The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets. (28th September 2020) - Li, Risong; Lu, Tianxiu
Abstract : In this paper, we study some chaotic properties of s -dimensional dynamical system of the form Ψ a 1, a 2, …, a s = g s a s, g 1 a 1, …, g s − 1 a s − 1, where a k ∈ H k for any k ∈ 1, 2, …, s,   s ≥ 2 is an integer, and H k is a compact subinterval of the real line ℝ = − ∞, + ∞ for any k ∈ 1, 2, …, s . Particularly, a necessary and sufficient condition for a cyclic permutation map Ψ a 1, a 2, …, a s = g s a s, g 1 a 1, …, g s − 1 a s − 1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy h Ψ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: g j ∘ g j − 1 ∘ ⋯ ∘ g 1 l∘ g s ∘ g s − 1 ∘ ⋯ ∘ g j + 1, if j = 1, …, s − 1 and g s ∘ g s − 1 ∘ ⋯ ∘ g 1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψ s is sensitive.