Density-Equicontinuity and Density-Sensitivity of Discrete Amenable Group Actions

Let ( X , G ) be a G -system which means that X is a compact metric space and G is an amenable group continuously acting on X . In this paper, we introduce the notions of density-equicontinuity with respect to a given Følner sequence and Banach density-equicontinuity for amenable group actions and we give some relations among Banach mean equicontinuity, Banach density- t -equicontinuity and Banach density-equicontinuity. Moreover, we introduce the concept of density n -sensitive tuple with respect to a given Følner sequence for amenable group actions and we prove that every topological sequence entropy n -tuple is a density n -sensitive tuple with respect to each tempered Følner sequence for an abelian group action which admits an ergodic measure with full support. For an invariant measure μ of ( X , G ), we introduce the concept of μ -density n -sensitive tuple with respect to a given Følner sequence and we show that if μ is ergodic and G is abelian, then every μ -sequence entropy n -tuple is a μ -density n -sensitive tuple with respect to any given tempered Følner sequence of G .