---
tag: R6QVS6P21N
kind: concept
name: Fixed point
msc: [37C25, 54H25, 37-01]
sources:
  - url: https://en.wikipedia.org/wiki/Fixed_point_(mathematics)
same_as:
  wikidata: Q217608
  wikipedia: Fixed point (mathematics)
---

## Summary

A fixed point of a map is a point that the map sends to itself.
In a dynamical system, a fixed point is a state that never changes, the discrete-time analogue of an [[equilibrium point|0N8NGKDZDZ]].
Many problems, such as solving an equation $g(x) = 0$, can be rewritten as finding a fixed point.
Theorems such as the [[Banach fixed point theorem|0VK6VEH345]] guarantee that one exists.
Near a fixed point, the behaviour of the iterates tells whether nearby states are attracted or repelled.

## Explanations

### L1 human

A fixed point is a point that a map leaves where it is.
For $f(x) = x^2$ on the real line, the fixed points are the solutions of $x^2 = x$, namely $0$ and $1$.
On a graph, the fixed points are where the graph of $f$ meets the diagonal line $y = x$.
Stir a cup of coffee and let it settle.
If the stirring is a continuous map of the disc to itself, at least one point of the coffee ends where it started.