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

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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Published 16 March 2017