  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.
