Asymptotic Differentiable Structure on Cantor Set

We study hyperbolic maps depending on a parameter ε. Each of them has an invariant Cantor set. As ε tends to zero, the map approaches the boundary of hyperbolicity. We analyze the asymptotics of scaling function of the invariant Cantor set as ε goes to zero. We show that there is a limiting scaling function of the limiting map and this scaling function has dense jump discontinuities because the limiting map is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting map with respect to a Ulam-von Neumann type metric.