### View Abstract

Abstract: Let $(X,\mathcal A,\mu)$ be a probability space and let $A_0,...,A_L\in\mathcal A$. In this talk we deal with the asymptotic independence of expressions of the form

$$\mu(A_0\cap T_1^{v_1(n)}A_1\cap \cdots\cap T_L^{v_L(n)}A_L),$$

where $T_1,...,T_L$ are commuting $\mu$-preserving transformations and $v_1,...,v_L$ are polynomials in $\mathbb Z[x]$, and some of their generalizations.

In the first part of this talk we introduce $\Sigma_\ell^*$-limits, $\ell\in\mathbb N$, a new notion of convergence that, as we will see, is strong enough to characterize strong mixing while, simultaneously, being weak enough for avoiding the obstructions to (and, hence, establishing) various strong mixing polynomial ergodic theorems. We will also provide examples which demonstrate that $\Sigma_\ell^*$-limits avoid these obstructions in an "optimal" way.

In the second part of the talk we will show how a slight modification to the method introduced by V. Bergelson in "Weakly mixing PET" can be employed to show that

$$\Sigma_{10}^*\text{-}\lim_{n\in\mathbb Z}\mu(A_0\cap T^{n^2}A_1\cap T^{2n^2}A_2)=\mu(A_0)\mu(A_1)\mu(A_2)$$

for any $A_0,A_1,A_2\in\mathcal A$ and any strongly mixing $T$. Then we will explain why such a method no longer works when dealing with an expression of the form $\mu(A_0\cap S^{n^2}A_1\cap T^{n^2}A_2)$ and briefly mention the new ideas and techniques required to prove that, under meager mixing assumptions on $S$ and $T$, one has that for some $m\in\mathbb N$,

$$\Sigma_{m}^*\text{-}\lim_{n\in\mathbb Z}\mu(A_0\cap S^{n^2}A_1\cap T^{n^2}A_2)=\mu(A_0)\mu(A_1)\mu(A_2).$$

This talk is based on joint work with V. Bergelson.