Definition:Weakly Mixing Measure-Preserving Transformation
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Definition
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 weakly mixing if and only if:
- $\ds \forall A, B \in \BB : \lim_{N \mathop\to \infty} \frac 1 N \sum _{n \mathop = n}^{N-1} \size {\map \mu {A \cap T^{-n} \sqbrk B} - \map \mu A \map \mu B} = 0$
where $T^{-n} \sqbrk B$ denotes the preimage of $B$ under the power $T^n$.
Definition 2
$T$ is said to be weakly mixing if and only if:
- $T \times T$ is ergodic with respect to $\mu \times \mu$
where $\mu \times \mu$ denotes the product measure on $\struct {X \times X, \BB \otimes \BB}$.
Definition 3
$T$ is said to be weakly mixing if and only if:
- $T \times T$ is weakly mixing in the sense of Definition 1 with respect to $\mu \times \mu$
where $\mu \times \mu$ denotes the product measure on $\struct {X \times X, \BB \otimes \BB}$.
Definition 4
$T$ is said to be weakly mixing if and only if:
- for every ergodic measure-preserving system $\struct {Y, \BB_Y, \nu, S}$, the system:
- $\struct {X \times Y, \BB \otimes \BB_Y, \mu \times \nu, T \times S}$
- is ergodic
where $\mu \times \nu$ denotes the product measure on $\struct {X \times Y, \BB \otimes \BB_Y}$
Also see
- Definition:Ergodic Measure-Preserving Transformation
- Definition:Strongly Mixing Measure-Preserving Transformation
- Results about weakly mixing measure-preserving transformations can be found here.
Sources
- 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory: Chapter $2$: Ergodicity, Recurrence and Mixing