混沌集的基数

Cardinalities of Scrambled Sets

在数学中,有许多关于混沌函数、混沌集及其基数结果。为此证明了对于任意基数α≥|R|,存在度量空间X上的连续映射f,使得f有一个基数为α的混沌集,但没有基数为β(>α)的混沌集。而后通过构造右移同胚的方式,证明了存在连通的紧致度量空间(X,d)及其上面的连续映射f:X→X,使得f有一个不可数混沌集但没有可数无限的c混沌集,其中c为任意正实数。

Chaotic phenomena have attracted a great deal of attention in recent years. There are many results about chaotic functions and scrambled sets in the mathematics field. For any cardinality α≥｜R｜ , where ｜ R ｜ denotes the cardinality of the real numbers, there exists a continuous map f from a metric space X to itself which has a scrambled set with cardinality a but has no scrambled set with greater cardinality. Then, it was shown that there exist compactly connected metric space （X, d ） and a continuous map f from X itself, so that f has a uncountable scrambled set but has no countable infinite positively scrambled set.