First-return maps as a unified renormalization scheme for dynamical systems

We propose to look at first-return maps into a specified region of phase space as a basis for a unified renormalization scheme for dynamical systems. The choice of the region for first return is dictated by the symbolic dynamics (e.g., kneading sequence) of the relevant trajectories. The renormalization group can be formulated on the symbolic level, but once translated to maps it yields the said renormalization scheme. We show how the well-studied examples of the onset of chaos via period doubling and quasiperiodicity fit into this approach, and argue that these problems get in fact unified. The unification leads also to a generalization that allows us to study the onset of chaos in maps that belong to larger spaces of functions than those usually considered. In these maps we discover a host of new scenarios for the onset of chaos. These scenarios are physically relevant since the maps considered are reductions of simple flows. We present a theoretical analysis of some of these new scenarios, and report universal results. Finally we show that all the available renormalization groups can be found using symbolic manipulations only.