Survey lecture on arithmetic dynamics

Abstract

Arithmetic dynamics is a relatively new field in which classical problems from number theory and algebraic geometry are reformulated in the setting of dynamical systems. Thus, for example, rational points on algebraic varieties become rational points in orbits, and torsion points on abelian varieties become points having finite orbits. Moduli problems also appear, where, for example, the complex multiplication points in the moduli space of abelian varieties correspond to the postcritically finite points in the moduli space of rational maps. In this article we give a survey of some of the major problems motivating the field of arithmetic dynamics, and some of the progress that has been made during the past 20 years.