The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

摘要

In this paper, we study some chaotic properties of s-dimensional dynamical system of the form Ψa1,a2,…,as=gsas,g1a1,…,gs?1as?1, where ak∈Hk for any k∈1,2,…,s,?s≥2 is an integer, and Hk is a compact subinterval of the real line ?=?∞,+∞ for any k∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs?1as?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 hΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj°gj?1°?°g1l°gs°gs?1°?°gj+1, if j=1,…,s?1and gs°gs?1°?°g1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψs is sensitive.