Lipschitz equivalence of self-similar sets with exact overlaps

TL;DR

This paper investigates the conditions under which certain self-similar sets with overlaps are Lipschitz equivalent to those with strong separation, providing algebraic criteria involving reducibility of specific polynomials.

Contribution

It establishes a necessary algebraic condition for Lipschitz equivalence between self-similar sets with overlaps and those satisfying the strong separation condition.

Findings

Identifies a polynomial reducibility condition linked to Lipschitz equivalence.

Shows that the number of overlaps must belong to a specific set of powers.

Provides a criterion involving the reducibility of $x^{2k}-mx^{k}+n$.

Abstract

In this paper, we study a class $\mathcal{A}(\lambda ,n,m)$ of self-similar sets with $m$ exact overlaps generated by $n$ similitudes of the same ratio $ \lambda $. We obtain a necessary condition for a self-similar set in $\mathcal{A}(\lambda ,n,m)$ to be Lipschitz equivalent to a self-similar set satisfying the strong separation condition, i.e., there exists an integer $ k\geq 2$ such that $x^{2k}-mx^{k}+n$ is reducible, in particular, $m$ belongs to $\{a^{i}:a\in \mathbb{N}$ with $i\geq 2\}.$