Rigidity of Groups of Circle Diffeomorphisms and Teichmuller Spaces

We consider deformations of a group of circle diffeomorphisms with Holder continuous derivative in the framework of quasiconformal Teichmuller 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 Mobius 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 Teichmuller space with the Weil-Petersson metric.