Some problems of arithmetic origin in rational dynamics

Abstract

These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the dynamics of rational maps on P^1. We first briefly introduce the basics of the iteration theory of rational maps on the projective line over C, as well as some elements of iteration theory over an arbitrary complete valued field and the construction of dynamically defined height functions for rational functions defined over \overline Q. The equidistribution of small points gives some original information on the distribution of preperiodic orbits, leading to some non-trivial rigidity statements. We then explain some consequences of arithmetic equidistribution to the study of the geometry of parameter spaces of such dynamical systems, notably pertaining to the distribution of special parameters and the classification of special subvarieties.