Lipschitz equivalence of self-similar sets with touching structures

TL;DR

This paper investigates the Lipschitz equivalence of self-similar fractal sets with touching structures in one dimension, extending known results to sets with arbitrarily many branches using a new geometric condition.

Contribution

It introduces the concept of substitutable self-similar sets and establishes Lipschitz equivalence results for sets with complex touching structures in R.

Findings

Established Lipschitz equivalence criteria for self-similar sets with touching structures in R.

Extended previous results from sets with up to 3 branches to arbitrarily many branches.

Introduced the geometric condition called 'substitutable' for analyzing self-similar sets.

Abstract

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much more challenging and intriguing at the same time. So far the only known results only cover self-similar sets in $\bR$ with no more than 3 branches. In this study we establish results for the Lipschitz equivalence of self-similar sets with touching structures in $\bR$ with arbitrarily many branches. Key to our study is the introduction of a geometric condition for self-similar sets called {\em substitutable}.