Three Types of Distributional Chaos for a Sequence of Uniformly Convergent Continuous Maps

Abstract

Let ${\left(h_s\right)}_{s=1}^{\infty }$ be a sequence of continuous maps on a compact metric space $W$ which converges uniformly to a continuous map $h$ on $W$ . In this paper, some equivalence conditions or necessary conditions for the limit map $h$ to be distributional chaotic are obtained (where distributional chaoticity includes distributional chaotic in a sequence, distributional chaos of type 1 (DC1), distributional chaos of type 2 (DC2), and distributional chaos of type 3 (DC3)).