  We uncover previously unknown properties of the family of periodic
superstable cycles in unimodal maps characterized each by a Lyapunov exponent
that diverges to minus infinity. Amongst the main novel properties are the
following: i) The basins of attraction for the phases of the cycles develop
fractal boundaries of increasing complexity as the period-doubling structure
advances towards the transition to chaos. ii) The fractal boundaries, formed by
the preimages of the repellor, display hierarchical structures organized
according to exponential clusterings that manifest in the dynamics as
sensitivity to the final state and transient chaos. iii) There is a functional
composition renormalization group (RG) fixed-point map associated to the family
of supercycles. iv) This map is given in closed form by the same kind of
$q$-exponential function found for both the pitchfork and tangent bifurcation
attractors. v) There is a final stage ultra-fast dynamics towards the attractor
with a sensitivity to initial conditions that decreases as an exponential of an
exponential of time.
