---
tag: B6DFK8HCB7
kind: concept
name: P-adic dynamics
msc: [37P20, 11S82]
specializes: [5AA9AJEARJ]
sources:
  - url: https://en.wikipedia.org/wiki/P-adic_analysis
---

## Summary

$p$-adic dynamics studies the iteration of maps defined over the [[$p$-adic numbers|BT6T702C28]] and their extensions, such as rational functions and convergent power series.
It is a [[discrete-time dynamical system|5AA9AJEARJ]] theory in which distances come from a [[non-Archimedean absolute value|1H5F1BP6XS]].
The strong triangle inequality makes the behaviour of orbits very different from the real or complex case.
It is a basic tool of [[arithmetic dynamics|BE4X0QG8XW]], where $p$-adic methods control orbits of points over number fields.

## Explanations

### L1 human

In $p$-adic numbers two numbers are close when their difference is divisible by a high power of $p$.
Iterating a map in this world means following a point step by step and watching how its $p$-adic size changes.
Discs either nest inside each other or do not meet, so orbits often move in very rigid patterns.
For example, the map $x \mapsto x + 1$ on the $p$-adic integers sends $0$ to $1, 2, 3, \dots$, and these points come arbitrarily close to every $p$-adic integer.