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arXiv · 2407.10161

Locally measure preserving property of bi-Lipschitz maps between Moran sets

Liang-yi Huang·Shishuang Liu
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Abstract

In literature it is shown that bi-Lipschitz maps between self-similar sets or self-affine sets enjoy a locally measure preserving property, namely, if $f:(E,μ)\to (F,ν)$ is a bi-Lipschitz map, then the Radon-Nykodym derivative $df^*ν/dμ$ is a constant function on a subset $E'\subset E$ with $μ(E')>0$, where $f^*ν(\cdot)=ν(f(\cdot))$. Indeed, this measure preserving property plays an important role in Lipschitz classification of fractal sets. In this paper, we show that such measure preserving property also holds for bi-Lipschitz maps between two Moran sets in a certain class.

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Liang-yi Huang, Shishuang Liu. 2024-07-28. Locally measure preserving property of bi-Lipschitz maps between Moran sets. https://arxiv.org/abs/2407.10161

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