Period-doubling cascades galore

The appearance of numerous period-doubling cascades is among the most prominent features of parametrized maps, that is, smooth one-parameter families of maps F : R x m -> m, where m is a smooth locally compact manifold without boundary, typically R(N). Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe that, whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic F.