## Institute for Mathematical Sciences

## Preprint ims92-6

** F. Przytycki and F. Tangerman**
* Cantor Sets in the Line: Scaling Function and the Smoothness of the Shift Map.*

Abstract: Consider $d$ disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps
each of these subintervals to the whole unit interval. The set
of points where all iterates of this expanding map are defined
is a Cantor set. Associated to the construction of this Cantor
set is the scaling function which records the infinitely deep
geometry of this Cantor set. This scaling function is an
invariant of $C^1$ conjugation. We solve the inverse problem
posed by Dennis Sullivan: given a scaling function, determine
the maximal possible smoothness of any expanding map which
produces it.

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