ergodic

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invariant

1 Ergodicity

Definition - Let $(X,\\mathfrak{B},\\mu)$ be a probability space and $T:X\\longrightarrow X$ a measure-preserving transformation. We say that $T$ is ergodic if all the subsets $A\\in\\mathfrak{B}$ such that $T^{-1}(A)=A$ have measure $0$ or $1$ .

In other words, if $A\\in\\mathfrak{B}$ is invariant (http://planetmath.org/InvariantByAMeasurePreservingTransformation) by $T$ then $\\mu(A)=0$ or $\\mu(A)=1$ .

1.0.1 Motivation

Suppose $(X,\\mathfrak{B},\\mu)$ is a probability space and $T:X\\longrightarrow X$ is a measure-preserving transformation. If $A\\in\\mathfrak{B}$ is an invariant (http://planetmath.org/InvariantByAMeasurePreservingTransformation) measurable subset, with $0<\\mu(A)<1$ ,then $X\\setminus A$ is also invariant and $0<\\mu(X\\setminus A)<1$ . Thus, in this situation, we can study the transformation $T$ by studying the two simpler transformations $T|_{A}$ and $T|_{X\\setminus A}$ in the spaces $A$ and $X\\setminus A$ , respectively.

The transformation $T$ is ergodic precisely when $T$ cannot be decomposed into simpler transformations. Thus, ergodic transformations are the measure-preserving transformations, in the sense described above.

Remark: When the invariant $A\\in\\mathfrak{B}$ has measure $\\mu(A)=0$ we can ignore it (as usual in measure theory), as its presence does not affect $T$ significantly. Thus, the study of $T$ is not simplified when restricting to $X\\setminus A$ .

2 Examples

•
The identity transformation in a probability space $(X,\\mathfrak{B},\\mu)$ is ergodic if (and only if) all measurable sets have measure $0$ or $1$ .

•
Let $\\mathbb{T}$ be the unit circle in $\\mathbb{C}$ , endowed with the arc length Lebesgue measure (or Haar measure). The transformation of $\\mathbb{T}$ given by $S(x)=ax$ , where $a\\in\\mathbb{T}$ , is ergodic if and only if $a$ is not a root of unity.

Title	ergodic
Canonical name	Ergodic
Date of creation	2013-03-22 12:19:38
Last modified on	2013-03-22 12:19:38
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	10
Author	asteroid (17536)
Entry type	Definition
Classification	msc 28D05
Classification	msc 37A25
Synonym	ergodicity
Synonym	ergodic transformation
Synonym	ergodic map
Related topic	Measure
Related topic	ErgodicTheorem
Related topic	MeasurePreserving