Given a ${\cal C}^{1+\gamma}$ 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 Hölder continuous set-valued
function defined
on Sullivan's dual Cantor set. We show the limit sets are themselves ${\cal
C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets,
with the highest degree of smoothness which occurs in the ${\cal
C}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the
following rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or
${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then the
conjugacy (with a different extension) is in fact already ${\cal
C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\cal
C}^{1+\gamma}$ 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 all
is a compact set which is the attractor for the semigroup: the collection of
limit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.
Regular Infinite Economies
