---
tag: FR49Y9GQHR
kind: concept
name: Topological transitivity
msc: [37B02, 37B05]
sources:
  - url: https://en.wikipedia.org/wiki/Mixing_(mathematics)
---

## Summary

A continuous map is topologically transitive when it carries every nonempty open set, at some time, into every other nonempty open set.
On a compact metric space without isolated points this is the same as having a point whose orbit is [[dense|0BSJSX7ME2]].
A transitive system cannot be split into two open invariant pieces, so it is the topological analogue of an [[ergodic transformation|AZRPHS478B]].
Transitivity is one of the three conditions in Devaney's definition of chaos.

## Explanations

### L1 human

Pick any two regions of the space, however small.
A transitive map, applied often enough, moves some point of the first region into the second.
So the motion links every part of the space with every other part.
A rotation of a circle by an irrational angle is transitive: every orbit winds around densely.
The identity map of an interval is not: nothing ever moves from one region to another.