---
tag: 5X6E97EMFR
kind: concept
name: Topological dynamics
msc: [37B02, 37B05]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_dynamics
same_as:
  wikidata: Q7825032
  wikipedia: Topological dynamics
---

## Summary

Topological dynamics is the part of the theory of [[dynamical systems|BBZP1R872N]] that studies continuous maps and group actions on topological spaces.
It uses only the topology, not a measure or a smooth structure.
Its questions concern the long-term behaviour of orbits: recurrence, density of orbits, minimal sets, mixing and the growth of complexity.
Typical objects are a compact metric space $X$ with a [[homeomorphism|0QT00PB5G2]] or a continuous map $T \colon X \to X$.
It sits beside ergodic theory, which studies the same questions with a measure in place of a topology.

## Explanations

### L1 human

Imagine a space of states and a rule that moves each state to the next one, where nearby states move to nearby states.
Topological dynamics asks qualitative questions about this motion without measuring anything.
Does a point come back near where it started?
Does some orbit pass near every point?
For a rotation of a circle by an irrational angle, every orbit comes back arbitrarily close to its start and passes near every point of the circle.