Equidistribution of Preimages in Nonarchimedean Dynamics

Abstract

One of the central results in holomorphic dynamics in several variables is the equidistribution of preimages theorem, which constructs invariant probability measures for a large class of interesting dynamical systems, namely the class of polarized dynamical systems. This opens the door to studying the dynamics of these systems via ergodic theoretic methods, an approach that has proved to be fruitful. In recent years, as interest in nonarchimedean dynamics has grown, it has become apparent that an analogue of the equidistribution of preimages theorem should hold for polarized dynamical systems over nonarchimedean fields, and, indeed, an analogue has been shown for endomorphisms of the projective line over these fields. In this dissertation we formulate a precise conjectural statement of a nonarchimedean equidistribution of preimages theorem, and supply further evidence for it by proving it for generic polarized nonarchimedean dynamical systems with good reduction, in any dimension. In the case when the nonarchimedean field is trivially valued, these results give a exact analogue of the equidistribution of preimages theorem for generic polarized dynamical systems in every dimension.