  We expose at a previously unknown level of detail the features of the
dynamics of trajectories that either evolve towards the Feigenbaum attractor or
are captured by its matching repellor. Amongst these features are the
following: i) The set of preimages of the attractor and of the repellor are
embedded (dense) into each other. ii) The preimage layout is obtained as the
limiting form of the rank structure of the fractal boundaries between attractor
and repellor positions for the family of supercycle attractors. iii) The joint
set of preimages for each case form an infinite number of families of
well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps
in each of these families can be ordered with decreasing width in accord to
power laws and are seen to appear sequentially in the dynamics generated by
uniform distributions of initial conditions. v) The power law with log-periodic
modulation associated to the rate of approach of trajectories towards the
attractor (and to the repellor) is explained in terms of the progression of gap
formation. vi) The relationship between the law of rate of convergence to the
attractor and the inexhaustible hierarchy feature of the preimage structure is
elucidated.
