May 2020 – Midwest dynamics and group actions seminar

06/08: Asaf Katz

Measure rigidity of Anosov flows via the factorization method

Abstract: Anosov flows are central objects in dynamics, generalizing the basic example of a geodesic flow over a negatively curved surface.

In the talk we will introduce those flows and their dynamical behavior. Moreover, we show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure rigidity for the moduli space of translation surfaces, can be adapted to smooth ergodic theory and in particular towards the study of Anosov flows. Using this adaption, we show that for a quantitatively non-integrable Anosov flow, every generalized u-Gibbs measure is absolutely continuous with respect to the whole unstable manifold. In the talk I will introduce the factorization method, the relations to previous works (Eskin-Mirzakhani, Eskin-Lindenstrauss) and the result together with some examples and applications. Technical details will be given on the second part.

Slides

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06/01: Kathryn Mann

Topological stability of actions on spheres at infinity

Abstract: The fundamental group of a negatively curved compact manifold acts by homeomorphisms on the visual boundary of its universal cover. In joint work with Jonathan Bowden, we show these actions are C^0 stable: any perturbation is a topological factor of the original action. One can do even better when the manifold is a surface and the visual sphere a circle, and along the same lines can also obtain a stronger result for actions “at infinity” of 3-manifold groups on circle that come from skew-Anosov foliations. However, in this talk, I will focus on the general rigidity result, applicable in all dimensions, and show some interesting slightly wild examples.

Slides

By awb1472 on Talks

05/25: Weikun He

Quantitative equidistribution of random walks on the torus.

Abstract: Under a proximality assumption, Bourgain, Furman, Lindenstrauss and Mozes established a quantitative equidistribution result for linear random walks on the torus. I will discuss some recent progress in this topic, including a joint work with Nicolas de Saxcé which relaxed the proximality assumption and a joint with Tsviqa Lakrec and Elon Lindenstrauss which generalizes (to some extent) the result to affine random walks.

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05/18: Or Landesberg

Horospherically invariant measures on geometrically infinite quotients

Abstract: We consider a locally finite (Radon) measure on $\mathrm{SO}^+(d,1)/\Gamma$ invariant under a horospherical subgroup of $\mathrm{SO}^+(d,1)/\Gamma$ where $\Gamma$ is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not admit any additional invariance properties then it must be supported on a set of points with geometrically degenerate trajectories under the corresponding contracting 1-parameter diagonalizable flow (geodesic flow). We deduce measure classification results. One such result in the context of finitely generated Kleinian groups in $\mathrm{PSL}(2, \mathbb{C})$ will be highlighted. Most of the talk will be based on joint work with Elon Lindenstrauss.

Slides