Annales Academiæ Scientiarum Fennicæ
Volumen 42, 2017, 649-692
New York University, Courant Institute of Mathematical Sciences
New York, NY 10012, U.S.A.; guydavid 'at' math.nyu.edu
Stony Brook University, Department of Mathematics
Stony Brook, NY 11794-3651, U.S.A.; schul 'at' math.sunysb.edu
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 Poincaré 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.
2010 Mathematics Subject Classification: Primary 28A75.
Key words: Beta numbers, metric space, traveling salesman, curvature.
Reference to this article: : The analyst's traveling salesman theorem in graph inverse limits. Ann. Acad. Sci. Fenn. Math. 42 (2017), 649-692.
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