# ergodic

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