---
tag: A5RCGG79JX
kind: definition
name: Discrete-time dynamical system
msc: [37B02, 37C05, 37-01]
defines: 5AA9AJEARJ
uses: [BBZP1R872N, AYVTR2ES5B, 0EPAT51DXQ, 0J2WAR83WP]
sources:
  - url: https://en.wikipedia.org/wiki/Dynamical_system
---

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a [[map|0EPAT51DXQ]].
The *discrete-time dynamical system* generated by $f$ is the pair $(X, f)$, viewed as the [[dynamical system|BBZP1R872N]] $\Phi \colon \N \times X \to X$ with $\Phi(n, x) = f^n(x)$.
Here $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] 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|0J2WAR83WP]] $f^{-1}$.

## 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.