Weak mixing - Knowlpedia

A probability-preserving transformation $T$ is weakly mixing if its Cartesian square $T\times T$ is ergodic on $(X\times X,\mu\otimes\mu)$ . A probability-preserving group action is weakly mixing if its diagonal action on the square is ergodic.

Correlation criterion for a transformation

Equivalently, for every $f,h\in L^2_0(X,\mu)$ ,

\frac1N\sum_{n=0}^{N-1}|\langle U_T^nf,h\rangle|^2\longrightarrow0.

Persistent oscillation cannot disappear merely by cancellation in this average. The spectral characterization identifies the obstruction as nonconstant eigenfunctions in the invertible case.

Comparison

Strong mixing implies weak mixing, which implies ergodicity. An irrational circle rotation is ergodic but not weakly mixing.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.2.2.