The Analyst's traveling salesman theorem in graph inverse limits

TL;DR

This paper extends Peter Jones' Traveling Salesman Theorem to complex non-Euclidean spaces formed as inverse limits of metric graphs, establishing criteria for rectifiability based on flatness measures.

Contribution

It introduces a version of the Traveling Salesman Theorem applicable to Laakso and Cheeger-Kleiner spaces, broadening the understanding of rectifiability in non-Euclidean metric spaces.

Findings

Sets are contained in rectifiable curves if flatness is controlled.

Provides a quantitative criterion for rectifiability in these spaces.

First analysis of rectifiability using Jones' theorem in inverse limit spaces.

Abstract

We prove a version of Peter Jones' Analyst's traveling salesman theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and generalized by Cheeger-Kleiner. These spaces are constructed as inverse limits of metric graphs, and include examples which are doubling and have a Poincare inequality. We show that a set in one of these spaces is contained in a rectifiable curve if and only if it is quantitatively "flat" at most locations and scales, where flatness is measured with respect to so-called monotone geodesics. This provides a first examination of quantitative rectifiability within these spaces.