Distributional Chaos and Sensitivity for a Class of Cyclic Permutation Maps

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p -dimensional dynamical systems of the form φ ( b 1 , b 2 , ⋯ , b p ) = ( u p ( b p ) , u 1 ( b 1 ) , ⋯ , u p − 1 ( b p − 1 ) ) , where b j ∈ H j ( j ∈ { 1 , 2 , ⋯ , p } ), p ≥ 2 is an integer, and H j ( j ∈ { 1 , 2 , ⋯ , p } ) are compact subintervals of the real line R = ( − ∞ , + ∞ ) . u j : H j → H j + 1 ( j = 1 , 2 , … , p − 1 ) and u p : H p → H 1 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.