Negative Schwarzian and cohomological inequality.
Abstract. We prove that analytic conjugacy class of
self map of a compact interval (circle)
with all critical points non-flat and all periodic points repelling
contains a map with negative Schwarzian derivative.
The main idea is to solve a cohomological inequality in the class of
essentially bounded measurable functions and then "smooth out" a
solution. In case of circle maps we need one more condition to solve the
cohomological inequality due to integral obstructions.