# mixing

Let $f$ be a measure-preserving transformation of a probability space $(X,\mathscr{A},\mu)$. We say that $f$ is mixing (or strong-mixing) if for all $A,B\in\mathscr{A}$,

 $\lim_{n\to\infty}\mu(f^{-n}(A)\cap B)=\mu(A)\mu(B),$

and $f$ is weakly mixing if

 $\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}|\mu(f^{-i}(A)\cap B)-\mu(A)\mu(B)% |=0$

for all $A,B\in\mathscr{A}$.

Every mixing transformation is weakly mixing, and every weakly mixing transformation is ergodic.

 Title mixing Canonical name Mixing Date of creation 2013-03-22 14:06:34 Last modified on 2013-03-22 14:06:34 Owner Koro (127) Last modified by Koro (127) Numerical id 6 Author Koro (127) Entry type Definition Classification msc 37A25 Defines strongly mixing Defines strong mixing Defines strong-mixing Defines weak-mixing Defines weakly mixing Defines weak mixing