Lipschitz equivalence of self-similar sets with exact overlaps

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 $k\ge 2$ such that $x^{2k}-m{x}^k+n$ is reducible, in particular, $m$ belongs to $\{a^i:a\in \mathbb{N}$ with $i\ge 2\}.$