Chaos to multiple mappings from a set-valued view

摘要

Let (X, d) be a compact metric space and F = {f(1), f(2), ..., f(m)} be an m-tuple of continuous maps from X to itself. In this paper, we investigate the multiple mappings dynamical system (X, F) with Hausdorff metric Li-Yorke chaos, distributional chaos and distributional chaos in a sequence properties from a set-valued view. On the basis of this research, we draw main conclusions as follows: (i) two topological conjugacy dynamical systems to multiple mappings have simultaneously Hausdorff metric Li-Yorke chaos or distributional chaos. (ii) Hausdorff metric Li-Yorke delta-chaos is equivalent to Hausdorff metric distributional delta-chaos in a sequence. (iii) By giving two examples, we show that there is non-mutual implication between that the multiple mappings F-i = {f(1), f(2), ..., f(m)} is Hausdorff metric Li-Yorke chaos and that each element f(i) (i = 1, 2, ..., m) in F is Li-Yorke chaos. (iv) For the multiple mappings, weakly mixing implies the Hausdorff metric strongly Li-Yorke chaos and Hausdorff metric distributional chaos in a sequence.