Density-Based Equicontinuity and Chaos

This paper studies density-based equicontinuity in time-varying discrete dynamical systems. It is shown that density-equicontinuity and almost density-equicontinuity are equivalent in minimal systems and both inhibit Devaney chaos. Two chaos criteria are established by combining nondensity-equicontinuity or nonalmost density-equicontinuity with typical chaotic conditions. Further, the equivalence between Banach density-equicontinuity and Banach mean equicontinuity is proved, and the preservation properties of several density-based equicontinuity notions in the finite products setting are verified. These results extend the theory of density-based equicontinuity for time-varying discrete dynamical systems.