Measure linearity of bi-Lipschitz maps of self-similar Cantor sets. (English) Zbl 0941.28006

Summary: 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 \(\mu _{C'}\), respectively. 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 \(\lambda >0\) and a subset \(A\) of \(C\) with \(\mu _C(A) >0\) 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 the Euclidean space.


28A80 Fractals
28A78 Hausdorff and packing measures
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets