  The stationary distributions of sums of positions of trajectories generated
by the logistic map have been found to follow a basic renormalization group
(RG) structure: a nontrivial fixed-point multi-scale distribution at the
period-doubling onset of chaos and a Gaussian trivial fixed-point distribution
for all chaotic attractors. Here we describe in detail the crossover
distributions that can be generated at chaotic band-splitting points that
mediate between the aforementioned fixed-point distributions. Self affinity in
the chaotic region imprints scaling features to the crossover distributions
along the sequence of band splitting points. The trajectories that give rise to
these distributions are governed first by the sequential formation of
phase-space gaps when, initially uniformly-distributed, sets of trajectories
evolve towards the chaotic band attractors. Subsequently, the summation of
positions of trajectories already within the chaotic bands closes those gaps.
The possible shapes of the resultant distributions depend crucially on the
disposal of sets of early positions in the sums and the stoppage of the number
of terms retained in them.
