Lipschitz classification of Bedford-McMullen carpets with uniform horizontal fibers

TL;DR

This paper characterizes when two Bedford-McMullen carpets with uniform horizontal fibers are Lipschitz equivalent by introducing a new structure tree concept for metric spaces.

Contribution

It introduces a novel structure tree framework to fully classify Lipschitz equivalence among certain self-affine carpets.

Findings

Complete characterization of Lipschitz equivalence in the specified class

Introduction of the structure tree concept for metric spaces

New criteria for carpet classification based on structure trees

Abstract

Let ${\cal M}_{t,v,r}(n,m)$, $2\leq m<n$, be the collection of self-affine carpets with expanding matrix $\diag(n,m)$ which are totally disconnected, possessing vacant rows and with uniform horizontal fibers. In this paper, we introduce a notion of structure tree of a metric space, and thanks to this new notion, we completely characterize when two carpets in ${\cal M}_{t,v,r}(n,m)$ are Lipschitz equivalent.