奇数分康托集与其平移并集的自相似性

Self-similarity of the union of odd-part Cantor set and its translation

考虑迭代函数系{fj(x)=(x+2j)/p}j(=p0-1)/2,其中p是奇数,且p≥3,则存在自相似集Ep,满足Ep=∪(p-1)/2j=0fj(Ep).讨论自相似集Ep与其平移Ep+β的并集的自相似性.证明方法从与奇数进制有关的引理入手,由特殊到一般给出一般表达式,最终得出β=2×pn为判断奇数分康托集与其平移的并集的自相似性提供了依据.

The iterated funtion system {fj（x）=（x＋2j）/p}j=0^（p-1）/2 is considered,p is an odd number,and p≥3,then there exists self-similar set Ep which satisfies Ep=∪j=0^（p-1）/2fj（Ep）.Self-similarity of the union of the self-similar set Ep and its translation was discussed.Start with the lemma of the odd band,method of proof is from special to general,and the general expression was given.To provide a basis for judging the self-similarity of the union of odd-part Cantor set and its translation,β=2×p^n is gained finally.