Measure theoretic rigidity for Mumford curves

Measure theoretic rigidity (in the style of Mostow) states
that two compact hyperbolic Riemann surfaces of the same genus are
isomorphic if and only if the "boundary map" associated to their
uniformizations is absolutely continuous. In this talk, we will
formulate the analog of this result for "p-adic Riemann surfaces",
i.e., for Mumford curves. In this case, the mere absolute continuity
of the boundary map implies only isomorphism of the special fibers of
the Mumford curves, and needs to be enhanced by a finite list of
conditions on the harmonic measures (in the sense of Schneider and
Teitelbaum) on the boundary to guarantee an isomorphism of the Mumford
curves.