Arithmetic dynamics

Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, $p$ -adic, and/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 refers to 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 classical dynamics in which one replaces the complex numbers $C$ by a $p$ -adic field such as $Q p$ or $C p$ 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:

Let $S$ be a set and let $F : S \to S$ be a map from $S$ to itself. The iterate of $F$ with itself $n$ times is denoted

A point $P \in S$ is periodic if $F (n) (P) = P$ for some $n > 1$ .

...
Wikipedia