Dynamical systems and ergodic theory Definition

Topological dynamics

Builds on Topological space Continuous map Topological group Group action

Definition

A topological dynamical system is a pair $(X, T)$ where $X$ is a topological space and $T \colon X \to X$ is a continuous map. More generally, it is a triple $(G, X, \alpha)$ where $G$ is a topological group and $\alpha \colon G \times X \to X$ is a continuous group action. 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 $1/2$ .

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Tag	`1Q901WTV5X`
URI	`m2://1Q901WTV5X`
Kind	definition
Writing	W0 stub
MSC	37B02, 37B05
Defines	Topological dynamics
Uses	Topological space, Continuous map, Topological group, Group action

Topological dynamics

Definition

Remarks

Reading path

Further reading

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