## Distortion estimates for negative Schwarzian mapsOne 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. |

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absolutely continuous ergodic absolutley continuous ADDRESS City area preserving properties attracting fixed point block number conformal maps consisting of monotone continuous with respect convex CORNELL UNIVERSITY defined dimensional maps disjoint union distortion and slope distortion dis(f DISTORTION ESTIMATES domain end of proof endpoint ergodic invariant measure ESTIMATES FOR NEGATIVE fm(Ji fn(J fn(x Folklore theorem following lemma follows from lemma fractional linear hence hyperbola h inflection point intervals of monotonicity je?J John Guckenheimer Lebesgue measure Lemma 1A let Jn Let ujt Markov Mathematical Sciences Institute measurable sets middle e-portion monotone branches mapping monotone sequence negative Schwarzian derivative NEGATIVE SCHWARZIAN MAPS number of branches OFFICE SYMBOL ORGANIZATION REPORT NUMBER(S ORGANIZATION U. S. Army partition points are iterated preimage proof Let proof The following repelling fixed point Research Triangle Park respect to Lebesgue SECURITY CLASSIFICATION sigma-finite absolutely continuous slope and maximal Stewart Johnson stopping time map SupiAi unit interval universal attractor weakly Bernoulli