Real Bounds, Ergodicity and Negative Schwarzian for Multimodal Maps

Over the last 20 years, many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps (or perturbations and complex extensions of such maps). Primarily, this is because in the one-dimensional case, much better distortion control can be obtained than for general dynamical systems. However, many of these spectacular results were obtained so far only for unimodal maps. The aim of this paper is to provide all the tools for studying general multimodal maps of an interval or a circle, by obtaining • real bounds controlling the geometry of domains of certain first return maps, and providing a new (and we believe much simpler) proof of absense of wandering intervals; • provided certain combinatorial conditions are satisfied, large real bounds implying that certain first return maps are almost linear; • Koebe distortion controlling the distortion of high iterates of the map, and negative Schwarzian derivative for certain return maps (showing that the usual assumption of negative Schwarzian derivative is unnecessary); • control of distortion of certain first return maps; • ergodic properties such as sharp bounds for the number of ergodic components. We will give historical comments below the statements of the theorems. There are many applications and potential applications of our bounds. For example, it is clear that any future renormalization results for multimodal maps (generalizing the unimodal results of Sullivan, McMullen, Lyubich, de Melo, Avila ...) would require real bounds. Our real bounds are one of the key ingredients in the proof that Axiom A maps are dense within the space of real polynomials with real critical points; see [7]. Let us now be more precise. Let M = [−1, 1] or M = S, and let f : M → M be a smooth map. This map is called multimodal if M has a partition into finitely many subintervals on which f is strictly monotone. Without loss of generality, we may and will assume that f(∂M) ⊂ ∂M . Let c1, . . . , cd be the critical points of f ,