Resumen de On Mean Sensitive Tuples of Discrete Amenable Group Actions

Xiusheng Liu, Jiandong Yin

• Let (X, G) be a G-system which means that X is a perfect compact metric space and G is a countable discrete infinite amenable group continuously acting on X. In this paper, for an invariant measureμof(X, G) and an integer n larger than 2, we introduce firstly the notions of μ-mean n-sensitive tuple with respect to a Følner sequence of G and μ-n-sensitive in the mean tuple with respect to a Følner sequence of G and we show that if μ is ergodic, then every measure-theoretic n-entropy tuple for μ is a μ-mean n-sensitive tuple with respect to each tempered Følner sequence of G. Then we introduce the concepts of mean n-sensitive tuple with respect to a Følner sequence of G and n-sensitive in the mean tuple with respect to a Følner sequence of G and we prove that each n-entropy tuple is a mean n-sensitive tuple with respect to each tempered Følner sequence of G for minimal G-systems. Finally, we introduce the notion of weakly n-sensitive in the mean tuple with respect to a Følner sequence of G and we obtain that the maximal mean equicontinuous factor with respect to a Følner sequence of G can be induced by the smallest invariant closed equivalence relation containing all weakly sensitive in the mean pairs with respect to the same Følner sequence of G.