---
tag: BE4X0QG8XW
kind: concept
name: Arithmetic dynamics
msc: [37P05, 37P15, 11G50]
sources:
  - url: https://en.wikipedia.org/wiki/Arithmetic_dynamics
same_as:
  wikidata: Q4791123
  wikipedia: Arithmetic dynamics
---

## Summary

Arithmetic dynamics studies the number-theoretic properties of points under iteration of polynomial and rational maps, and more generally of self-maps of algebraic varieties.
Typical questions ask which rational or algebraic points are [[periodic|T75CZ5H47N]] or preperiodic.
Others ask how the [[height|F1FF505PWH]] of a point grows along its orbit, and how orbits behave modulo primes or over $p$-adic fields.
It combines complex dynamics with Diophantine geometry.
Many of its conjectures are dynamical analogues of results on elliptic curves and abelian varieties, where torsion points play the role of preperiodic points.

## Explanations

### L1 human

Take a simple rule like $x \mapsto x^2 - 1$ and apply it again and again to a fraction.
Some starting values, such as $0$, run in a cycle: $0 \mapsto -1 \mapsto 0$.
Most starting values, such as $2$, produce numbers with more and more digits: $2, 3, 8, 63, \ldots$.
Arithmetic dynamics asks which starting values give cycles, how fast the digits grow, and what such patterns look like over other number systems.