Analysis and dynamics on the Berkovich projective line

Abstract

This is a set of expanded lecture notes from the Berkovich Space seminar held at the University of Georgia during Spring, 2004. The purpose of the notes is to provide a non-technical introduction to Berkovich spaces, and to develop the foundations for analysis on the Berkovich projective line, with a view toward applications in dynamics. After describing the underlying topological space and the sheaf of functions on the Berkovich line, we introduce the Hsia kernel, the fundamental kernel for potential theory. We develop a theory of capacities, define a Laplacian operator, and construct a theory of harmonic functions. We then develop the theory of subharmonic functions and give applications to dynamics, including a construction of the Lyubich measure attached to a rational function.