Point-wise doubling indices of measures and its application to bi-Lipschitz classification of Bedford-McMullen carpets

TL;DR

This paper introduces a point-wise doubling index for measures, calculates it for Bedford-McMullen carpets, and uses it to classify carpets bi-Lipschitzly, revealing invariants under such transformations.

Contribution

It defines a new measure-theoretic index for non-doubling measures and applies it to classify Bedford-McMullen carpets up to bi-Lipschitz equivalence.

Findings

Point-wise doubling indices calculated for Bedford-McMullen carpets.

Bi-Lipschitz equivalence implies identical fiber sequences for most carpets.

Introduces a new invariant for measure classification in fractal geometry.

Abstract

Doubling measure was introduced by Beurling and Ahlfors in 1956 and now it becomes a basic concept in analysis on metric space. In this paper, for a measure which is not doubling, we introduce a notion of point-wise doubling index, and calculate the point-wise doubling indices of uniform Bernoulli measures on Bedford-McMullen carpets. As an application, we show that, except a small class of Bedford-McMullen carpets, if two Bedford-McMullen carpets are bi-Lipschitz equivalent, then they have the same fiber sequence up to a permutation.