Gap sequences and Topological properties of Bedford-McMullen sets

Zhen Liang, Jun Jie Miao, Huo-Jun Ruan

Published 2020-03-18Version 1

In this paper, we study the topological properties and the gap sequences of Bedford-McMullen sets. First, we introduce a topological condition and a geometric condition: component separation condition (CSC) and exponential rate condition (ERC). Then we prove that the CSC implies the ERC, while both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences of Bedford-McMullen sets. Finally, we explore topological properties of Bedford-McMullen sets, and prove that all nonlinear Bedford-McMullen sets satisfy the CSC, which implies that the asymptotic estimate of gap sequences holds without any artificial restriction.