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×M → M, where M is a locally compact manifold without boundary, typically R . In particular, we investigate F (μ, ·) for μ ∈ J = [μ1, μ2], when F (μ1, ·) has only finitely many periodic orbits while F (μ2, ·) 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 μ2. Furthermore, all but finitely many of these regular orbits at μ2 are tethered to their own period-doubling cascade. Specifically, each orbit ρ at μ2 lies in a connected component C(ρ) of regular orbits in J ×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 ρ is “tethered” or “tied” to a unique cascade. When F (μ2) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in J ×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. Mathematics Subject Classification (2010). 37G, 37B.