Author:
Liang Zhen,Miao Jun Jie,Ruan Huo-Jun
Abstract
Abstract
In this paper, we study the topological properties and the gap sequences of Bedford–McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford–McMullen sets and prove that all normal Bedford–McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedford–McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence.
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
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