Category:Definitions/Weakly Mixing Measure-Preserving Transformations

This category contains definitions related to Weakly Mixing Measure-Preserving Transformations.
Related results can be found in Category:Weakly Mixing 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 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 $\struct {X \times Y, \BB \otimes \BB_Y}$