Gap sequences and Topological properties of Bedford–McMullen sets*

Abstract In this paper, we study the topological properties and gap sequences of Bedford–McMullen sets. First, introduce a condition, component separation condition (CSC), geometric exponential rate (ERC). Then prove that CSC implies ERC, both them are sufficient conditions for obtaining asymptotic estimate sequences. We also explore sets all normal with infinitely many connected components satisfy CSC, from which obtain without any restrictions. Finally, apply our result to Lipschitz equivalence.