Teichmüller spaces of piecewise symmetric homeomorphisms on the unit circle

Abstract

We interpolate a new family of Teichmüller spaces $T_{\sharp}^X$ between the universal Teichmüller space $T$ and its little subspace $T_0$, which we call the Teichmüller space of piecewise symmetric homeomorphisms. This is defined by prescribing a subset $X$ of the unit circle. The inclusion relation of $X$ induces a natural inclusion of $T_{\sharp}^X$, and an approximation of $T$ is given by an increasing sequence of $T_{\sharp}^X$. In this paper, we discuss the fundamental properties of $T_{\sharp}^X$ from the viewpoint of the quasiconformal theory of Teichmüller spaces. We also consider the quotient space of $T$ by $T_{\sharp}^X$ as an analog of the asymptotic Teichmüller space.