Berkovich space dynamics

Definition

Let $K$ be an algebraically closed field that is complete for a nontrivial non-Archimedean absolute value $\abs{\cdot}$, such as $\C_p$. The Berkovich affine line $\mathbb{A}^1_{\mathrm{Berk}}$ is the set of maps $x \colon K[T] \to [0, \infty)$, written $f \mapsto [f]_x$, such that:

1. $[f g]_x = [f]_x [g]_x$ for all $f, g \in K[T]$;
2. $[f + g]_x \le \max([f]_x, [g]_x)$ for all $f, g \in K[T]$;
3. $[c]_x = \abs{c}$ for every constant $c \in K$.

It carries the weakest topology for which $x \mapsto [f]_x$ is continuous for every $f \in K[T]$. The Berkovich projective line is $\mathbb{P}^1_{\mathrm{Berk}} = \mathbb{A}^1_{\mathrm{Berk}} \cup \set{\infty}$, the one-point compactification. Each point $a \in K$ gives the point $[f]_a = \abs{f(a)}$, so $\mathbb{P}^1(K)$ is a dense subset of $\mathbb{P}^1_{\mathrm{Berk}}$. Every rational map $\varphi \in K(z)$ of degree $d \ge 1$ extends uniquely to a continuous map $\varphi \colon \mathbb{P}^1_{\mathrm{Berk}} \to \mathbb{P}^1_{\mathrm{Berk}}$; for a polynomial $\varphi$ it is given by $[f]_{\varphi(x)} = [f \circ \varphi]_x$. Berkovich space dynamics is the study of the iterates $\varphi^n$ of this extended map on $\mathbb{P}^1_{\mathrm{Berk}}$, and more generally of endomorphisms of Berkovich analytic spaces.