---
tag: 1Q901WTV5X
kind: definition
name: Topological dynamics
msc: [37B02, 37B05]
defines: 5X6E97EMFR
uses: [0CQHFPGJ45, 0VGFHTM75Y, 92700QAB5Y, 0WCY42EV9V]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_dynamics
---

## Definition

A *topological dynamical system* is a pair $(X, T)$ where $X$ is a [[topological space|0CQHFPGJ45]] and $T \colon X \to X$ is a [[continuous map|0VGFHTM75Y]].
More generally, it is a triple $(G, X, \alpha)$ where $G$ is a [[topological group|92700QAB5Y]] and $\alpha \colon G \times X \to X$ is a continuous [[group action|0WCY42EV9V]].
*Topological dynamics* is the study of the properties of such systems that are preserved by topological conjugacy, that is, by homeomorphisms $h \colon X \to Y$ with $h \circ T = S \circ h$.

## Remarks

In most of the theory $X$ is a compact metric space, and $T$ is a homeomorphism or $G$ is $\Z$ or $\R$.
A map $T$ gives an action of the monoid $\N$ by $n \cdot x = T^n(x)$; a homeomorphism gives an action of $\Z$; a continuous flow gives an action of $\R$.
Example: the circle $X = \R / \Z$ with $T(x) = x + \alpha$ is a topological dynamical system, since $T$ is continuous.
Non-example: $X = [0, 1]$ with $T(x) = 0$ for $x < 1/2$ and $T(x) = 1$ for $x \ge 1/2$ is not one, because $T$ is not continuous at $1/2$.

## Explanations

### L1 human

A topological dynamical system is a space together with a continuous rule of motion.
Topological dynamics studies the features of the motion that survive a continuous change of coordinates.