Arithmetic Dynamics

Arithmetic Dynamics is the intersection of Number Theory and Dynamical Systems.

 

In the ‘classical’ study of diophantine equations, one looks for integer or rational solutions to polynomial equations, whilst in complex dynamics, one studies the behaviour of iterated polynomial or rational maps over the field of the complex numbers.

 

The natural way to combine the two areas is to study the behaviour of iterated rational maps over a number field K (e.g. the rational numbers), and to ask questions such as, ‘Are there infinitely many K-rational (pre)periodic points of a given rational map?’ or ‘When does the orbit of a point contain infinitely many integers?’

 

This reading group will be a gentle introduction to this highly interesting area. We will assume only a basic knowledge of Number Theory and Dynamical Systems and will spend the first few talks covering the required pre-requisite knowledge of these areas.

 

The authoritative text on the subject is Joseph Silverman’s ‘The Arithmetic of Dynamical Systems’, and we will tentatively aim to cover the first three chapters of this book, which cover an introduction to classical dynamics, dynamics over local fields (good reduction) and dynamics over global fields.

 

Depending on interest and how quickly we move through the book, we can then move onto the later chapters.