---
tag: EY2W34JZA4
kind: concept
name: Non-wandering set
msc: [37B20]
sources:
  - url: https://en.wikipedia.org/wiki/Wandering_set
---

## Summary

The non-wandering set of a continuous map consists of the points $x$ whose every neighbourhood meets one of its own later images.
It is the complement of the set of wandering points from the notion of [[wandering set|VNW5FHCQ2M]].
It is closed and invariant, and it contains every [[periodic orbit|0AA97XA70K]], every [[recurrent point|4SQHHRPKMV]] and every $\omega$-[[limit set|YF4XH2BEG5]].
In smooth dynamics, hyperbolicity conditions such as Smale's Axiom A are imposed on the non-wandering set.

## Explanations

### L1 human

Some points of a system are only passing through: a small region around them drifts away and never overlaps itself again.
The non-wandering set is what is left when all such passing points are removed.
It is where the motion keeps coming back to, so it holds all the long-term behaviour.
For a pendulum with friction, only the rest positions are non-wandering; every other state slowly drifts toward rest.