ergodic
\PMlinkescapephrase
invariant
1 Ergodicity
Definition  Let $(X,\U0001d505,\mu )$ be a probability space and $T:X\u27f6X$ a measurepreserving transformation^{}. We say that $T$ is ergodic if all the subsets $A\in \U0001d505$ such that ${T}^{1}(A)=A$ have measure $0$ or $1$.
In other words, if $A\in \U0001d505$ is invariant (http://planetmath.org/InvariantByAMeasurePreservingTransformation) by $T$ then $\mu (A)=0$ or $\mu (A)=1$.
1.0.1 Motivation
Suppose $(X,\U0001d505,\mu )$ is a probability space and $T:X\u27f6X$ is a measurepreserving transformation. If $A\in \U0001d505$ is an invariant (http://planetmath.org/InvariantByAMeasurePreservingTransformation) measurable subset, with $$, then $X\setminus A$ is also invariant and $$. 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 measurepreserving transformations, in the sense described above.
Remark: When the invariant $A\in \U0001d505$ 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

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The identity transformation in a probability space $(X,\U0001d505,\mu )$ is ergodic if (and only if) all measurable sets^{} have measure $0$ or $1$.

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Let $\mathbb{T}$ be the unit circle^{} in $\u2102$, 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  20130322 12:19:38 
Last modified on  20130322 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 