Teichmüller spaces of piecewise symmetric homeomorphisms on the unit circle

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