ON THE MAGNIFICATION OF CANTOR SETS AND THEIR LIMIT MODELS

For a C-1+y hyperbolic (cookie-cutter) Canter set C we consider the li mits of sequences of closed subsets of R obtained by arbitrarily high magnifications around different points of C. It is shown that a well d efined set of limit models exists for the infinitesimal geometry, or s cenery, in the Cantor set. If (C) over tilde is a diffeomorphic copy o f C then the set of limit models of (C) over tilde is the same as that of C. Furthermore every limit model is made of Canter sets which are C-1+y diffeomorphic with C (for some gamma>0, gamma is an element of(0 , 1)), but not all such C-1+y copies of C occur in the limit models. W e show the relation between this approach to the asymptotic structure of a Canter set and Sullivan's ''scaling function''. An alternative de finition of a Fractal is discussed.