Infinite hierarchies of nonlinearly dependent periodic orbits

Quadratic maps are used to show explicitly that the skeleton of unstable periodic orbits underlying classical and quantum dynamics is stratified into a doubly infinite hierarchy of orbits inherited from a set of basic "seeds" through certain nonlinear transformations T(alpha)(x). The hierarchy contains nonunique substructurings which arise from the different possibilities of sequencing the transformations T(alpha)(x). The structuring of the orbital skeleton is shown to be generic for Abelian equations, i.e., for all dynamical systems generated by iterating rational functions.