Notes for my lecture on multiple recurrence theorem for weakly mixing systems – Part 1

So…It’s finally my term to lecture on the ergodic theory seminar! (background, our goal is to go through the ergodic proof of Szemerédi’s theorem as in Furstenberg’s book). My part is the beginning of the discussion on weak mixing and prove the multiple recurrence theorem in the weak mixing case, the weak mixing assumption shall later be removed (hence the theorem is in fact true for any ergodic system) and hence prove Szemerédi’s theorem via the correspondence principal discussed in the last lecture.

Given two measure preserving systems $(X_1, \mathcal{B}_1, \mu_1, T_1)$ and $(X_2, \mathcal{B}_2, \mu_2, T_2)$ , we denote the product system by $(X_1 \times X_2, \mathcal{B}_1 \times \mathcal{B}_2, \mu_1 \times \mu_2, T_1 \times T_2)$ where $\mathcal{B}_1 \times \mathcal{B}_2$ is the smallest $\sigma$ -algebra on $X_1 \times X_2$ including all products of measurable sets.

Definition: A m.p.s. $(X, \mathcal{B}, \mu, T)$ is weakly mixing if $(X \times X, \mathcal{B} \times \mathcal{B}, \mu \times \mu, T \times T)$ is ergodic.

Note that weak mixing $\Rightarrow$ ergodic

as for non-ergodic systems we may take any intermediate measured invariant set $\times$ the whole space to produce an intermediate measured invariant set of the product system.

For any $A, B \in \mathcal{B}$ , let $N(A, B) = \{ n \in \mathbb{N} \ | \ \mu(A \cap T^{-n}(B)) > 0 \}$ .

Ergodic $\Leftrightarrow$ for all $A,B$ with positive measure, $N(A,B) \neq \phi$

Weakly mixing $\Rightarrow$ for all $A,B,C,D$ with positive measure, $N(A \times C, B \times D) \neq \phi$ .

Since $n \in N(A \times C, B \times D)$

$\Leftrightarrow \ \mu \times \mu(A \times C \cap T^{-n}(B \times D)) > 0$

$\Leftrightarrow \ \mu \times \mu(A \cap T^{-n}(B) \times C \cap T^{-n}(D)) > 0$

$\Leftrightarrow \ \mu(A \cap T^{-n}(B)) > 0$ and $\mu(C \cap T^{-n}(D)) > 0$

$\Leftrightarrow \ n \in N(A,B)$ and $n \in N(C,D)$

Hence $T$ is weakly mixing $\Rightarrow$ for all $A,B,C,D$ with positive measure, $N(A,B) \cap N(C,D) \neq \phi$ . We’ll see later that this is in fact $\Leftrightarrow$ but let’s say $\Rightarrow$ for now.

As a toy model for the later results, let’s look at the proof of following weak version of ergodic theorem:

Theorem 1: Let $(X, \mathcal{B}, \mu, T)$ be ergodic m.p.s., $f, g \in L^2(X)$ then

$\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int f \cdot g \circ T^n \ d\mu = \int f \ d\mu \cdot \int g \ d\mu$ .

Proof: Let $\mathcal{U}: f \mapsto f \circ T, \ \mathcal{U}$ is unitary on $L^2(X)$ .

Hence $\{ \frac{1}{N+1} \sum_{n=0}^N g \circ T^n \ | \ N \in \mathbb{N} \} \subseteq \overline{B( \overline{0}, ||g||)}$

Any weak limit point of the above set is $T$ -invariant, hence ergodicity implies they must all be constant functions.

Suppose $\lim_{i \rightarrow \infty} \frac{1}{N_i+1} \sum_{n=0}^N g \circ T^n \equiv c$

then we have $c = \int c \ d\mu = \lim_{i \rightarrow \infty} \frac{1}{N_i+1} \sum_{n=0}^N \int g \circ T^n \ d\mu = \int g \ d \mu$

Therefore the set has only one limit point under the weak topology.

Since the closed unit ball in Hilbert space is weakly compact, hence $\frac{1}{N+1} \sum_{n=0}^N g \circ T^n$ converges weakly to the constant valued function $\int g \ d \mu$ .

Therefore $\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int f \cdot g \circ T^n \ d\mu$

$= \int f \cdot ( \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int g \circ T^n) \ d\mu$

$= \int f \cdot (\int g \ d\mu) d\mu = \int f \ d\mu \cdot \int g \ d\mu$ .

Next, we apply the above theorem on the product system and prove the following:

Theorem 2: For $(X, \mathcal{B}, \mu, T)$ weakly mixing,

$\lim_{N \rightarrow \infty} \frac{1}{1+N} \sum_{n=0}^N (\int f \cdot (g \circ T^n) \ d \mu - \int f \ d \mu \int g \ d \mu)^2$

$= 0$

Proof: For $f_1, f_2: X \rightarrow \mathbb{R}$ , let $f_1 \otimes f_2: X \times X \rightarrow \mathbb{R}$ where $f_1 \otimes f_2 (x_1, x_2) = f_1(x_1) f_2(x_2)$

By theorem 1, we have

$\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N(\int f \cdot g \circ T^n \ d \mu)^2$

$= \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int (f \otimes f) \cdot ((g \otimes g) \circ (T \times T)^n) \ d \mu \times \mu$

$= (\int f \otimes f \ d\mu \times \mu) \cdot (\int g \otimes g \ d\mu \times \mu)$

$= (\int f \ d\mu)^2 (\int g \ d\mu)^2 \ \ \ \ ( \star )$

Set $a_n = \int f \cdot (g \circ T^n) \ d \mu, \ a = \int f \ d \mu \int g \ d \mu$ hence by theorem 1, we have

$\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N a_n = a$ ;

By $( \star )$ , we have

$\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N a_n^2 = a^2$

Hence $\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N (a_n-a)^2$

$= \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N (a_n^2 - 2a \cdot a_n + a^2)$

$= \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N a_n^2 - 2a \cdot \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N a_n + a^2$

$= a^2 - 2 a \cdot a + a^2 = 0$

This establishes theorem 2.

We now prove that the following definition of weak mixing is equivalent to the original definition.

Theorem 3: $(X, \mathcal{B}, \mu, T)$ weakly mixing iff for all $(Y, \mathcal{D}, \nu, S)$ ergodic, $(X \times Y, \mathcal{B} \times \mathcal{D}, \mu \times \nu, T \times S)$ is ergodic.

proof:“ $\Leftarrow$ ” is obvious as if $(X, \mathcal{B}, \mu, T)$ has the property that its product with any ergodic system is ergodic, then $(X, \mathcal{B}, \mu, T)$ is ergodic since we can take its product with the one point system.
This implies that the product of the system with itself $(X \times X, \mathcal{B} \times \mathcal{B}, \mu \times \mu, T \times T)$ is ergodic, which is the definition of being weakly mixing.

“ $\Rightarrow$ ” Suppose $(X, \mathcal{B}, \mu, T)$ weakly mixing.

$T \times S$ is ergodic iff all invariant functions are constant a.e.

For any $g_1, g_2 \in L^2(X), \ h_1, h_2 \in L^2(Y)$ , let $C = \int g_1 \ d \mu$ , let $g_1' = g_1-C$ ; hence $\int g_1' \ d \mu = 0$ .

$\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int g_1 \cdot (g_2 \circ T^n) \ d \mu \cdot \int h_1 \cdot (h_2 \circ S^n) \ d \nu$

$= \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int C \cdot (g_2 \circ T^n) \ d \mu \cdot \int h_1 \cdot (h_2 \circ S^n) \ d \nu$

$+ \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int g_1' \cdot (g_2 \circ T^n) \ d \mu \cdot \int h_1 \cdot (h_2 \circ S^n) \ d \nu$

Since $\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int C \cdot (g_2 \circ T^n) \ d \mu \cdot \int h_1 \cdot (h_2 \circ S^n) \ d \nu$

$= C \cdot \int g_2 \ d \mu \cdot \lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int h_1 \cdot (h_2 \circ S^n) \ d \nu$

By theorem 1, since $S$ is ergodic on $Y$ ,

$=\int g_1 \ d \mu \cdot \int g_2 \ d \mu \cdot \int h_1 \ d \nu \cdot \int h_2 \ d \nu$

On the other hand, let $a_n = \int g_1' \cdot (g_2 \circ T^n) \ d \mu, \ b_n = \int h_1 \cdot (h_2 \circ S^n) \ d \nu$

By theorem 2, since $T$ is weak mixing $\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N (a_n - 0 \cdot \int g_2 \ d \mu)^2 = 0$ hence $\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N a_n^2 = 0 \ \ \ (\ast)$

$(\sum_{n=0}^N a_n \cdot b_n)^2 \leq ( \sum_{n=0}^N a_n^2) \cdot ( \sum_{n=0}^N b_n^2)$ by direct computation i.e. subtract the left from the right and obtain a perfect square.

Therefore $\lim_{N \rightarrow \infty} (\frac{1}{N+1} \sum_{n=0}^N a_n \cdot b_n)^2$

$\leq (\frac{1}{N+1} \sum_{n=0}^N a_n^2) \cdot (\frac{1}{N+1} \sum_{n=0}^N b_n^2)$

Approaches to $0$ as $N \rightarrow \infty$ by $(\ast)$ .

Therefore, $\lim \frac{1}{N+1} \sum_{n=0}^N \int g_1' \cdot (g_2 \circ T^n) \ d \mu \cdot \int h_1 \cdot (h_2 \circ S^n) \ d \nu = 0$

Combining the two parts we get

$\lim_{N \rightarrow \infty} \frac{1}{N+1} \sum_{n=0}^N \int g_1 \cdot (g_2 \circ T^n) \ d \mu \cdot \int h_1 \cdot (h_2 \circ S^n) \ d \nu$

$= \int g_1 \ d \mu \cdot \int g_2 \ d \mu \cdot \int h_1 \ d \nu \cdot \int h_2 \ d \nu$ .

Since the linear combination of functions of the form $f(x, y) = g(x)h(y)$ is dense in $L^2(X \times Y)$ (in particular the set includes all characteristic functions of product sets and hence all simple functions with basic sets being product sets)

We have shown that for a dense subset of $f \in L^2(X \times Y)$ the sequence of functions $\frac{1}{N+1}\sum_{n=0}^N f(T^n(x), S^n(y))$ converge weakly to the constant function. (Since it suffice to check convergence a dense set of functional in the dual space)

Hence for any $f \in L^2(X \times Y)$ , the average weakly converges to the constant function $\int f \ d \mu \times \nu$ .

For any $T \times S$ -invariant function, the average is constant, hence this implies all invariant functions are constant a.e. Hence we obtain ergodicity of the product system.

Establishes the theorem.

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Notes for my lecture on multiple recurrence theorem for weakly mixing systems – Part 1

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