Fixed points indices and period-doubling cascades

Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, \({F : \mathbb{R}\times\mathfrak{M} \rightarrow \mathfrak{M},}\) where \({\mathfrak{M}}\) is a locally compact manifold without boundary, typically \({\mathbb{R}^N}\). In particular, we investigate F(μ, ·) for \({\mu \in J = \mu_{1}, \mu_{2}]}\), when F(μ ₁, ·) has only finitely many periodic orbits while F(μ 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 μ ₂. Furthermore, all but finitely many of these regular orbits at μ ₂ are tethered to their own period-doubling cascade. Specifically, each orbit ρ at μ ₂ lies in a connected component C(ρ) of regular orbits in \({J \times \mathfrak{M}}\); 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(μ ₂) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in \({J \times \mathfrak{M}}\).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.