Fixed points indices and period-doubling cascades

Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, F : R x M -> M, where M is a locally compact manifold without boundary, typically R-N. In particular, we investigate F(mu, .) for mu is an element of J - [mu(1), mu(2)], when F(mu(1), .) has only finitely many periodic orbits while F(mu(2), .) has exponential growth of the number of periodic orbits as a function of the period. For generic F, under additional hypotheses, we use a fixed point index argument to show that there are infinitely many "regular" periodic orbits at mu(2). Furthermore, all but finitely many of these regular orbits at mu(2) are tethered to their own period-doubling cascade. Specifically, each orbit rho at mu(2) lies in a connected component C(rho) of regular orbits in J x M; different regular orbits typically are contained in different components, and each component contains a period-doubling cascade. These components are one-manifolds of orbits, meaning that we can reasonably say that an orbit rho is "tethered" or "tied" to a unique cascade. When F(mu(2)) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in J x M. As corollaries of our main results, we give several examples, we prove that the map in each example has infinitely many cascades, and we count the cascades.