March 2021 – Midwest dynamics and group actions seminar

04/05: Adam Kanigowski

On ergodic and statistical properties of smooth systems

Abstract: We will recall some basic ergodic and statistical properties such as: ergodicity, (quantitative) mixing, K property, Bernoullicity, central limit theorem. We will be interested in rigidity and flexibility of these properties for smooth diffeomorphisms preserving a smooth measure. Our main rigidity result is that $C^{1+\alpha}$ smooth diffeomorphisms which are exponentially mixing are Bernoulli (joint with D. Dolgopyat and F.Rodriguez-Hertz). For flexibility results we show existence of C^r smooth systems which satisfy the (non-trivial) central limit theorem and are of zero entropy. Moreover we show that there are smooth K, non-Bernoulli systems which satisfy (non-trivial) central limit theorem (joint with D. Dolgopyat, C. Dong, P.Nandori).

Slides

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03/29: Barbara Schapira

Critical exponents and amenability of covers

Abstract: In this joint work with R Coulon, R Dougall and S Tapie, we prove the following result. Let $\Gamma'<\Gamma$ be two discrete groups acting properly isometrically on a hyperbolic space $X$ . Then their critical exponents coincide if and only if the small group $\Gamma'$ is coamenable in $\Gamma$ . In this talk, I will explain the statement and sketch the main steps of the proof, which involves a strange construction of so-called twisted Patterson-Sullivan measures.

Slides

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03/15: Mikolaj Fraczyk

Injectivity radius of discrete subgroups of higher rank Lie groups.

Abstract: Let $G$ be a simple higher rank Lie group and let $X$ be the associated symmetric space. Margulis conjectured that any discrete subgroup $\Gamma$ of $G$ such that $X/\Gamma$ has uniformly bounded injectivity radius must be a lattice. I will present the proof of this conjecture and explain how stationary random subgroups play the central role in the argument. The talk is be based on a recent joint work with Tsachik Gelander.

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03/08: Thang Nguyen

Sub-Riemannian dynamics and local rigidity of higher hyperbolic rank

Abstract: Symmetric and locally symmetric spaces are important classes of Riemannian manifolds of nonnegative curvatures. We study a rigidity question whether a less symmetric geometric condition leads to symmetric and locally symmetric space structures. The condition we consider is that every geodesic in the universal cover is contained in a totally geodesic constant curvature hyperbolic plane (such condition is called higher hyperbolic rank). We are able to obtain a local rigidity for perturbations of locally symmetric spaces. The novelty of the work is a new idea in using geometric analysis on sub-Riemannian manifolds to study dynamics of geodesic flows. We also obtain a local rigidity for some boundary actions using this new idea. This is based on a joint work with Chris Connell and Ralf Spatzier.

Slides