On Teichmüller space of circle diffeomorphisms with Hölder continuous derivative

Matsuzaki [M1] introduced the Teichmuller space $$T_{0}^{\alpha }$$ of diffeomorphisms of the unit circle with Holder continuous derivatives and investigated its Schwarzian derivative model. This paper deals with the pre-Schwarzian derivative model $$T_{0}^{\alpha }(1)$$ of the Teichmuller space $$T_{0}^{\alpha }$$ . It is shown that $$T_{0}^{\alpha }(1)$$ is a connected open subset of $${\mathcal {B}}_{0}^{\alpha }(\Delta )$$ and the pre-Bers projection is a holomorphic split submersion in $$T_{0}^{\alpha }$$ .