## Institute for Mathematical Sciences

## Preprint ims90-10

** G. Swiatek**
* One-Dimensional Maps and Poincare Metric*

Abstract: Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian
derivative and others whose sum of distortions is bounded. If
the assumptions of the Koebe principle hold, we show that the
joint distortion of the composition is bounded. On the other
hand, if all maps with possibly non-negative Schwarzian
derivative are almost linear-fractional and their
nonlinearities tend to cancel leaving only a small total, then
they can all be replaced with affine maps with the same domains
and images and the resulting composition is a very good
approximation of the original one. These technical tools are
then applied to prove a theorem about critical circle maps.
(AMS subject code 26A18)

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