Gap sequences and Topological properties of Bedford-McMullen sets

TL;DR

This paper investigates the topological and geometric properties of Bedford-McMullen sets, establishing conditions for gap sequence estimates and applying these results to Lipschitz equivalence.

Contribution

It introduces the component separation condition (CSC) and exponential rate condition (ERC), proving their implications and applying them to characterize Bedford-McMullen sets.

Findings

CSC implies ERC, both ensure gap sequence estimates

Normal Bedford-McMullen sets with infinitely many components satisfy CSC

Results are applied to Lipschitz equivalence of sets

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.