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.

Meeting ID: 938 8598 9739

Recorded Talk:

Seminar 04.14.22 Yang

Title: Entropy rigidity for 3D Anosov flows

Speaker:  Yun Yang – Virginia Tech

Abstract: Anosov systems are among the most well-understood dynamical systems. Special among them are the algebraic systems. In the diffeomorphism case, these are automorphisms of tori and nilmanifolds. In the flow case, the algebraic models are suspensions of such diffeomorphisms and geodesic flows on negatively curved rank one symmetric spaces. In this talk, we will show that given an integer k ≥ 5, and a C^k Anosov flow Φ on some compact connected 3-manifold preserving a smooth volume, the measure of maximal entropy is the volume measure if and only if Φ is C^{k−ε}-conjugate to an algebraic flow, for ε > 0 arbitrarily small. This is a joint work with Jacopo De Simoi, Martin Leguil and Kurt Vinhage.