---
tag: 6RM4RWWSSX
kind: concept
name: Berkovich space dynamics
msc: [37P50, 37P40, 14G22]
specializes: [5AA9AJEARJ]
sources:
  - url: https://en.wikipedia.org/wiki/Berkovich_space
---

## Summary

Berkovich space dynamics studies the iteration of a rational map over a complete non-Archimedean field on the Berkovich projective line.
More generally, it studies endomorphisms of Berkovich analytic spaces.
The field itself, such as $\C_p$, is totally disconnected and not locally compact, which makes analysis of orbits awkward.
The Berkovich projective line adds extra points to $\mathbb{P}^1(\C_p)$ and is compact, Hausdorff and path-connected.
Every rational map extends to it continuously.
On this space one gets [[Julia sets|5M4S7P04GR]], [[Fatou sets|C0N0XGC6GK]] and invariant measures that behave much like those of [[complex dynamics|BQQZ0ZCGQF]].

## Explanations

### L1 human

Over the $p$-adic numbers, discs never overlap partially and the space falls apart into tiny pieces.
Berkovich added one new point for each closed disc, together with a few limit points, so that the discs are joined together into a tree-like space.
A rational map moves discs to discs, so it moves these new points too.
For the map $z \mapsto z^2$, the point that stands for the closed unit disc stays fixed.
This point is the whole Julia set, while on the ordinary $p$-adic points the Julia set is empty.