Rigidity of groups of circle diffeomorphisms and teichmüller spaces

We consider deformations of a group of circle diffeomorphisms with Hölder continuous derivative in the framework of quasiconformal Teichmüller theory and showcertain rigidity under conjugation by symmetric homeomorphisms of the circle. As an application, we give a condition for such a diffeomorphism group to be conjugate to a Möbius group by a diffeomorphism of the same regularity. The strategy is to find a fixed point of the group which acts isometrically on the integrable Teichmüller space with the Weil–Petersson metric.