---
tag: T75CZ5H47N
kind: concept
name: Periodic point
msc: [37C25, 37B02, 37-01]
sources:
  - url: https://en.wikipedia.org/wiki/Periodic_point
same_as:
  wikidata: Q116301
  wikipedia: Periodic point
---

## Summary

A periodic point of a map is a point that comes back to itself after some positive number of steps.
Its orbit is a finite cycle that repeats forever.
[[Fixed points|R6QVS6P21N]] are the periodic points that return after one step.
Periodic points are the simplest kind of long-time behaviour, and counting them says a lot about a system.
For flows the analogue is a [[periodic orbit|0AA97XA70K]].

## Explanations

### L1 human

Start at a point, apply the map again and again, and watch whether the point ever comes back.
If it comes back after $n$ steps, it will repeat the same loop of $n$ points forever.
For $f(x) = -x$ on the real line, the point $1$ goes to $-1$ and then back to $1$, so it is periodic with period $2$.
For the rotation of a clock face by one hour, every point comes back after $12$ steps.