区间上重叠迭代函数系统的多重域的豪斯多夫维数

Pub Date : 2021-04-01 DOI:10.18910/79428

Kengo Shimomura

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引用次数: 1

摘要

我们考虑由两个相似率相同的压缩相似变换生成的单位区间上的迭代函数系统。当比率大于或等于$1/2$时，限制集是区间本身，并且代码映射不是一对一的。我们研究了具有唯一地址的极限集的点集。应用图有向马尔可夫系统的概念，得到了当相似率属于一定区间时集合的Hausdorff维数的一个公式。

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THE HAUSDORFF DIMENSION OF THE REGION OF MULTIPLICITY ONE OF OVERLAPPING ITERATED FUNCTION SYSTEMS ON THE INTERVAL

We consider iterated function systems on the unit interval generated by two contractive similarity transformations with the same similarity ratio. When the ratio is greater than or equal to $1/2$, the limit set is the interval itself and the code map is not one-to-one. We study the set of points of the limit set having unique addresses. We obtain a formula for the Hausdorff dimension of the set when the similarity ratio belongs to certain intervals by applying the concept of graph directed Markov system.

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