Distortion Estimates for Negative Schwarzian Maps.

Abstract

Abstract : One dimensional maps with a negative schwarzian derivative are shown to have area preserving properties: the distortion dis(f) = f'/(f')2 varies inversely proportional to the vertical distance from a critical point for maps f with negative Schwarzian; maps consisting of monotone branches mapping across an interval either have a sigma-finite absolutely continuous ergodic measure or a universal attractor at the ends of the interval. The Schwarzian derivative was defined H.A. Schwartz in connection with the study of conformal maps of the complex plane. The derivative has found an interesting application in the study of one dimensional maps where the assumption of a negative Schwarzian has been used to establish topological conjugacy between unimodal maps with identical kneading sequences. This and other one dimensional applications arise from inherent measure preserving properties of maps with a negative Schwarzian derivative. The authors attempt in this paper to make these properties more explicit.