Density-equicontinuity and Density-sensitivity

In this paper we introduce the notions of (Banach) density-equicontinuity and density-sensitivity. On the equicontinuity side, it is shown that a topological dynamical system is density-equicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measure-theoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure; and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is density-sensitive for this measure.