On the topology of fractal squares with controlled overlaps

Abstract

This paper establishes a complete topological trichotomy for fractal squares characterized by controlled overlap structures, thereby extending the foundational work of Lau, Luo and Rao [14]. Building on this framework, we apply a novel methodology rooted in hyperbolic graph theory to analyze the Lipschitz equivalence of a family of totally disconnected fractal squares. It is shown that the Lipschitz classification within this family is uniquely determined by the number of controlled overlaps.