The main focus of my research is arithmetic dynamics. Consider a rational function f in one variable with rational coefficients. Such a function can be viewed as a morphism mapping the projective line to itself. (More generally, one can allow f to be defined over a number field or other global or local field. Even more generally, one can work with endomorphisms of projective N-space or even of arbitrary varieties.)
Let f n denote f composed with itself n times (the nth iterate of f). One of the main problems in arithmetic dynamics is to understand the set of (rational) preperiodic points; that is, numbers x for which f n(x)=f m(x) for some nonnegative integers n>m. A related problem is to understand the points of small canonical height, which are, in a certain technical sense, almost preperiodic.
To study these problems, it is usually crucial to understand the dynamics over the associated metric fields; that is, over the real (or complex) numbers, and over every (non-archimedean) p-adic field. Much of my research, especially early in my career, has concerned non-archimedean Fatou and Julia sets, which are the regions of equicontinuity and chaos, respectively.
List of papers (includes abstracts and dvi files)