Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set$C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down towarda point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to anergodic Cantor set valued process. The values of this process, calledlimit sets, are indexed by a Hölder continuous set-valuedfunction definedon Sullivan's dual Cantor set. We show the limit sets are themselves ${\calC}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets,with the highest degree of smoothness which occurs in the ${\calC}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to thefollowing rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then theconjugacy (with a different extension) is in fact already ${\calC}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\calC}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold,which is acted on by the semigroup given by rescaling subintervals.Smoothness classes nest down, and contained in the intersection of them allis a compact set which is the attractor for the semigroup: the collection oflimit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.