  One can describe isomorphism of two compact hyperbolic Riemann surfaces of
the same genus by a measure-theoretic property: a chosen isomorphism of their
fundamental groups corresponds to a homeomorphism on the boundary of the
Poincar\'e disc that is absolutely continuous for Lebesgue measure if and only
if the surfaces are isomorphic.
  In this paper, we find the corresponding statement for Mumford curves, a
nonarchimedean analog of Riemann surfaces. In this case, the mere absolute
continuity of the boundary map (for Schottky uniformization and the
corresponding Patterson-Sullivan measure) only implies isomorphism of the
special fibers of the Mumford curves, and the absolute continuity needs to be
enhanced by a finite list of conditions on the harmonic measures on the
boundary (certain nonarchimedean distributions constructed by Schneider and
Teitelbaum) to guarantee an isomorphism of the Mumford curves.
  The proof combines a generalization of a rigidity theorem for trees due to
Coornaert, the existence of a boundary map by a method of Floyd, with a
classical theorem of Babbage-Enriques-Petri on equations for the canonical
embedding of a curve.
