Gap sequences and Topological properties of Bedfordâ€"McMullen sets.

In this paper, we study the topological properties and the gap sequences of Bedfordâ€"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 Bedfordâ€"McMullen sets and prove that all normal Bedfordâ€"McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedfordâ€"McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence. [ABSTRACT FROM AUTHOR]