A deformation of Penner's simplicial coordinate

Abstract: We produce a one-parameter family of coordinates ${\Psi_h}{h\in\mathbb{R}}$ of the decorated Teichm\"{u}ller space of an ideally triangulated punctured surface $(S,T)$ with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If $h\geqslant0$, the decorated Teichm\"{u}ller space in the $\Psi_h$ coordinate becomes an explicit convex polytope $P(T)$ independent of $h$, and if $h<0$, the decorated Teichm\"{u}ller space becomes an explicit bounded convex polytope $P_h(T)$ so that $P_h(T)\subset P{h'}(T)$ if $h<h'$. As a consequence, Bowditch-Epstein and Penner's cell decomposition of the decorated Teichm\"{u}ller space is reproduced.