RUI: Galois Action and Entropy in Non-archimedean Dynamics

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This project concerns a number of open problems in non-archimedean dynamics, a field on the interface between number theory and traditional (archimedean) dynamical systems. This project joins together the very different realms of dynamical systems and of number theory. Diophantine problems, which ask about the set rational number solutions to polynomial equations, have been a major theme in number theory from ancient times to the present day. On the other hand, the study of dynamical systems has arisen far more recently, exhibiting not only a purely mathematical beauty but also spectacular computer drawings of fractals and related sets. This project draws on, builds on, and joins together both fields. In addition, as in three earlier successful projects, the investigator plans to supervise some students in an REU summer research project to aid in their mathematical training. Any computational data produced in the REU 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.One key class of problems concerns Galois actions on non-archimedean dynamical systems, related to the central number theory problem of understanding the absolute Galois group of the rational numbers. On the one hand, non-archimedean dynamics provides the local information needed in arithmetic dynamics, which in turn realizes itself as a particular kind of Diophantine problem. A second class of problems concerns the ergodic properties, especially the entropy, of such dynamical systems; the entropy is a number that measures the amount of chaos and unpredictability in the system. These two topics are tied together by the study of Julia sets in Berkovich spaces, which are technical objects that, in the past decade, have proven to be of central importance in the study of non-archimedean dynamics. The project also draws on tools from complex dynamics, ergodic theory, and non-archimedean analysis. The problems to be studied branch into new areas but are continuations of rich theories with long and storied histories. In particular, the Galois action problem promises to provide new (dynamical) tools for attacking the study of absolute Galois groups, while the study of the associated entropy issues promise to provide new examples in the study of the ergodic theory of dynamical systems.