Abstract

This thesis has three main subjects. The first subject is Measure-theoretic rigidity of Mumford Curves. 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
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disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this thesis, we find the corresponding statement for Mumford curves, a non-Archimedean 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 minimimal models of Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary 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. The second subject is the gonality of curves. We present a method to control gonality of nonarchimedean curves based on graph theory. Let K denote the fraction field of an excellent discrete valuation ring. We first prove a lower bound for the gonality of a curve over the algebraic closure of K in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinementof the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some ``volume'' of the original graph; this can be seen as a substitute for graphs of the Li--Yau inequality from differential geometry, although we also prove that the strict analogue of this conjecture fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups that is linear in the index, with a constant that only depends on the residue field degree and the degree of the chosen ``infinite'' place. This is a function field analogue of a theorem of Abramovich for classical modular curves. The third subject is Dynamics measured in a non-Archimedean field. We study dynamical systems using measures taking values in a non-Archimedean field. The underlying space for such measure is a zero-dimensional topological space. In this chapter we elaborate on the natural translation of several notions, e.g., probability measures, isomorphic transformations, entropy, from classical dynamical systems to a non-Archimedean setting
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