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

Scaling of distributions of sums of positions for chaotic dynamics at band-splitting points