Graphs, Curves and Dynamics

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 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 settin