p-adic geometry and dynamics

Complex dynamics in a single variable is a well-studied field pioneered by Gaston Julia and Pierre Fatou.It deals with the iteration of rational maps on the Riemann sphere.Many beautiful fractal images arise such as the Mandelbrot set or the Julia sets of functions.As explored in [Sil07], dynamical systems has applications to arithmetic problems.The idea prompting this thesis was to look at the dynamics of p-adic correspondences and derive arithmetic information.We start our study of dynamics with the classical complex case.A quick diversion into nonarchimedean analysis is then needed before we move on to nonarchimedean dynamics, initially over P 1 (K) where K is a nonarchimedean field.Many results similar to those in the complex setting hold in this nonarchimedean situation.After illustrating some of these similarities, as well as some differences, we move on to build the theory of nonarchimedean geometry.Starting from the basics, we construct the category of rigid analytic spaces and give some important examples.These spaces, while interesting in their own right, are not true topological spaces and hence not apt spaces on which to study dynamics.In light of this, we construct Berkovich spaces (this time truly topological spaces) in the next section and give many examples.In our penultimate section, we look at how to define dynamics on P 1 Berk (K) by taking advantage of the tree structure.Finally, we explore some questions about dynamics of nonarchimedean correspondences, focusing specifically on the quadratic case.for their help with this thesis and more generally my development as a mathematician and researcher.They both have spent many hours over the past few years explaining things to me as well as working together on problems.In a field that can make it hard to maintain self-confidence, my supervisors were always kind, supportive, and intelligent.Their wealth of knowledge about technical details and support for my research career were incredibly useful.The idea for this thesis formed in the summer of 2014 when I was doing summer research with Professors Goren and Kassaei as an undergraduate student.Two other undergraduate students, Eric Stubley and Nicolas Resch, were working with me at the time.They were both wonderful to work with, very passionate, and taught me a ton of math.I wouldn't be the same caliber mathematician if it wasn't for their help.I would like to thank all my fellow mathematics students at McGill for working on problems with me, hanging out when I didn't want to work, and generally creating a nice community.Helen, the mathematics graduate program coordinator, provided invaluable help with my degree.I also received financial support during my Master's program from NSERC, my co-supervisors, and the McGill mathematics department in the form of a

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