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 Poincaré 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 non-Archimedean analogue of Riemann surfaces. In this case, the mere absolute continuity of

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... 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 non-Archimedean 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 and Petri on equations for the canonical embedding of a curve. G. Cornelissen and J. Kool There exist versions of this result for more general Fuchsian groups; i.e., Schottky groups, where, instead of Lebesgue measure, the more general Patterson-Sullivan measure on the boundary (=limit set) is used [19, 25] . There are also analogues for more general subgroups of isometry groups of hyperbolic space that, for example, apply to graphs, cf. Bowen [4], Coornaert [5], Hersonsky and Paulin [12], Tukia [28] and Yue [29]. As is well known, there is a p-adic, or more general non-Archimedean, analogue of uniformization of Riemann surfaces by Schottky groups, namely, Mumford's theory of uniformization by non-Archimedean Schottky groups (cf., for example, Mumford [11, 18] ).