---
tag: 4SQHHRPKMV
kind: concept
name: Recurrent point
msc: [37B20]
sources:
  - url: https://en.wikipedia.org/wiki/Recurrent_point
same_as:
  wikidata: Q7303259
  wikipedia: Recurrent point
---

## Summary

A recurrent point of a continuous map is a point whose orbit returns arbitrarily close to it again and again.
Formally, the point lies in its own $\omega$-[[limit set|YF4XH2BEG5]].
Recurrence is the topological form of the idea that a system comes back near its initial state.
Every point of a [[minimal set|CRGZZKQ7Z9]] is recurrent, so on a nonempty compact metric space every continuous map has recurrent points.

## Explanations

### L1 human

Follow a point under repeated application of a map.
The point is recurrent if, however small a window you draw around it, the orbit enters that window at infinitely many later times.
A periodic point is recurrent, since it comes back exactly.
For a rotation of the circle by an irrational angle, every point is recurrent without ever coming back exactly.
For the map $x \mapsto x/2$ on the real line, the point $1$ is not recurrent: its orbit $1, 1/2, 1/4, \ldots$ runs off to $0$.