Arithmetic dynamics[1] is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
Global arithmetic dynamics is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while local arithmetic dynamics, also called p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers C by a p-adic field such as Qp or Cp and studies chaotic behavior and the Fatou and Julia sets.
The following table describes a rough correspondence between Diophantine equations, especially abelian varieties, and dynamical systems:
| Diophantine equations | Dynamical systems |
|---|---|
| Rational and integer points on a variety | Rational and integer points in an orbit |
| Points of finite order on an abelian variety | Preperiodic points of a rational function |