In R.C.Penner "Decorated Teichmuller theory of boarded surface", on Page 7 and 8, it says that (without proof) the Teichmuller space of surface with $s$ labelled punctures and $r$ labelled boundary components and one marked point on each boundary is homeomorphic to an open ball of dimension $6g-6+2s+4r$, where is the proof of that?I never see the version of T space that has marked points on the boundary, is also that they all homeomorphic to a ball like usual? Is there any reference about this?Thank you!
You could check the following nice and rather elementary paper, which I believe contains answers to your your questions as well as nice coordinate systems on this Teichmüller space (if there is at least one marked point).
Dual Teichmuller and lamination spaces, V.V. Fock, A.B. Goncharov, arXiv:math/0510312
(However as pointed out by Kevin Lin those are well-known questions and I'm sure there are many other possible references.)