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.

Please contact the organizers, Andreas Koutsogiannis and Dan Thompson for a Zoom link.

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 01st: Dmitry Kleinbock (Brandeis University)

Apr 15th: Joel Moreira (University of Warwick)

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

Seminar 12.3.20 Frantzikinakis

Title: Furstenberg Systems of Bounded Sequences

Speaker: Nikos Frantzikinakis – University of Crete

Abstract: Furstenberg systems are measure preserving systems that are used to model statistical properties of bounded sequences of complex numbers. They offer a different viewpoint for a variety of problems for which progress can be made by a partial or complete description of suitably chosen Furstenberg systems. In this lecture I will give several examples of this principle and in the process we will see several structural properties of Furstenberg systems arising from smooth functions and bounded multiplicative functions.

Zoom link here

Seminar 11.19.20 Hafuta

Title: Limit theorems for time dependent expanding dynamical systems

Speaker: Yeor Hafuta – Ohio State University

Abstract: Some of the results like the Berry-Esseen theorem and moderate deviations principle hold true for general sequences of maps when the variance of the underlying partial sums grows faster than n^{2/3}, while other results such as the local central limit theorem hold true for certain classes of random not necessarily stationary transformations. The results also include a certain type of stability theorem in a complex version of the sequential Rulle-Perron-Frobenius theorem, which yields that the variance grows linearly fast when the underlying maps are close enough to a single expanding map.

Zoom link here

Pdf of slides available here

Seminar 11.12.20 Mirzadeh

Title: On the dimension drop conjecture for diagonal flows on the space of lattices

Speaker: Shahriar Mirzadeh – Michigan State University

Abstract: (see attached pdf for better formatting): Consider the set of points in a homogeneous space X=G/Γ whose gt-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when has real rank 1.

In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: G=SLm+n(R), Γ=SLm+n(Z)and gt=\diag(et/m,⋯,et/m,e−t/n,⋯,e−t/n) We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.

Zoom recording available here

Pdf of slides available here

Seminar 10.29.20 Kucherenko

Title: Multiple phase transitions on compact symbolic systems

Speaker: Tamara Kucherenko – City College of New York

Abstract: A first-order phase transition refers to a loss of differentiability of the pressure function with respect to a parameter regarded as the inverse temperature. Such non-differentiability necessarily implies coexistence of several equilibrium states, although the converse is not true. In the case of H\”older continuous potentials on transitive SFTs the pressure is real analytic and there are no phase transitions. Therefore, in order to allow the possibility of phase transitions one needs to consider potentials that are merely continuous. Note that the convexity of the pressure implies that a continuous potential has at most countably many phase transitions. We show that the case of infinitely many phase transitions can indeed be realized. In this talk we present a method to explicitly construct a continuous potential on a full shift with an infinite number of first order phase transitions occurring at any increasing sequence of predetermined points. This is based on joint work with Anthony Quas and Christian Wolf.

Zoom recording available here

Seminar 10.22.20 Best

Title: The Furstenberg-Sárközy theorem and asymptotic total ergodicity

Speaker: Andrew Best – Ohio State University

Abstract: The Furstenberg-Sárközy theorem asserts that the difference set E-E of a subset E of the natural numbers with positive upper density contains a (nonzero) square. Furstenberg’s approach relies on a correspondence principle and a version of the Poincaré recurrence theorem along squares; the latter is shown via the result that for any measure-preserving system $(X,\mathcal{B},\mu,T)$ and set A with positive measure, the ergodic average $\frac{1}{N} \sum_{n=1}^N \mu(A \cap T^{-n^2}A)$ has a positive limit c(A) as N tends to infinity. Motivated — by what? we shall see — to optimize the value of c(A), we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings (Z/N_m Z) is asymptotically totally ergodic if and only if the least prime factor of N_m grows to infinity. From this fact, we derive some combinatorial consequences. These results are based on joint work with Vitaly Bergelson.

Zoom recording available here

Seminar 10.16.20 Wang

Title: Central Limit Theorem for equilibrium measures in dynamical systems

Speaker: Tianyu Wang – Ohio State University

Abstract: Central limit theorem of certain class of equilibrium measures is a heavily studied statistical property in smooth dynamics. In the first half of the talk, I will briefly introduce some common strategies to study CLT that are useful in many classic settings, e.g. Anosov flows, expanding maps on the unit circle, (countable) Markov shift, etc. In the second half, I will show how specification can be applied to derive an asymptotic version of CLT for the equilibrium measures in the case of geodesic flow on non-positively curved rank-one manifold. This method is first introduced by Denker, Senti, Zhang and the result is based on a recent joint work with Dan Thompson.

Zoom recording available here

Seminar 10.9.20 Demers

Title: Thermodynamic Formalism for Sinai Billiards

Speaker: Mark Demers – Fairfield University

Abstract: While the ergodic properties of Sinai billiards with respect to the SRB measure are well understood, there have been few studies of other invariant measures and equilibrium states. As a step in this direction, we study the family of geometric potentials $– t \log (J^uT)$, $t>0$. For any finite horizon Sinai billiard map $T$, we find $t_* >1$ such that for each $t \in (0, t_*)$, there exists a unique equilibrium state $\mu_t$ for the potential. We show that $\mu_t$ is exponentially mixing for H\”older observables, and that the pressure function $P(t)$ is analytic on $(0,t_*)$. This extends our recent results for the case $t=0$, corresponding to the measure of maximal entropy. This is joint work with Viviane Baladi.

Zoom recording available here

Pdf of slides available here

Seminar 10.2.20 Das

Title: Successive minima of lattice trajectories and topological games to compute fractal dimensions

Speaker: Tushar Das – University of Wisconsin – La Crosse

Abstract: We present certain sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establishes a new connection to the metric theory via a variational principle that computes the fractal dimensions of various sets of number-theoretic interest. Applications of our results include computing the Hausdorff and packing dimensions of the set of singular systems of linear forms and showing they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of the correspondence principle (initiated by Dani, and deepened by Kleinbock and Margulis), the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Highlights of our program include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry. I hope to present a sampling of open questions and directions that have yet to be explored, some of which may be pursued by either following or adapting the technology described in my talk.

Zoom recording available here

Pdf of slides available here

Seminar 9.18.20 Kao

Title: Pressure Metrics for Deformation Spaces of Quasifuchsian Groups with Parabolics

Speaker: Lien-Yung “Nyima” Kao – George Washington University

Abstract: Thurston pointed out that one can use variations of lengths of closed geodesics on hyperbolic surfaces to construct a Riemannian metric on the Teichmueller space. When the surface is closed, Wolpert showed that Thurston’s construction recovers the Weil-Petersson metric. Using thermodynamic formalism, McMullen proposed a new perspective to this Riemannian metric, and called it the pressure metric. In this talk, I will discuss how to extend this dynamical construction to spaces of quasiconformal deformations of (non-compact) finite area hyperbolic surfaces. This is a joint work with Harry Bray and Dick Canary.

Zoom recording available here

Pdf of slides available here


New Ohio State Online Ergodic Theory Seminar


UPDATE: We will continue our program in Spring 2021. However, we are taking a brief Winter hiatus. We expect to resume in February.

We are pleased to announce that we will be running an online seminar program in Fall 2020. The seminar will take place in our usual time slot unless otherwise noted – Thursdays 3.00pm (EST). Some seminars are scheduled at an alternate time of Friday 12.40pm (EST).

Please contact the organizers for a Zoom link.

Our current schedule for the semester follows:

Sept 17: Lien-Yung “Nyima” Kao (George Washington University)

Oct 2 (Friday, 1pm EST): Tushar Das (University of Wisconsin)

Oct 9 (Friday, 12.40pm EST): Mark Demers (Fairfield University)

Oct 16 (Friday, 12.40pm EST): Tianyu Wang (Ohio State)

Oct 22: Andrew Best (Ohio State)

Oct 29: Tamara Kucherenko (City College of New York)

Nov 12: Shahriah Mirzadeh (Michigan State)

Nov 19: Yeor Hafuta (Ohio State)

Dec 3: Nikos Frantzikinakis (University of Crete)

Online seminars Spring 2020

We are not currently running an online seminar in place of our regular seminar. However, we recommend the excellent programs being run by our friends in other dynamics and ergodic theory groups including:

University of Maryland Dynamics seminar, Thursdays at 2.00pm EST

University of Utah-based working ergodic theory seminar, Tuesdays at 4.00pm EST

West Coast Dynamics seminar, Tuesdays at 5.00pm EST

Resistencia Dinamica, Rio de Janeiro, Fridays 12.00pm EST

ETH Zurich Ergodic theory and dynamical systems

Seminar 3.9.20 Climenhaga

Title: Non-adapted measures for billiards and other systems with singularities

Speaker: Vaughn Climenhaga – University of Houston

Abstract: The extension of smooth ergodic theory to systems with singularities, such as billiards, generally requires one to work with “adapted” measures, which do not give too much weight to the neighborhoods of the singularities of the system. For hyperbolic systems such as the Sinai billiard, it is often the case that natural invariant measures, such as the SRB measure and the measure of maximal entropy (MME), are adapted. More generally one can ask about equilibrium measures, and it becomes important to understand how large the entropy of a non-adapted measure can be. I will describe some simple examples illustrating some of the possible behaviors for interval maps, as well as an example of a billiard system with a positive entropy non-adapted measure (joint work with Mark Demers, Yuri Lima, and Hongkun Zhang). Finally, I will formulate some conjectures and describe work in progress towards realizing them.

Seminar 2.27.20 Zelada Cifuentes

Title: Odd polynomials, Diophantine approximations and applications to ergodic theory

Speaker: Rigo Zelada Cifuentes – The Ohio State University

Abstract: Let v(x)=Nj=1ajx2j1 be an odd real polynomial. We will start by describing new Diophantine results pertaining to sets of the form {n|v(n)<ϵ}, where || || denotes the distance to the closest integer. The second part of the talk will be devoted to applications of these Diophantine results (and the techniques behind them) to ergodic theory. Among other things, we will discuss a new version of Khintchine’s recurrence theorem, a new characterization of weakly mixing systems and a result on strong mixing of all orders. The talk is based on a joint work with Dr. Bergelson.

Seminar 2.20.20 Call

Title: The K Property for Equilibrium States of Flows with an Application to Geodesic Flows in Nonpositive Curvature

Speaker: Benjamin Call – The Ohio State University

Abstract: I will present some easy to state assumptions to show that a wide class of equilibrium states have the K property, which is a mixing property stronger than mixing of all orders and weaker than Bernoulli. I will then discuss an application to the setting of geodesic flows on Riemannian manifolds with nonpositive curvature for the class of equilibrium states studied by Burns-Climenhaga-Fisher-Thompson. This is joint work with Dan Thompson.

Seminar 2.6.20 Ferre Moragues

Title: Combinatorial notions of largeness and their interaction with Ergodic Theory

Speaker: Andreu Ferre Moragues – The Ohio State University

Abstract: A theorem due to Hindman states that for any set E⊆ℕ withd∗(E):=limsupN−M→∞|E∩{M,…,N−1}|/(N−M) >0, and any ε>0 there exists some N∈ℕ such that d∗(⋃N i=0(E−i))>1−ε. Hindman’s theorem, a guiding theme for the talk, will allow us to distinguish between two notions of largeness: upper density (d¯) and upper Banach density (d∗).

We will also see how Hindman’s theorem allows for a deeper understanding of Furstenberg’s correspondence principle. Indeed, one can show that an appropriate version of Furstenberg’s correspondence principle yields a dynamical proof of this theorem which is simpler than the original combinatorial one and can be generalized to amenable semigroups.

Moreover, a general version of Hindman’s theorem helps characterize WM groups (i.e., groups with the property that any ergodic measure preserving action (Tg)g∈G on a probability space (X,B,μ) is weakly mixing). Time permitting, we will discuss the strategy of the proofs and how far the results can be extended. The talk is based on a joint work with Dr. Bergelson.

Seminar 1.23.20 Potrie

Title: Partial hyperbolicity and foliations in 3-manifolds

Speaker: Rafael Potrie – CMAT (Uruguay)

Abstract: I’ll explain a beautiful old result by Margulis and Plante-Thurston stating that if a 3-manifold admits an Anosov flow then its fundamental group has exponential growth (as well as explaining what these things mean). I will then explain how some ideas can be pushed further to understand the topological structure of partially hyperbolic diffeomorphisms.

Seminar for Spring 2020

Here is a list of visiting speakers currently scheduled for Spring semester 2020. All talks are on Thursdays in MW154 at 3.00pm-4.00pm (unless otherwise indicated). More talks will be announced soon.

Jan 23: Rafael Potrie (CMAT, Uruguay)

Feb 6: Andreu Ferre Moragues (Ohio State)

Feb 20: Ben Call (Ohio State)

Feb 27: Rigo Zelada Cifuentes (Ohio State)

Mar 9: Vaughn Climenhaga (Houston) (Note unusual day, MONDAY)

The following talks were scheduled but are now postponed due to the pandemic:

Mar 19: Federico Rodriguez Hertz (Penn State)

Mar 26: Ayse Sahin (Wright State)

Apr 2: Pengfei Zhang (Oklahoma)

Seminar 11.21.19 Park

Title: Thermodynamic formalism of fiber-bunched GL(d,R)-cocycles

Speaker: Kiho Park – University of Chicago

Abstract: We study subadditive thermodynamic formalism of H\”older and fiber-bunched GL(d,R)-cocycles over subshift of finite types. Here, fiber-bunched cocycles refer to cocycles that are nearly conformal. Unlike additive thermodynamic formalism where any H\”older continuous potential has a unique equilibrium state, there are examples of H\”older continuous matrix cocycles with multiple equilibrium states. Restricted to fiber-bunched cocycles, we show that there exists an open and dense subset of cocycles with unique equilibrium states; such open and dense subset consists of typical cocycles first introduced by Bonatti and Viana. The unique equilibrium states of typical cocycles follow from a property known as quasi-multiplicativity, and they have the subadditive Gibbs property.When d=2, we have complete description of cocycles with unique equilibrium states. In particular, irreducible cocycles necessarily have unique equilibrium states, and we provide characterization for reducible cocycles with more than one equilibrium states.

Seminar 11.14.19 Vinhage

Title: New Progress on the Katok-Spatzier conjecture

Speaker: Kurt Vinhage – Pennsylvania State University

Abstract: We will discuss recent progress on the Katok-Spatzier conjecture, which aims to classify Anosov actions of higher-rank abelian groups under the assumption that there are no nontrivial smooth rank one factors. We develop new techniques to build homogeneous structures from dynamical ones. The remarkable features of the techniques are their low regularity requirements and their use of metric geometry over differential geometry to build group actions. We apply these techniques to obtain a classification result in the totally Cartan setting, where bundles associated to the hyperbolic structure are one-dimensional. Joint with Ralf Spatzier.