关于周期吸附系统的分布混沌的注记

A NOTE ON DISTRIBUTIONAL CHAOS OF PERIODICALLY ADSORBING SYSTEMS

设X是一个紧致度量空间,f:X→X是一个连续映射.若存在f的一个m-周期点p和另一个m′-周期点q(p≠q),使得对任意非空开集V(?)X,都有{p,q}(?)f^n(V),则称动力系统(X,f)是一个(m,m′)型周期吸附系统.证明了:1)若(X,f)是一个(m,m′)型周期吸附系统且X是自密的,则对任一给定的正整数k,存在一个f^k的的分布混沌集S,使得S与X的任一非空开集之交均含有一个Cantor集;2)若(X,f)是一个(m,m′)型周期吸附系统且拓扑共轭于(X′,f′),则(X′,f′)也是一个(m,m′)型周期吸附系统.改进和推广了已有结果.

Let X be a compact metric space and f：X→X be a continuous map. A dynamical system（X,f）is called a periodically adsorbing system of type（m,m＇）if there exist a periodic point p of f with period m and another periodic point q of f with period m＇ satisfying {p,q}（？）f^n（V）for any nonempty open set V C X,where p≠q.In this paper,it is proved that：1）if（X,f）is a periodically adsorbing system of type（m,m＇）and X is perfect, then for a given integer k0,there exists a distributionally chaotic set S（？）X of f^k such that the intersection of S and every nonempty open set V（？）X contains a Catsror set;and that 2）if（X,f）is a periodically adsorbing system of type（m,m＇）and is topologically conjugate to a system（X＇,f＇）,then the system（X＇,f＇）is also a periodically adsorbing system of type （m,m＇）.Some corresponding results in the literatures are improved and extended.