Weak and strong mixing

28 Jan 2018

I had a useful conversation with Jon Chaika today where he explained some elementary ergodic theory to me. I really ought to know these things better, so I decided to write it down and work some of the details out.

Given any dynamical system $(\OFP,T)$, we can define an $L^2$ unitary operator

$$\begin{align} Uf(x) = f(Tx). \end{align}$$

The unitarity follows from the fact that $T$ preserves measure. We express the ergodic average as

$$S_n(f) = \frac1{n} \sum_{i=1}^n U^i f$$

The ergodic theorem says that for any $L^1$ function, $S_nf \to \overline{f}$ almost surely and in $L^1$. Here, $\overline{f}$ is the conditional expectation with respect to the invariant $\sigma$-algebra of $T$. We will assume ergodicity; i.e., the invariant $\sigma$-algebra is trivial.

The usual notions of ergodicity, and mixing can be described in terms of the spectrum of the unitary operator $U$. For example, we know that $T$ is ergodic if and only if the eigenvalue $1$ is simple. In other words, the only eigenvectors corresponding to $1$ are constant functions.

A system is weak-mixing if it has no eigenvalues other than $1$. In terms of the operator $U$, a system is weak-mixing iff for each $f \in L^2$, and all $g \in L^2$,

$$\E[ g S_n(f)] \to \E[ g \overline{f}]$$

From this it is easy to argue as follows. Suppose there is a $\lambda \neq 1$ (but $\lvert\lambda\rvert = 1$) such that

$$Uf = \lambda f$$

Then from the characterization of weak-mixing above, we get

$$\E[ g S_n(f)] = \frac{1}n \frac{1 - \lambda^{n+1}}{1 - \lambda}\E[g f] \to 0.$$

Since $g$ can be any $L^2$ function, this completes the proof.

There is a spectral description of strong mixing as well, but it’s in terms of the spectral measure of $U$. Apparently, the requirement is that the spectral measure is Reichmann. I cannot reproduce the condition because I can’t quite remember it, but it seems like it’s saying something about the decay of the moments of the spectral measure.

The dyadic machine

One frequently hears that the obstructions to weak-mixing are compact group rotations. My impression was that one should think of systems that are not weak-mixing as containing a copy of the standard rotation on a unit-circle or a product of such rotations. This is not correct, and systems that are not weak-mixing can contain really complicated compact group rotations.

A canonical example of a compact group rotation is the so-called dyadic machine. Dyadic machines can have really complicated spectra.

The system is defined on $\W = \{0,1\}^{\Z}$. Suppose $x = (0,1,1,0,0,\ldots) \in \W$. Then, $Tx = (1,0,0,1,0,\ldots)$. That is, you add $1\mod2$ to the first coordinate and carry over the remainder to the next coordinate. Precisely,

$$\begin{align} (Tx)_0 & = (x_0 + 1) \mod 2 \quad \AND x'_1 := 1 - (Tx)_0\\ (Tx)_i & = (x_i + x'_i) \mod 2 \quad \AND x'_{i+1} := 1 - (Tx)_i \end{align}$$

where the $x’_i$ is simply used to track the “carrying over” operation. You can think of $\W$ as the unit interval by mapping each point to its binary expansion. In this case, the map acts quite weirdly on the unit interval.

The $\Bernoulli(1/2)$ measure is uniquely ergodic for this map. The unique ergodicity of this iid measure is quite suprising to me. It means that there are no bad sets to consider for the ergodic theorem; things converge “surely” and not almost surely.

Now, one notices that

$$T 1_{x_0 = 0} = 1_{x_0 = 1} \AND T 1_{x_0 = 1} = 1_{x_0 = 0}$$

and therefore $1_{x_0 = 0} - 1_{x_0 = 1}$ has eigenvalue $-1$. Similarly,

$$\begin{align*} T 1_{x_0 = 0,x_1 = 1} & = 1_{x_0 = 1,x_1 = 0}\\ T 1_{x_0 = 1,x_1 = 0} & = 1_{x_0 = 0,x_1 = 0}\\ T 1_{x_0 = 0,x_1 = 0} & = 1_{x_0 = 1,x_1 = 1}\\ T 1_{x_0 = 1,x_1 = 1} & = 1_{x_0 = 0,x_1 = 1} \end{align*}$$

This implies that, for example

$$T 1_{x_0 = 0,x_1 = 1} + s 1_{x_0 = 1,x_1 = 0} + s^2 1_{x_0 = 0,x_1 = 0} + s^3 1_{x_0 = 1,x_1 = 1} + s^4 1_{x_0 = 1,x_1 = 1}$$

is an eigenvector when $s^5 = 1$ is a root of unity! There are a whole host of such eigenvectors and corresponding eigenvalues. One can change the space to $\W = \{0,\ldots,k\}^{\Z}$ and change the operation to addition mod $k$ with carryover to get other spectra. Despite its complexity, the dyadic machine is not weak-mixing. Moreover, it’s a compact group rotation.

This note is clearly incomplete: I haven’t really specified what a compact group rotation is, and I haven’t proved that the dyadic machine is a compact group rotation. I intend to revisit this at some point in the future. Some of these concepts can be found in T. Austin’s notes. The notes also prove that the obstructions to weak mixing are such compact group rotations.