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arXiv · 2501.02426

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

Hui Rao·Yan-Li Xu·Yuan Zhang
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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.

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Hui Rao, Yan-Li Xu, Yuan Zhang. 2025-01-05. Point-wise doubling indices of measures and its application to bi-Lipschitz classification of Bedford-McMullen carpets. https://arxiv.org/abs/2501.02426

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