  Given a $C^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence
$C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show
that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process.
The values of this process, called limit sets, are indexed by a H\"older
continuous set-valued function defined on D. Sullivan's dual Cantor set. We
show the limit sets are themselves $C^{k+\gamma}, C^\infty$ or $C^\omega$
hyperbolic Cantor sets, with the highest degree of smoothness which occurs in
the $C^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the
following rigidity theorem: if two $C^{k+\gamma}, C^\infty$ or $C^\omega$
hyperbolic Cantor sets are $C^1$-conjugate, then the conjugacy (with a
different extension) is in fact already $C^{k+\gamma}, C^\infty$ or $C^\omega$.
  Within one $C^{1+\gamma}$ 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 $C^1$ norm.
