Title: Polynomial Ergodic Theorems for Strongly Mixing Commuting Transformations

Speaker:  Rigo Zelada Cifuentes – University of Maryland

Abstract: We present new polynomial ergodic theorems dealing with probability measure preserving $\mathbb Z^L$-actions having at least one strongly mixing element. We prove that, under different conditions, the set of $n\in\mathbb Z$ for which the multi-correlation expressions $$\mu(A_0\cap T_{\vec v_1(n)}A_1\cap \cdots\cap T_{\vec v_L(n)}A_L)$$ are $\epsilon$-independent, must be $\Sigma_m^*$. Here $\vec v_1,…,\vec v_L$ are $\mathbb Z^L$-valued polynomials in one variable and $\Sigma_m^*$, $m\in\N$, is one of a family of notions of largeness intrinsically connected with strongly mixing. We will also present two examples showing the limitations of our results. The existence of these examples suggests further questions dealing with the weakly, mildly, and strongly mixing properties of a multi-correlation sequence along a polynomial path.  This talk is based in joint work with Vitaly Bergelson.