On Mean Sensitive Tuples of Discrete Amenable Group Actions

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 measure μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} of (X, G) and an integer n larger than 2, we introduce firstly the notions of μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-mean n-sensitive tuple with respect to a Følner sequence of G and μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-n-sensitive in the mean tuple with respect to a Følner sequence of G and we show that if μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is ergodic, then every measure-theoretic n-entropy tuple for μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is a μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-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.