Open Access
1996/1997 Measure linearity of bi-Lipschitz maps of self-similar Cantor sets
Hung Nhon Vuong
Author Affiliations +
Real Anal. Exchange 22(2): 574-589 (1996/1997).


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.


Download Citation

Hung Nhon Vuong. "Measure linearity of bi-Lipschitz maps of self-similar Cantor sets." Real Anal. Exchange 22 (2) 574 - 589, 1996/1997.


Published: 1996/1997
First available in Project Euclid: 22 May 2012

zbMATH: 0941.28006
MathSciNet: MR1460973

Keywords: biLipschitz , Cantor set , linearity

Rights: Copyright © 1996 Michigan State University Press

Vol.22 • No. 2 • 1996/1997
Back to Top