On the Separation Structure of Lalley-Gatzouras Fractals

TL;DR

This paper characterizes when Lalley-Gatzouras fractals are uniformly disconnected, identifies those quasisymmetrically equivalent to the Cantor set, and examines their gap sequence behavior.

Contribution

It provides a necessary and sufficient condition for uniform disconnection and classifies Lalley-Gatzouras sets based on quasisymmetric equivalence.

Findings

Identified conditions for uniform disconnection in Lalley-Gatzouras sets

Classified sets quasisymmetrically equivalent to the Cantor set

Analyzed the limit behavior of gap sequences

Abstract

We obtain a necessary and sufficient condition for Lalley-Gatzouras sets to be uniform disconnected. This enable us to find all Lalley-Gatzouras sets which are quasisymmetrically equivalent to the Cantor ternary set. As another application, we also study the limit behavior of the gap sequence of Lalley-Gatzouras sets.