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

TL;DR

This paper extends the understanding of measure-preserving properties of bi-Lipschitz maps from self-similar and self-affine sets to Moran sets, showing such maps preserve measure locally in this broader context.

Contribution

It demonstrates that bi-Lipschitz maps between Moran sets also exhibit local measure-preserving properties, expanding the class of fractal sets where this property holds.

Findings

Bi-Lipschitz maps between Moran sets preserve measure locally.

The measure-preserving property extends from self-similar to Moran sets.

Supports Lipschitz classification of a broader class of fractal sets.

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 $f:(E,\mu)\to (F,\nu)$ is a bi-Lipschitz map, then the Radon-Nykodym derivative $df^*\nu/d\mu$ is a constant function on a subset $E'\subset E$ with $\mu(E')>0$, where $f^*\nu(\cdot)=\nu(f(\cdot))$. 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.