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

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)→ (F,ν) is a bi-Lipschitz map, then the Radon-Nykodym derivative df^*ν/dμ is a constant function on a subset E'⊂ E with μ(E')>0, where f^*ν(·)=ν(f(·)). 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.