Category:Ergodic Measure-Preserving Transformations

This category contains results about Ergodic Measure-Preserving Transformations.
Definitions specific to this category can be found in Definitions/Ergodic Measure-Preserving Transformations.

Let $\struct {X, \BB, \mu}$ be a probability space.

Let $T: X \to X$ be a measure-preserving transformation.

Definition 1

$T$ is said to be ergodic if and only if:

for all $A \in \BB$:

$T^{-1} \sqbrk A = A \implies \map \mu A \in \set {0, 1}$

Definition 2

$T$ is said to be ergodic if and only if:

for all $A \in \BB$:

$\map \mu {T^{-1} \sqbrk A \symdif A} = 0 \implies \map \mu A \in \set {0, 1}$

where $\symdif$ denotes the symmetric difference.

Definition 3

$T$ is said to be ergodic if and only if:

for all $A \in \BB$:

$\ds \map \mu A > 0 \implies \map \mu {\bigcup_{n \mathop = 1}^\infty T^{-n} \sqbrk A} = 1$

Definition 4

$T$ is said to be ergodic if and only if:

for all $A, B \in \BB$:

$\map \mu A \map \mu B > 0 \implies \exists n \ge 1 : \, \map \mu {T^{-n} \sqbrk A \cap B} > 0$

Definition 5

$T$ is said to be ergodic if and only if:

for all measurable $f: X \to \C$:

$f \circ T = f$ holds $\mu$-almost everywhere

$\implies \exists c \in \C:\, f = c$ holds $\mu$-almost everywhere