1994/05/31 by Bedford, Tim, Fisher, Albert M.
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/9405217
Given a C1+γ hyperbolic Cantor set C, we study the sequence Cn,x of Cantor subsets which nest down toward a point x in C. We show that Cn,x is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves Ck+γ, C^∞ or Cω hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the C1+γ conjugacy class of C. The proof of this leads to the following rigidity theorem: if two Ck+γ, C^∞ or Cω hyperbolic Cantor sets are C1-conjugate, then the conjugacy (with a different extension) is in fact already Ck+γ, C^∞ or Cω. Within one C1+γ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Conjugacy 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 C1 norm.