Locally measure preserving property of bi-Lipschitz maps between Moran 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 $d{f}^{\ast }\nu /d\mu$ is a constant function on a subset $E^{\prime }\subset E$ with $\mu (E^{\prime })>0$, where $f^{\ast }\nu (\cdot )=\nu (f(\cdot ))$. 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.