BGU Probability and Ergodic Theory (PET) seminar Ical Atom

We study slow entropy invariants for abelian unipotent actions U on any finite volume homogeneous space $G/\Gamma$. For every such action we show that the topological complexity can be computed directly from the dimension of a special decomposition of Lie(G) induced by Lie(U). Moreover, we are able to show that the metric complexity of the action coincides with its topological complexity, which provides a classification of these actions in isomorphic class. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of G. This generalizes our previous rank one results from to higher rank abelian actions. This is a joint work with Adam Kanigowski, Philipp Kunde and Kurt Vinhage.

Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space. In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.

Oseledets’ multiplicative ergodic theorem (MET) is an important tool in smooth ergodic theory. It may be viewed as a generalization of Birkhoff’s pointwise ergodic theorem where numbers are replaced by matrices and arithmetic means are replaced by geometric means. Starting from Ruelle in 1982, many infinite-dimensional generalizations of the MET have been produced; however, these results assume quasi-compactness conditions and so do not deal with continuous spectrum. In a different direction Karlsson-Margulis obtained a geometric generalization of the MET, which we apply in our work to obtain an MET with operators in von Neumann algebras with semi-finite trace. We do not assume any compactness conditions on the operators. Joint work with Lewis Bowen and Ben Hayes.