  We explain how specific dynamical properties give rise to the limit
distribution of sums of deterministic variables at the transition to chaos via
the period-doubling route. We study the sums of successive positions generated
by an ensemble of initial conditions uniformly distributed in the entire phase
space of a unimodal map as represented by the logistic map. We find that these
sums acquire their salient, multiscale, features from the repellor preimage
structure that dominates the dynamics toward the attractors along the
period-doubling cascade. And we explain how these properties transmit from the
sums to their distribution. Specifically, we show how the stationary
distribution of sums of positions at the Feigebaum point is built up from those
associated with the supercycle attractors forming a hierarchical structure with
multifractal and discrete scale invariance properties.

Chaotic Dynamics