RATIO GEOMETRY, RIGIDITY AND THE SCENERY PROCESS FOR HYPERBOLIC CANTOR SETS

Given a C1+y hyperbolic Canter set C, we study the sequence C-n,C-x of Canter subsets which nest down toward a point x in C. We show that C- n,C-x is asymptotically equal to an ergodic Canter set valued process. The values of this process, called limit sets, are indexed by a Holde r continuous set-valued function defined on Sullivan's dual Canter set . We show the limit sets are themselves Ck+y, C-infinity or C-w hyperb olic Canter sets, with the highest degree of smoothness which occurs i n the C1+y conjugacy class of C. The proof of this leads to the follow ing rigidity theorem: if two Ck+y, C-infinity or C-w hyperbolic Canter sets are C-1 conjugate, then the conjugacy (with a different extensio n) is in fact already Ck+y, C-infinity Or C-w. Within one C1+y conjuga cy class, each smoothness class is a Banach manifold, which is acted o n by the semigroup given by rescaling subintervals. Smoothness classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the C-1 norm.