Asian Journal of Mathematics

Volume 21 (2017)

Number 1

A criterion for a finite union of intervals to be a self-similar set satisfying the open set condition

Pages: 185 – 196

DOI: http://dx.doi.org/10.4310/AJM.2017.v21.n1.a6

Authors

Zhi-Ying Wen (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Xuan Zhao (National Education Examinations Authority, Beijing, China)

Abstract

Let $k_1, k_2, \dotsc , k_m; \lambda_1, \lambda_2, \dotsc , \lambda_{m-1}$ be positive numbers. Let $K$ $(k_1, k_2, \dotsc ,k_m; \lambda_1, \lambda_2, \dotsc , \lambda_{m-1})$ be the union of $m$ closed intervals of lengths $k_1, k_2, \dotsc , k_m$ and gap lengths $ \lambda_1, \lambda_2, \dotsc , \lambda_{m-1}$. In this paper, we will give a characterization over $k_1, k_2, \dotsc , k_m$ and $ \lambda_1, \lambda_2, \dotsc , \lambda_{m-1}$ such that $K(k_1, k_2, \dotsc , k_m; \lambda_1, \lambda_2, \dotsc , \lambda_{m-1})$ is a self-similar set satisfying the open set condition.

Keywords

self-similar sets, open set condition, multiple word, non-negative matrices, common eigenvector

2010 Mathematics Subject Classification

Primary 28A80. Secondary 28A75.

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