Measure linearity of bi-Lipschitz maps of self-similar Cantor sets

Abstract

Let \(C\) and \(C'\) be Cantor sets in \(\mathbb{R}^n\) generated by Euclidean similarities, called clone Cantor sets. There are associated Hausdorff measures \(\mu_C\) and \(\mu_{C'}.\) We show that if there is a bi-Lipschitz map \(\phi\) of \(\mathbb{R}^n\) which maps \(C\) onto a clopen subset of \(C'\) then there exists a constant \(\lambda \gt 0\) and a subset \(A\) of \(C\) with \(\mu_C(A)\gt 0\) and such that for all \(\mu_C\)-measurable sets \(B\) of \(A\) we have \(\phi(B)\) is \(\mu_{C'}\)-measurable and \(\mu_{C'}(\phi B)=\lambda\mu_C(B)\). This result leads to an almost complete classification of clone Cantor sets up to bi-Lipschitz maps of Euclidean space.