The main focus of my research has been p-adic dynamics. I consider a rational function f in one variable defined over a p-adic field (or more generally, any non-archimedean field). Such a function can be viewed as a morphism mapping the projective line to itself. Letting fn denote f composed with itself n times (the nth iterate of f), I study the behavior of the family {fn}. The objects I usually study are the resulting Fatou and Julia sets of f, which are the regions of equicontinuity and chaos, respectively. I've developed notions of non-archimedean ``connected'' components to study the Fatou set; as a result, I've been able to parallel the theory of dynamics of connected components of the Fatou set of complex functions.
The goal of this research is twofold. First, I'd like to compare and contrast dynamics in the archimedean and non-archimedean settings; so far, both the similarities and the differences have been striking. And second, I hope that a good understanding of p-adic dynamics will lead to a better understanding of dynamical problems over global fields, like the counting of rational periodic points of a function defined over a number field.
List of papers (includes abstracts and dvi files)