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arXiv · 1603.03077

The Analyst's traveling salesman theorem in graph inverse limits

Guy C. David·Raanan Schul
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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.

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Guy C. David, Raanan Schul. 2016-03-09. The Analyst's traveling salesman theorem in graph inverse limits. https://arxiv.org/abs/1603.03077

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