Cell decompositions of Teichm\"uller spaces of surfaces with boundary

A family of coordinates ${\psi }_h$ for the Teichm"uller space of a compact surface with boundary was introduced in \cite{l2}. In the work \cite{m1}, Mondello showed that the coordinate ${\psi }_0$ can be used to produce a natural cell decomposition of the Teichm"uller space invariant under the action of the mapping class group. In this paper, we show that the similar result also works for all other coordinate ${\psi }_h$ for any $h\ge 0$.