When an Infinitely-Renormalizable Endomorphism of the Interval Can Be Smoothed

*** NOTE - For full text of paper contact Charles Tresser (tresser@watson.ibm.com). *** Let K be a closed subset of a smooth manifold M, and let f:K-->K be a continuous self-map of K. We say that f is smoothable if it is conjugate to the restriction of a smooth map by a homeomorphism of the ambient space M. We give a necessary condition for the smoothability of the faithfully infinitely interval-renormalizable homeomorphisms of Cantor sets in the unit interval. This class contains, in particular, all minimal homeomorphisms of Cantor sets in the line which extend to continuous maps of an interval with zero topological entropy.