RUI: Families, Ramification, and Berkovich Spaces in Non-archimedean Dynamics

Amherst College, Amherst MA

Investigators

Abstract

This project concerns a number of open questions in non-archimedean dynamics, a field bridging the interface between number theory and traditional (archimedean) dynamical systems. In the past ten years, it has become clear that Berkovich spaces, a certain class of technical objects in non-archimedean analysis, are essential for the non-archimedean theory. In particular, the construction of a probability measure invariant under the dynamical system requires the use of Berkovich spaces. Also in the past decade, our understanding of ramification of functions on Berkovich spaces has grown, and with it has grown our ability to apply the existence of invariant measures to certain problems in non-archimedean dynamics. In addition, the use of one-parameter families has been used to great effect in constructing pathological examples of non-archimedean dynamical systems. Using all these newly available tools in tandem, the PI plans to study several open questions that have resisted previous less sophisticated attacks. The proposal draws on tools from both complex dynamics and non-archimedean analysis, and the problems to be studied have applications to the theory of arithmetic dynamics over global fields and hence to certain problems in Diophantine geometry. This project joins together the very different realms of dynamical systems and of number theory. On the one hand, non-archimedean dynamics is a subfield of arithmetic dynamics, which concerns a particular class of Diophantine geometry problems. Such problems, i.e., understanding the set of rational number solutions to a naturally arising set of polynomial equations, have been a major theme in number theory from the ancient Greeks through Fermat and into the present day. On the other hand, the study of dynamical systems, and especially of complex dynamics, has arisen far more recently, exhibiting not only a purely mathematical beauty but also spectacular computer drawings of fractals and related sets. This project's proposed study of non-archimedean dynamics thus draws on, builds on, and joins together both fields, the ancient and modern alike. In addition, as in two earlier successful projects, the PI plans to supervise some students in an REU summer research project to aid in their mathematical training. Depending on the interest of the students, the REU may involve some intensive computer computations to generate interesting examples; if so, any relevant data generated will be published or posted on the web, for the benefit of the larger research community. Naturally, any results will also be disseminated via websites such as ArXiv and publication in mathematical journals. In addition, the PI is currently writing a graduate-level textbook on dynamics in one non-archimedean variable, as the field has too few expository texts today.