Lipschitz equivalence of self-similar sets

In 1997 David and Semmes asked whether there exists a bilipschitz map between the two compact self-similar subset M and $M^{\prime }$ of the real line defined by the relations $M=(M/5)\cup (M/5+2/5)\cup (M/5+4/5)$ and $M^{\prime }=(M^{\prime }/5)\cup (M^{\prime }/5+3/5)\cup (M^{\prime }/5+4/5)$. We answer this question positively. To cite this article: H. Rao et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).