关于周期吸附系统的分布混沌的注记

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(C)X,都有{p,q}(C)∞ Un=0fn(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fk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.

A NOTE ON DISTRIBUTIONAL CHAOS OF PERIODICALLY ADSORBING SYSTEMS

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 k>0,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.