---
tag: 6MX5R482KJ
kind: concept
name: Chain recurrence
msc: [37B20, 37B35]
sources:
  - url: https://en.wikipedia.org/wiki/Conley%27s_fundamental_theorem_of_dynamical_systems
---

## Summary

A point is chain recurrent when it can return to itself along a path that follows the dynamics up to arbitrarily small jumps.
Such paths are called $\varepsilon$-chains or $\varepsilon$-pseudo-orbits.
The chain recurrent set is the weakest of the usual notions of recurrence: it contains the [[non-wandering set|EY2W34JZA4]].
By [[Conley's fundamental theorem|D07MBM0S3V]], outside the chain recurrent set the dynamics is gradient-like.

## Explanations

### L1 human

Imagine following the orbit of a point, but allowing yourself a tiny error at each step, as a computer with rounding would.
The point is chain recurrent if, for every allowed error size, you can get back to the starting point this way.
A recurrent point is chain recurrent: its orbit comes back close to it, and one small final jump closes the loop.
For the map $x \mapsto x/2$ on $[0, 1]$, a point $x > 0$ is not chain recurrent: with small errors you still cannot climb back from $x/2$ to $x$.