Applied Mathematics Seminar

Integrable dynamical systems have a distinguished history and model many natural phenomena. These systems have no chaos and the powerful KAM theory tells us that some of their quasiperiodic dynamics persists under non-integrable perturbation. In the past decade, there has been intense interest in integrable systems where time is discrete, i.e. integrable difference equations and integrable maps. The following question has attracted much attention: how do we know a priori whether a discrete system might be integrable, and how do we distinguish integrable from near integrable ?
We present a new method for testing integrability in rational maps of the real plane. We exploit the idea that possession of a (rational) integral by a map is an algebraic property that survives if we consider the reduction of the map over a finite field. Results from arithmetic geometry, in particular the celebrated Hasse-Weil bound on the number of points on an algebraic curve, explain why we find markedly different orbit length statistics depending upon whether the original map over the real plane was integrable or not. We discuss possible extensions to higher dimensional maps and to other dynamical properties. We will assume little knowledge of dynamical systems or arithmetic geometry.