Limit Properties of Folded Sums of Chaotic Trajectories

We investigate the statistical properties of a process defined by summing the subsequent values assumed by the state of a chaotic map, and by constraining the result within a finite domain by means of a folding operation. It is found that the limit distribution is always uniform regardless of the chaotic map, that the folded sums tend to be independent of the future evolution of the chaotic trajectory, and that, whenever the map state is multidimensional, the folded sum vectors tend to be made of independent components. Numerical simulations are employed to show that practical finite-time behaviors are correctly approximated by the limit results herein provided. Finally, the theory is applied to give a formal ground to some key steps in the derivation of the spectrum of signals that are chaotically frequency modulated.