# ergodicity of a map in terms of its induced operator

Let $(X,\mathfrak{B},\mu)$ be a probability space and $T:X\longrightarrow X$ a measure-preserving transformation. The following statements are equivalent:

1. 1.

- $T$ is ergodic.

2. 2.

- If $f$ is a measurable function and $f\circ T=f$ a.e. (http://planetmath.org/AlmostSurely), then $f$ is constant a.e.

3. 3.

- If $f$ is a measurable function and $f\circ T\geq f$ a.e., then $f$ is constant a.e.

4. 4.

- If $f\in L^{2}(X)$ and $f\circ T=f$ a.e., then $f$ is constant a.e..

5. 5.

- If $f\in L^{p}(X)$, with $p\geq 1$, and $f\circ T=f$ a.e., then $f$ is constant a.e.

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Let $U_{T}$ denote the operator induced by $T$ (http://planetmath.org/OperatorInducedByAMeasurePreservingMap), i.e. the operator defined by $U_{T}f:=f\circ T$. The statements above are statements about $U_{T}$. The above theorem can be rewritten as follows:

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Theorem - Let $(X,\mathfrak{B},\mu)$ be a probability space and $T:X\longrightarrow X$ a measure-preserving transformation. The following statements are equivalent:

1. 1.

- $T$ is ergodic.

2. 2.

- The only fixed points of $U_{T}$ are the functions that are constant a.e.

3. 3.

- If $f$ a measurable function and $U_{T}f\geq f$ a.e., then $f$ is constant a.e.

4. 4.

- The eigenspace of $U_{T}$ (seen as an operator in $L^{p}(X)$, with $p\geq 1$) associated with the eigenvalue $1$, is one-dimensional and consists of functions that are constant a.e.

Title ergodicity of a map in terms of its induced operator ErgodicityOfAMapInTermsOfItsInducedOperator 2013-03-22 17:59:22 2013-03-22 17:59:22 asteroid (17536) asteroid (17536) 6 asteroid (17536) Theorem msc 47A35 msc 37A30 msc 37A25 msc 28D05