Real bounds, ergodicity and negative Schwarzian for multimodal maps

Abstract

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