设$X$是一个紧致度量空间, $f:X\to X$是一个连续映射.若存在$f$的一个$m$-周期点$p$和另一个$m^{\prime }$-周期点$q(p\ne q)$,使得对任意非空开集$V\subset X$,都有$\{p,q\}\subset \bigcup _{n=0}^{\mathrm{\infty }}{f}^n(V)$,则称动力系统$(X,f)$是一个 $(m,m^{\prime })$型周期吸附系统. 证明了:$1)$ 若$(X,f)$是一个$(m,m^{\prime })$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$(m,m^{\prime })$型周期吸附系统且拓扑共轭于$(X^{\prime },f^{\prime })$, 则 $(X^{\prime },f^{\prime })$也是一个$(m,m^{\prime })$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} ⊂∞Sn=0 fn(V ) for any nonempty open set V ⊂ X, where p 6= 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 fk 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.