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I recently heard a talk which depended on the dynamics of a system being mixing. I was told, and Wikipedia confirms, that "mixing" is a stronger condition than "ergodic", but after comparing the definitions I don't have much intuition for the distinction.

Are there any simple examples of a system that is ergodic but not mixing?

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2 Answers

up vote 3 down vote accepted

My impression is that the difference is mostly about periodic and periodic-like behavior. I will use the following definitions:

Definitions. Let $(X,\mu)$ be a measure space, and let $T\colon X\to X$ be a measure-preserving transformation.

  1. We say that $T$ is strongly mixing if $\lim_{n\to\infty} \mu(A\cap T^{-n}B) = \mu(A)\mu(B)$ for all measurable sets $A$ and $B$.

  2. We say that $T$ is ergodic if, for every measurable set $E$ satisfying $T^{-1}(E)=E$, either $\mu(E)=0$ or $\mu(E)=1$.

Now consider the following examples:

  1. Let $X = \{0,1\}$, where $\mu(\{0\}) = \mu(\{1\}) = 1$, and let $T\colon X\to X$ be the function $T(0) = 1$ and $T(1)=0$. Then $T$ is clearly ergodic. However, $T$ is not strongly mixing, since the limit $\lim_{n\to\infty} \mu\bigl(\{0\}\cap T^{-n}\{0\}\bigr)$ does not exist.

  2. More generally, suppose that $X$ is a disjoint union $X_1\uplus\cdots\uplus X_k$ of subsets with equal measure, and let $T\colon X\to X$ be a measure-preserving transformation that cyclically permutes the $X_i$. Then $T$ cannot possibly be strongly mixing, since $\lim_{n\to\infty} \mu(X_1\cap T^{-n}X_1)$ does not exist. However, $T$ will be ergodic if and only if the $k$th iterate $T^k\colon X_1\to X_1$ is ergodic on $X_1$.

  3. Here is an example whose behavior is "periodic-like'' as opposed to periodic. Let $S^1$ be the unit circle, and let $T\colon S^1\to S^1$ be a rotation by an irrational angle. Again, it should be intuitively obvious that $T$ is ergodic, though of course this requires some proof. However, $T$ is again not strongly mixing, since the limit does not exist when $A$ and $B$ are open intervals.

In the same way that example (2) is a generalization of example (1), it ought to be possible to generalize example (3) to yield much more complicated ergodic systems with periodic-like behavior. For example, if $T\colon S^1\to S^1$ is the map from example (3) and $T'\colon X\to X$ is any strongly mixing map, then presumably the product map $T\times T'$ on $S^1\times X$ is ergodic but not strongly mixing.

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Well, we don't have to insist on any particular measure (such as the Lebesgue measure for the case of rotation on $S^1$). For instance, one could take rational rotation on $S^1$ of period $n$ and define an equidistributed measure on some $n$-cycle. The system would be ergodic, but neither strongly nor weakly mixing. –  William Feb 9 '12 at 20:11
    
Thank you both for your answers! –  yep Feb 10 '12 at 2:34
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Consider, for example, a shift $T$ on bi-infinite sequences $\Sigma$ on, say, two letters $\left\{a, b \right\}$. Let $\left\{p_1, p_2, \dots, p_n\right\}$, with $n\geq 3$ be a periodic $n$-cycle (i.e. $T: p_1\mapsto p_2\mapsto\cdots\mapsto p_n\mapsto p_1$). Assign the measure $\mu$ on $\Sigma$ by $\mu(\left\{p_i\right\}) = 1/n$ and for any $A\subset\Sigma$ not containing any $p_i$, $i = 1,\dots,n$, $\mu(A) = 0$. Now take two sets $A = \left\{p_1 \right\}$ and $B = \left\{p_2\right\}$. Now verify that

$$ \lim_{k\rightarrow\infty} \mu(A\cap T^k(B)) = \mu(A)\mu(B) $$

fails. Thus the system is not strongly mixing. Also verify that

$$ \lim_{k\rightarrow\infty}\frac{1}{k}\sum_{j=0}^k\left|\mu(A\cap T^j(B)) - \mu(A)\mu(B)\right| = 0 $$

also fails. Thus the system is not weakly mixing.

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