Existence of asymptotic sequence in ergodic measure-preserving transformations

Question

Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \subseteq B$, then $B = T^{-1}B \, (\text{mod } \mu)$. Consider a set $A \in \mathcal{F}$ with $0 < \mu(A) < +\infty$ for which the following holds: $$ \int_A \left( \sum_{k=0}^{n-1} \chi_A \circ T^k \right)^2 d\mu \leq \lambda \left( \int_A \left( \sum_{k=0}^{n-1} \chi_A \circ T^k \right) d\mu \right)^2, \quad \forall n \geq 1, $$ for some constant $\lambda > 0$. Under these assumptions, we aim to establish the existence of a sequence of constants $(\theta_n)_n$ increasing to infinity such that: $$ \lim_{n \to \infty} \frac{1}{\theta_n} \sum_{k=0}^{n-1} \mu(B \cap T^{-k} C) = \mu(B) \mu(C), \quad \forall B,C \in \mathcal{F}, \text{ with } B,C \subseteq A. $$ I believe this relates to the concept of weak mixing. While attempting to construct a sequence $(\theta_n)_n$, I considered using $\theta_n = \frac{1}{\mu(A)^2}\sum_{k=0}^{n-1}\mu(A\cap T^{-k}A)$. However, I am having difficulty proving the limit. Could you please assist me with this? Thank you!

Answer 1

You can define $\phi_n=S_n(1_A)/\theta_n$ where $S_n(1_A)=\sum_{k=0}^{n-1}1_A\circ T^k$ and view $\phi_n$ as funcitons in $L^2(A,\mu)$.

The condition you mentioned gives that $\|\phi_n\|_2\leq \lambda$ for all $n\in\mathbb{N}$. Aaronson uses this and the Banach Zaks theorem to establish that the every (weak) sub-sequential limit of $\phi_n$ is $T$ invariant and since $T$ is ergodic the only possible limit is constant. In addition, the constant has to satisfy $$\mu(A)^2=\lim_{n\to\infty}\int_A \phi_nd\mu=c\int_Ad\mu=c\mu(A).$$

He concludes from this that $\phi_n$ converges weakly to $\mu(A)$ on $L^2(A,\mu)$.

Since $T$ is measure preserving, your condition also applies with $T$ replaces by $T^{-1}$ and thus $\psi_n=\frac{1}{\theta_n}\sum_{k=0}^{n-1}1_A\circ T^{-k}$ also converges weakly in $L^2(A,\mu)$ to $\mu(A)$.

Now let $\mathcal{B}\ni B\subset A$ and write $\varphi_n=S_n(1_B)/\theta_n$. We calculate, $$\frac{1}{\theta_n}\sum_{k=0}^{n-1}\mu(A\cap T^{-k}B)=\int_A \varphi_n d\mu=\int 1_B\cdot \frac{\sum_{k=0}^{n-1}1_A\circ T^{-k}}{\theta_n}d\mu \xrightarrow[n\to\infty]{} \mu(A)\mu(B).$$

As $B\subset A$, for all $n\in\mathbb{N}$, $S_n(1_B)\leq S_n(1_A)$, thus $\sup_n\|\varphi_n\|_2\leq \lambda$. A similar application of the Banach- Saks theorem and ergodicity as in $\phi_n$ would give that $\psi_n$ converges weakly in $L^2(A,\mu)$ to $\lim_{n\to\infty}\int_A\varphi_nd\mu$, which we have shown to equal $\mu(B)$.

This gives that for all $\mathcal{B}\ni C\subset A$, $$\frac{1}{\theta_n}\sum_{k=0}^{n-1}\mu(C\cap T^{-k}B)=\int 1_C\cdot \varphi_n d\mu\xrightarrow[n\to\infty]{}\mu(B)\mu(C).$$