Dynamical systems and ergodic theory Definition

Discrete-time dynamical system

Builds on Dynamical system Iterated function Function Inverse function

Definition

Let $X$ be a set and let $f \colon X \to X$ be a map. The discrete-time dynamical system generated by $f$ is the pair $(X, f)$ , viewed as the dynamical system $\Phi \colon \N \times X \to X$ with $\Phi(n, x) = f^n(x)$ . Here $f^n$ is the $n$ -th iterate of $f$ , with $f^0 = \id_X$ . If $f$ is a bijection, the system is invertible, and $\Phi(n, x) = f^n(x)$ is defined for all $n \in \Z$ . Here $f^{-n}$ is the $n$ -th iterate of the inverse map $f^{-1}$ .

Remarks

The forward orbit of $x$ is the sequence $x, f(x), f^2(x), \dots$ . Every dynamical system with time $\N$ arises in this way, with $f = \Phi(1, \cdot)$ , since clause 2 of the definition of a dynamical system forces $\Phi(n, \cdot) = f^n$ .

Example: $X = \R$ and $f(x) = 3x$ . Then $\Phi(n, x) = 3^n x$ , and since $f$ is a bijection the system is invertible with $\Phi(-n, x) = 3^{-n} x$ .

Non-example of an invertible system: $f(x) = x^2$ on $\R$ generates a discrete-time system, but it is not invertible, because $f(-1) = f(1)$ .

Usually $X$ is a topological space and $f$ is continuous, or $X$ is a measure space and $f$ is measurable.

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Dynamical system concept
Function concept
Inverse function concept
Iterated function concept
Discrete-time dynamical system definition

Tag	`A5RCGG79JX`
URI	`m2://A5RCGG79JX`
Kind	definition
Writing	W0 stub
MSC	37B02, 37C05, 37-01
Defines	Discrete-time dynamical system
Uses	Dynamical system, Iterated function, Function, Inverse function

Discrete-time dynamical system

Definition

Remarks

Reading path

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