Strong mixing - Knowlpedia

A probability-preserving transformation $T$ is strongly mixing if

\mu(T^{-n}A\cap B)\longrightarrow\mu(A)\mu(B) \quad(n\to\infty)

for every measurable $A,B$ . For a noncompact locally compact group action, replace $T^n$ by $T_g$ and let $g$ leave every compact subset of the group.

Observables

Equivalently, $\langle U_T^nf,h\rangle\to0$ for every mean-zero $f,h\in L^2$ . This follows from the event formulation by approximation with simple functions. Hence $U_T^nf$ converges weakly to zero on $L^2_0$ , even though $\|U_T^nf\|_2=\|f\|_2$ . Unitarity and mixing are compatible.

Examples

The cat map, Bernoulli shifts, and the doubling map are mixing. Rotations of a nontrivial compact abelian group are not mixing.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §§1.7 and 2.2.