The set of maps \(F_{a,b}:x\mapsto x+a+{b\over 2\pi}\sin(2\pi x)\) with any given rotation interval is contractible (Q1901812)

The rotation interval \([\alpha, \beta]\) of a degree 1 endomorphism of the circle is a generalization of an invariant introduced by Poincaré, the rotation number, which is defined for degree 1 homeomorphisms of the circle. The family of maps \(F_{a,b} : x \mapsto x + a + {b \over 2 \pi} \sin (2 \pi x)\) is known as the standard family of maps of the circle. For \(|b |> 1\), these maps are not homeomorphisms. The authors consider the set \(S_{\alpha, \beta}\) of pairs \((a,b)\) such that the maps \(F_{a,b}\) have a given rotation interval \([\alpha, \beta]\) and show that this set is contractible. They give further information about the structure of the set according to whether \(\alpha\) and \(\beta\) are rational or irrational. The proof is in two parts, the first of which is based on real analysis, using the Schwarzian derivative and combinatorial techniques of one-dimensional dynamics. The second part is complex analytic and makes use of the techniques of quasiconformal mappings and Teichmüller theory.