Wasserstein Distance and the Rectifiability of Doubling Measures: Part II

TL;DR

This paper investigates the structure of doubling measures using Wasserstein distances, demonstrating that self-similar measures have flat tangent measures and their support can be decomposed into rectifiable components.

Contribution

It introduces a novel approach using Wasserstein distances to analyze self-similarity and rectifiability of doubling measures, providing new structural insights.

Findings

Measures with self-similarity have unique flat tangent measures

Support decomposes into rectifiable pieces of various dimensions

Wasserstein distance estimates are key to structural analysis

Abstract

We study the structure of the support of a doubling measure by analyzing its self-similarity properties, which we estimate using a variant of the $L^1$ Wasserstein distance. We show that measure satisfying certain self-similarity conditions admits a unique (up to multiplication by a constant) flat tangent measure at almost every point. This allows us to decompose the support into rectifiable pieces of various dimensions.