Arithmetic dynamics

Definition

Let $K$ be a field, let $N \ge 1$, and let $\phi \colon \mathbb{P}^N \to \mathbb{P}^N$ be a morphism of projective space defined over $K$. For $N = 1$ such a morphism is given by a rational function $\phi \in K(z)$. Write $\phi^n$ for the $n$-fold composition of $\phi$ with itself and $\phi^0 = \id$. For a point $P \in \mathbb{P}^N(K)$:

1. the forward orbit of $P$ is $\mathcal{O}_\phi(P) = \set{\phi^n(P) : n \ge 0}$;
2. $P$ is periodic if $\phi^n(P) = P$ for some $n \ge 1$;
3. $P$ is preperiodic if $\phi^m(P)$ is periodic for some $m \ge 0$, equivalently if $\mathcal{O}_\phi(P)$ is finite.

Arithmetic dynamics is the study of these orbits when $K$ is a field of arithmetic interest. Examples of such fields are an algebraic number field, a $p$-adic field or a finite field.