Mixing

Remarks

Example: let $T(x) = 2x \bmod 1$ on $[0,1)$ with Lebesgue measure $\mu$. Let $A$ and $B$ be dyadic intervals of length $2^{-k}$, that is, of the form $[j 2^{-k}, (j+1) 2^{-k})$. For $n \ge k$, the set $T^{-n} A$ is a union of $2^n$ intervals of length $2^{-n-k}$, one in each interval $[i 2^{-n}, (i+1) 2^{-n})$. Exactly $2^{n-k}$ of them lie in $B$, so $\mu(T^{-n} A \cap B) = 2^{-2k} = \mu(A) \mu(B)$. Dyadic intervals generate the Borel sets, and an approximation argument gives the limit for all $A$ and $B$.

Non-example: let $T(x) = x + \alpha \bmod 1$ with $\alpha$ irrational, and let $A = B = [0, 1/4)$. The numbers $n \alpha \bmod 1$ come arbitrarily close to $0$ for infinitely many $n$, and for those $n$ the measure $\mu(T^{-n} A \cap A)$ is close to $1/4$. So it does not tend to $1/16$, and the rotation is not mixing, although it is ergodic.

Taking $A = B$ with $T^{-1} A = A$ shows that a mixing map is ergodic: then $\mu(A) = \mu(A)^2$, so $\mu(A) \in \set{0, 1}$. In terms of functions, $T$ is mixing if and only if $\int (f \circ T^n) \, g \, d\mu \to \int f \, d\mu \int g \, d\mu$ for all $f, g \in L^2(\mu)$.