Distributional Chaos and Sensitivity for a Class of Cyclic Permutation Maps

Abstract

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form φ(b1,b2,⋯,bp)=(up(bp),u1(b1),⋯,up−1(bp−1)), where bj∈Hj (j∈{1,2,⋯,p}), p≥2 is an integer, and Hj (j∈{1,2,⋯,p}) are compact subintervals of the real line R=(−∞,+∞). uj:Hj→Hj+1(j=1,2,…,p−1) and up:Hp→H1 are continuous maps. Necessary and sufficient conditions for a class of cyclic permutation maps to have Li–Yorke chaos, distributional chaos in a sequence, distributional chaos, or Li–Yorke sensitivity are given. These results extend the existing ones.