1 Ergodicity

Definition - Let (X,𝔅,μ) be a probability space and T:XX a measure-preserving transformationPlanetmathPlanetmath. We say that T is ergodic if all the subsets A𝔅 such that T-1(A)=A have measure 0 or 1.

In other words, if A𝔅 is invariant ( by T then μ(A)=0 or μ(A)=1.

1.0.1 Motivation

Suppose (X,𝔅,μ) is a probability space and T:XX is a measure-preserving transformation. If A𝔅 is an invariant ( measurable subset, with 0<μ(A)<1, then XA is also invariant and 0<μ(XA)<1. Thus, in this situation, we can study the transformationMathworldPlanetmath T by studying the two simpler transformations T|A and T|XA in the spaces A and XA, 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 A𝔅 has measure μ(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 XA.

2 Examples

  • Let 𝕋 be the unit circlePlanetmathPlanetmath in , endowed with the arc length Lebesgue measureMathworldPlanetmath (or Haar measure). The transformation of 𝕋 given by S(x)=ax, where a𝕋, 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