September 2020 – Midwest dynamics and group actions seminar

10/05: Davi Obata

Open sets of partially hyperbolic systems having a unique SRB measure

Abstract: For a dynamical system, a physical measure is an ergodic invariant measure that captures the asymptotic statistical behavior of the orbits of a set with positive Lebesgue measure. A natural question in the theory is to know when such measures exist.

It is expected that a “typical” system with enough hyperbolicity (such as partial hyperbolicity) should have such measures. A special type of physical measure is the so-called hyperbolic SRB (Sinai-Ruelle-Bowen) measure. Since the 70`s the study of SRB measures has been a very active topic of research.

In this talk, we will see some new examples of open sets of partially hyperbolic systems with two dimensional centers having a unique SRB measure. One of the key features for these examples is a rigidity result for a special type of measure (the so-called u-Gibbs measure) which allows us to conclude the existence of the SRB measures.

Slides

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09/28: Alireza Salehi Golsefidy

Two new concepts for compact groups: Spectral independence and local randomness

Abstract: I will explain two new concepts for compact groups mentioned in the title. Their basic properties and their connections with the FAb property, quasi-randomness, and super-approximation will be outlined. I will present how these ideas help us show that a Borel probability measure m on the product of compact open subgroups of two locally non-isomorphic simple analytic groups has spectral gap when its projection to each factor has. (Joint work with Keivan Mallahi-Karai and Amir Mohammadi)

Slides

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09/21: Wouter van Limbeek

Commensurators and arithmeticity of thin groups

Abstract: The commensurator of a discrete, Zariski-dense subgroup $\Gamma$ of a simple Lie group $G$ contains information on the arithmetic nature of $\Gamma$ : For example, in 1974, Margulis proved that if $\Gamma$ is a lattice and its commensurator is dense, then $\Gamma$ is arithmetic. In 2011, Shalom asked if the same is true only assuming $\Gamma$ is Zariski-dense in $G$ . I will report on recent progress on this question that uses ideas from infinite ergodic theory, profinite actions, representation theory, Brownian motion, random walks and the structure of locally compact groups. I will show how these combine to give information on commensurators and applications to problems in group theory, hyperbolic 3-manifolds, and structure of arithmetic lattices in Lie groups. This is joint work with D. Fisher and M. Mj.

Slides