分形列的收敛特性

Convergent Characteristic of Sequence of Fractals

对于完备度量空间 (X ,d)及相应的分形空间 (H(X) ,h) ,我们曾得到结论 :如果 (H(X) ,h)中的分形列 {An}收敛 ,则∪∞n =1An 为 (X ,d)的完全有界集 ,{An}的极限limn→∞ An=∩∞n =1∪∞m =nAm,并且 ,当{An}单调增加时 ,∪∞n =1An 的完全有界性亦成为 {An}收敛的充分条件。本文进一步研究了在除去 {An}的单调性情况下 ,借助于∩∞n=1∪∞m =nAm来寻找 {An}收敛的充分必要条件问题 ,得到了其收敛的特征 ,即 {An}收敛当且仅当∪∞n =1An 为 (X ,d)的完全有界集且∩∞n =1∪∞m =nAm =∩{Ani}∈S ∪∞i=1Ani,其中S为 {An}的子列全体 ,“———”表示闭包。

Let (X, d) be a complete metric space and (H(X), h) be the corresponding fractal space. The following facts were already known to us: (1) If a sequence of fractals {An} in (H(X), h) is convergent, then ∪n = 1∞An is totally bounded, and limn&rarr∞An = ∩n = 1∞ ∪m = n∞Am; (2) If {An} is monotone increasing, the totally bound of ∪n = 1∞An also become the sufficient condition for convergence of {An}. Under the condition of omitting the monotonicity of {An}, the problem of seeking necessary and sufficient condition for convergence of {An} was further studied in this paper, and the convergent characteristic of {An} was obtained, i.e., {An} is convergent if and only if ∪n = 1∞An is totally bounded in (X, d) and ∩n = 1∞ ∪m = n∞Am = ∩{Ani}ΕS ∪i = 1∞Ani, where S is the set of all subsequence of {An} and ' - ' denotes the closure of a set.