Workshop on Dynamical Systems and Related Topic Session I: Parabolic Dynamics April 9-11 2021 Department of Mathematics University of Maryland SCHEDULE Friday, April 9: 10:30-11:20 am Alex Gorodnik (University of Zurich) Multiple equidistribution for unstable leaves 11:30-12 BREAK 12- 12:50 Taylor McAdam (Yale University) Basepoint-independent density of almost-primes in horospherical orbits in SL(3,Z)\SL(3,R) 1 pm - 2 pm LUNCH BREAK 2- 2:50 Samuel Edwards (Yale University) Horospheres in geometrically finite manifolds 3: 15- 4:15 Corinna Ulcigrai (University of Zurich) (Colloquium) Slow chaos in flows on surfaces Saturday, April 10: 10:30 - 11:20 am Zhiren Wang (Penn State University) Möbius disjointness for almost reducible analytic quasi-periodic cocycles 11:30 - 12 BREAK 12 - 12:50 Kurt Vinhage (Penn State University) Classification and complexity for unipotent group actions 1 - 2 pm LUNCH BREAK 2 - 4:30 Brin Prize session 2 - 2:15 Presentation of the Award 2: 15 - 3:15 Mariusz Lemanczyk Spectral theory, Ratner's properties and Furstenberg disjointness 3:30-4:30 Stefano Marmi Some arithmetical aspects of renormalization in Teichmuller dynamics Sunday, April 11: 10:00 -10:50 am Maria Isabel Cortez (Pontificia Universidad Católica de Chile) Classification of group actions on the Cantor set 11-11:20 BREAK 11:20 - 11:50 Shrey Sanadhya (University of Iowa) Substitution on infinite alphabet and generalized Bratteli-Vershik model. 12 - 12:50 pm Dmitri Scheglov (Universidade Federal Fluminense) Ergodic properties of G-extensions over translation flows and IETs ABSTRACTS María Isabel Cortez Title: Classification of group actions on the Cantor set Abstract: Every countable group admits continuous actions on the Cantor set. The dynamical behaviour of these actions may be useful to answer questions about the properties of the acting group, and conversely, the properties of some algebraic invariants of these actions characterize some of their dynamical properties. In this talk we will explore this kind of questions, giving some examples, including the results of different joint works with Paulina Cecchi, Kostya Medynets and Samuel Petite. Samuel Edwards Title: Horospheres in geometrically finite manifolds Abstract: I will discuss ongoing work studying various equidistribution properties of horospheres in unit tangent bundles of geometrically finite hyperbolic (d+1)-manifolds. Particular focus will be on the case when the critical exponent of the manifold is greater than d/2, allowing the application of representation-theoretic methods to obtain precise relations between rates of equidistribution for expanding translates of pieces of horospheres, the rate of mixing of the geodesic flow on the unit tangent bundle of the manifold, and the spectrum of the Laplace-Beltrami operator on the manifold. Alexander Gorodnik Title: Multiple equidistribution for unstable leaves Abstract: We explore equidistribution for translates of measures supported on proper submanifolds of unstable leaves. Motivated by some arithmetic questions, we investigate multiple correlations for such translated measures. This is a joint work with M. Bjorklund. Taylor McAdam Title: Basepoint-independent density of almost-primes in horospherical orbits in SL(3,Z)\SL(3,R) Abstract: Inspired by the work of Sarnak and Ubis [1] in SL(2,Z)\SL(2,R), we prove that almost- prime times (i.e. integer times having fewer than a fixed number of prime factors) in horospherical orbits of generic points in SL(3,Z)\SL(3,R) are dense in the whole space, where the number of prime factors allowed in the almost-primes is independent of the basepoint. This is in contrast to previous work [2] in which the number of prime factors depends on a Diophantine property of the basepoint. The proof involves a case-by-case analysis of the different ways in which a basepoint can fail the Diophantine property. If a basepoint fails to equidistribute rapidly in the whole space with respect to the continuous time flow, then there exists a sequence of nearby periodic orbits of increasing volume that approximate the original orbit up to larger and larger time scales, and which equidistribute in the whole space as the volume grows. Given an open set, one can find a large enough periodic orbit such that almost-primes of a fixed order in the periodic orbit land inside that set, and this property can then be transported to the nearby orbit of the original basepoint. This is joint work-in-progress with Manuel Luethi. [1] Sarnak, Peter, and Adrián Ubis. "The horocycle flow at prime times." Journal de mathématiques pures et appliquées 103.2 (2015): 575-618. [2] McAdam, Taylor. "Almost-prime times in horospherical flows on the space of lattices." Journal of Modern Dynamics 15 (2019): 277. Shrey Sanadhya Title : Substitution on infinite alphabet and generalized Bratteli-Vershik model. Abstract: We consider substitutions on countably infinite alphabet as Borel dynamical systems and build their Bratteli-Vershik models. We prove two versions of Rokhlin’s lemma for such substitution dynamical systems. Using the Bratteli-Vershik model we give an explicit formula for a shift-invariant measure (finite and infinite) and provide a criterion for this measure to be ergodic (or uniquely ergodic). This is joint work with Sergii Bezuglyi and Palle Jorgensen. Dmitri Scheglov Title: Ergodic properties of G-extensions over translation flows and IETs Abstract: We will discuss our recent results on ergodicity and weak mixing of G-extensions over translation flows on a compact surface, where G is a compact connected Lie group. Also we will talk on the ongoing project with G. Forni on the spectral gap for twisted Kontsevich-Zorich cocycle over Teichmuller flow on character variety and applications to the effective weak mixing of G-extensions. Corinna Ulcigrai Title: Slow chaos in flows on surfaces Abstract: Flows on surfaces describe many systems of physical origin and are one of the most fundamental examples of dynamical systems, studied since Poincaré. These systems often display a 'slow' form of butterfly effect, which makes them important model of 'slowly chaotic' behaviour. In the last decade, there have been a lot of advances in our understanding of the fine chaotic properties of smooth area-preserving flows on higher genus surfaces. During the talk, we will survey some of these properties and results and hint at some of the geometric and arithmetic mechanisms which explain them. Kurt Vinhage Title: Classification and complexity for unipotent group actions Abstract: I will discuss unipotent group actions on homogeneous spaces, and entropy-like invariants of isomorphism and Kakutani equivlance. The slow entropy of such actions will be described explicitly in terms of the adjoint action, and associated Jordan block-like structures. I will emphasize a specific subclass of actions for which the computation is connected to homogeneous space. Joint work with Adam Kanigowski, Philipp Kunde and Daren Wei. Zhiren Wang Title: Möbius disjointness for almost reducible analytic quasi-periodic cocycles. Abstract: In this talk we will discuss Möbius disjointness for one-frequency analytic quasi-periodic SL(2,R)-cocycles which are almost reducible. This is a joint work with Wen Huang, Jing Wang and Qi Zhou. .