## Institute for Mathematical Sciences

## Preprint ims91-12b

** Y. Jiang**
* Dynamics of Certain Smooth One-Dimensional Mappings IV: Asymptotic Geometry of Cantor Sets.*

Abstract: We study hyperbolic mappings depending on a parameter $\varepsilon $. Each of them has an invariant Cantor set. As
$\varepsilon $ tends to zero, the mapping approaches the
boundary of hyperbolicity. We analyze the asymptotics of the gap
geometry and the scaling function geometry of the invariant
Cantor set as $\varepsilon $ goes to zero. For example, in the
quadratic case, we show that all the gaps close uniformly with
speed $\sqrt {\varepsilon}$. There is a limiting scaling
function of the limiting mapping and this scaling function has
dense jump discontinuities because the limiting mapping is not
expanding. Removing these discontinuities by continuous
extension, we show that we obtain the scaling function of the
limiting mapping with respect to the Ulam-von Neumann type metric.

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