Category:Ergodic Measure-Preserving Transformations
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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
- $f \circ T = f$ holds $\mu$-almost everywhere
Subcategories
This category has only the following subcategory.
Pages in category "Ergodic Measure-Preserving Transformations"
The following 2 pages are in this category, out of 2 total.