## Seminar 04.29.21 Kleinbock

Title: Some remarks on the `eventually always hitting’ property

Speaker: Dmitry Kleinbock – Brandeis University

Abstract: Eventually always hitting (EAH) points are those whose long orbit segments eventually hit the corresponding shrinking targets for all future times. This is a uniform version of the classical hitting property in ergodic theory with shrinking targets; the terminology is due to Dubi Kelmer. Unlike its classical counterpart, not much is known about conditions on the targets for which almost all vs. almost no points are EAH. I will talk about systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes. For such systems tight conditions on the shrinking rate of the targets can be stated so that the set of eventually always hitting points is null or co-null. This is a joint work with Ioannis Konstantoulas (Upsala) and Florian Richter (Northwestern, formerly OSU).

Meeting ID: 980 3359 0349

## Seminar 04.22.21 Cantrell

Title: Rough similarity, rigidity and the Manhattan Curve for metrics on
hyperbolic groups

Speaker: Steve Cantrell – The University  of Chicago

Abstract: Consider a hyperbolic group equipped with two hyperbolic metrics
that are left invariant and are quasi-isometric to a word metric. A
natural question to ask is: when are these metrics roughly similar, i.e.
when are they within bounded distance after scaling by a positive
constant? In this talk we’ll discuss rigidity statements that characterize
rough similarity in terms of the properties of the so-called Manhattan
Curve. We’ll see how to study this curve using a blend of ideas coming
from ergodic theory and geometric group theory. This is based on joint
work with Ryokichi Tanaka.

Meeting ID: 980 3359 0349

Recorded Talk: https://osu.zoom.us/rec/play/oojIKeNTtJYj2_BDzTMoq_B838qwHXhn_iL8nILR-obhxHVOVdSq9i-yd1-KTAT3QgRolQBWdTVY5Nx_.tCVmCXOuKzaXbsN_?continueMode=true&_x_zm_rtaid=Vd768-G5T-qQzFi8nIZqcw.1619131951571.08574cec3bd38c984e60711df5327a21&_x_zm_rhtaid=146

## Seminar 04.15.21 Moreira

Title: Multiplicative recurrence with additive averaging

Speaker: Joel Moreira – University  of Warwick

Abstract: Motivated by the question of whether Pythagorean triples are partition regular, one is naturally led to study sets of recurrence in the semigroup of natural numbers under multiplication. However, for sets with “additive structure”, the usual tools (such as the van der Corput trick) don’t seem to be useful in this context. As an alternative, we propose to study sets of averaging recurrence, where the averaging is taken additively. We present some results in this direction, and some applications to number theory. This is based on joint work with Sebastian Donoso, Anh Le and Wenbo Sun.

Meeting ID: 980 3359 0349

Recorded Talk: https://osu.zoom.us/rec/play/J1MkxyEaOGUHGcdGM84cNOQXt7thTW47Im6oWulA6EIbn4c5tpI0vGFW1eR7u_vDHqpnpLQCKVQso5SK.86O6eYU5IyOTjLxX?continueMode=true

## Seminar 04.08.21 DeWitt

Title: Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds

Speaker: Jonathan DeWitt – The University of Chicago

Abstract: Suppose that M is a closed isotropic Riemannian manifold and that R_1,…,R_m generate the isometry group of M. Let f_1,…,f_m be smooth perturbations of these isometries. We show that the f_i are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian from S^n to real, complex, and quaternionic projective spaces.

Meeting ID: 980 3359 0349

## Seminar 03.25.21 Lemańczyk

Title: On Furstenberg systems of some aperiodic multiplicative functions

Speaker: Mariusz Lemańczyk – Nicolaus Copernicus University in Toruń

Abstract: Studying arithmetic properties of multiplicative functions through the so called Furstenberg systems became a powerful and fruitful ergodic tool when dealing with the Sarnak and Chowla conjectures, cf. Frantzikinakis-Host’s theorem on the validity of logarithmic Sarnak’s conjecture for systems having not too many ergodic measures.
The Chowla conjecture, originally formulated for the Liouville function, was expected to hold for a much larger class of multiplicative functions in the sense that it has precisely one Furstenberg system, and this system is “maximally random”.
In 2015‪ Matomäki‬ , Radziwiłł and Tao gave a counterexample to Elliot’s conjecture by constructing aperiodic multiplicative functions (bounded by 1) for which (already) the Chowla conjecture of order 2 fails.
During the talk I will try to describe recent results concerning a variety of Furstenberg systems for ‪Matomäki‬, Radziwiłł, Tao’s functions, in particular, showing that the Chowla conjecture holds for them along some subsequences. The talk is based on my joint work with Alex Gomilko and Thierry de la Rue.

Meeting ID: 980 3359 0349

## Seminar 03.18.21 Robertson

Title: Uniform Distribution of Saddle Connection Lengths

Speaker: Donald Robertson – University of Manchester

Abstract: Saddle connections on flat surfaces are those straight line trajectories connecting singular points. In this talk I will explain what that means and discuss work with Jon Chaika and Benjamin Dozier on the uniform distribution mod 1 of the lengths of saddle connections.

Meeting ID: 980 3359 0349

Recorded Talk: https://osu.zoom.us/rec/play/Gn_hXP0BBP7r3HPdodAJuEUxk3ed9ZUfUstA9aS6gKBrFBiLuyOmp6Y8tdA4zHta_Yk0zox-lIuk2iUR._SxZ3acW_MWm4WXD?continueMode=true&_x_zm_rtaid=8CfvQXwLTHOXESID46FWow.1616204141275.c79ea67121104fa26c652ee4a2cdd174&_x_zm_rhtaid=272

## Seminar 03.11.21 Kwietniak

Title: Dbar-approachability, entropy density and B-free shifts

Speaker: Dominik Kwietniak – Jagiellonian University in Krakow

Abstract: Let dbar denote the pseudometric on the full shift over a
finite alphabet A given by the upper asymptotic density of the set of
positions at which two A-valued sequences differ. Write H-dbar for the
associated Hausdorff pseudometric for subsets of the full shift. We
study which properties of shift spaces (shifts) are closed with
respect to H-dbar. In particular, we study shifts, which are H-dbar
limits of their Markov approximations. We call these shifts
dbar-approachable. We provide a topological characterization of chain
mixing dbar-approachable shifts analogous to Friedman-Ornstein’s
characterization of Bernoulli processes.

We prove that many specification properties imply
dbar-approachability. It follows that mixing shifts of finite type,
mixing sofic shifts, and beta-shifts are dbar-approachable. We
construct minimal and proximal examples of mixing dbar-approachable
shifts. We also show that dbar-approachability and chain-mixing imply
dbar-stability, a property recently introduced by Tim Austin. This
leads to examples of minimal or proximal dbar-stable shift spaces,
answering a question posed by Austin. Finally, we show that the set of
shifts with entropy-dense ergodic measures is H-dbar closed. Note that
entropy-density of ergodic measures is known to follow from the
specification property, but the minimal or proximal examples are far
from having any specification. Finally, we show entropy-density for a
class of shifts that includes many interesting B-free shifts. These
shift spaces are not dbar-approachable, but they are H-dbar limits of
sequences of transitive sofic shifts, and this implies
entropy-density.

This is a joint work with Jakub Konieczny and Michal Kupsa.

Meeting ID: 980 3359 0349

Recorded talk: https://osu.zoom.us/rec/play/_DXkoWtXTB92Pui6F7zl4eoVstNWH1rMUdb2a8NjFe61zd2BC9dTZP4UnuUKAC9behs6MQs88XEToF8A.vhVYz7t7fI5_en7U?continueMode=true&_x_zm_rtaid=HImgc_KTTByZM_8W1gjyuA.1615523637945.3021b693a8b3eeed460d5a4c44061f1c&_x_zm_rhtaid=457

## Seminar 03.04.21 Merriman

Title: Using modular surfaces to generate continued fractions

Speaker: Claire Merriman – The Ohio State University

Abstract: Continued fractions are frequently studied in number theory, but they can also be described geometrically. I will give both pictorial and algebraic descriptions of the flows that describe continued fraction expansions. This talk will focus on continued fractions of the form $a_1\pm\frac{1}{a_2\pm\frac{1}{a_3\pm\ddots}}$, where the $a_i$ are odd. I will show how to describe these continued fractions as geodesic on the hyperbolic plane, and how they cross cells of the Farey tessellation.

Meeting ID: 980 3359 0349

Recorded talk: https://osu.zoom.us/rec/play/FNCFPum1mokl6Bnf8uJ76iRehQRPNq5Op3VMXBDbNz7lAPb5qGWwnud4KJJucCuZQhrufoMV3d7X7MbK.icd0xSMxEksuitng?continueMode=true&_x_zm_rtaid=TxJ_aekJTv69MOkCQOL-dA.1614920606908.f7398a7dd87a1fb116574333eca30d89&_x_zm_rhtaid=489

## Seminar 02.25.21 Richter

Title: Additive and geometric transversality of fractal sets in the integers

Speaker: Florian K. Richter – Northwestern University

Abstract: Using the language of fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the 1960s that explore the relationship between digit expansions of real numbers in distinct prime bases. While his famous x2 x3 conjecture remains open, recent solutions to some of his “transversality conjectures” have shed new light on old problems. In this talk we explore analogues questions in the discrete setting of the integers, with the aim of understanding the independence of sets of integers that are structured with respect to different prime bases. This is based on joint work with Daniel Glasscock and Joel Moreira.

Meeting ID: 980 3359 0349

Recorded talk: https://osu.zoom.us/rec/play/Rkd9fe3_RRhHi58HtHE0Gjv6Ls831YiiVREOlnvsMCZh1MAQ2pwLNXdLTF04YV1bASAUr0xCj6JIoVO_.ukxDRCKORfRDV69r?continueMode=true&_x_zm_rtaid=SeyN37XMRTCLsJefPOamKg.1614300556080.85186464ed81cb1aa81402c607077979&_x_zm_rhtaid=566

## Seminar 02.18.21 Glasscock

Title: Recent progress on a question of Katznelson concerning topological recurrence

Speaker: Daniel Glasscock – UMass Lowell

Abstract: Katznelson’s question is a longstanding open question at the intersection of topological dynamics, combinatorial number theory, and harmonic analysis: Is every set of Bohr recurrence a set of topological recurrence?  Equivalently, does the set of differences A-A of a set of integers A with bounded gaps contain the iterated difference set (B-B)-(B-B) of a set B of positive upper density?  In this talk, I will survey what little is known about Katznelson’s question and explain some recent progress achieved in joint work with Andreas Koutsogiannis and Florian Richter.

Meeting ID: 980 3359 0349

## Seminar 02.11.21 Donoso

Title: Topological and combinatorial aspects of finite topological rank systems

Speaker: Sebastián Donoso – University of Chile

Abstract: In this talk, I will review recent results in the class of finite topological rank minimal subshifts. Such systems are the ones that can be represented with a Bratteli diagram (and a Vershik map on it) where the number of vertices at each level is uniformly bounded. I will analyze their correspondence with the $\mathcal{S}$-adic subshifts and their complexity word function.

Meeting ID: 980 3359 0349

Recorded talk: https://osu.zoom.us/rec/play/AISzeWPNTTxkC-dy_ZtZM76wDoDJofw6-fpqPsslISf34ULhGvUbGjgI4mf3_h2jCNruS9vOPdplxCzR.hNo2Yi9bD6SHkiE3?continueMode=true&_x_zm_rtaid=KvM_SXzBT0qkkWkrmUHZJg.1613079456544.ff00d7a1110cfbae291de3cd17886910&_x_zm_rhtaid=402

## Seminar program for Spring 2021

We are pleased to resume our online seminar program. As usual, we meet on Thursdays at 3.00pm EST unless otherwise noted.

The following is our current schedule, and more talks will be announced soon.

Feb 4th: No seminar due to the one-day workshop ‘Hyperbolic Day Online‘ organized by Andrey Gogolev (Ohio State) and Rafael Potrie (Universidad de la Republica)

Feb 11th: Sebastian Donoso (University of Chile)

Feb 18th: Daniel Glasscock (UMass Lowell)

Feb 25th: Florian Richter (Northwestern)

Mar 04th: Claire Merriman (The OSU)

Mar 11th: Dominik Kwietniak (Jagiellonian University in Krakow)

Mar 18th: Donald Robertson (University of Manchester)

Mar 25th: Mariusz Lemańczyk (Nicolaus Copernicus University)

Apr 1st: Break

April 8th: Jonathan DeWitt (The University of Chicago)

Apr 15th: Joel Moreira (University of Warwick)

Apr 22nd: Steve Cantrell (The University of Chicago)

Apr 29th: Dmitry Kleinbock (Brandeis University)