal system acting by an amenable group, under the local Bronstein condition, we establish parallel results regarding weak mean sensitivity and establish that every mean-sensitive tuple is an IT-tuple.