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ergodic

(Definition)

Ergodicity

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

In other words, if $A \in \mathfrak{B}$ is invariant by then $\mu(A) = 0$ or $\mu(A)=1$ .

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 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 by studying the two simpler transformations $T\vert _{A}$ and $T\vert _{X \setminus A}$ in the spaces and $X \setminus A$ , respectively.

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

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

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 .

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 , where $a \in \mathbb{T}$ , is ergodic if and only if is not a root of unity.

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Other names:

ergodicity, ergodic transformation, ergodic map

Keywords:

ergodic

Attachments:: ergodicity of a map in terms of its induced operator (Theorem) by asteroid

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Cross-references: root of unity, Haar measure, Lebesgue measure, arc length, unit circle, measurable sets, identity, theory, transformations, transformation, measurable, measure, subsets, measure-preserving transformation, probability space
There are 7 references to this entry.

This is version 7 of ergodic, born on 2002-02-14, modified 2008-05-20.
Object id is 1949, canonical name is ErgodicTransformation.
Accessed 7614 times total.

Classification:

AMS MSC:	28D05 (Measure and integration :: Measure-theoretic ergodic theory :: Measure-preserving transformations)
	37A25 (Dynamical systems and ergodic theory :: Ergodic theory :: Ergodicity, mixing, rates of mixing)

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