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 $T_{\alpha }\mathrm{}\left(x\right).$ The hierarchy contains nonunique substructurings which arise from the different possibilities of sequencing the transformations $T_{\alpha }\mathrm{}\left(x\right).$ 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.