## Institute for Mathematical Sciences

## Preprint ims94-9

** T. Bedford & A. Fisher**
* Ratio Geometry, Rigidity and the Scenery Process for Hyperbolic Cantor Sets*

Abstract: 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.

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