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 by $T$ 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 $T$ 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 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 $T$ significantly. Thus, the study of $T$ 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 $1$ .

• 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.