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.

Seminar 11.7.19 Katz

Title: Measure Rigidity for Anosov Flows via the Factorization Method

Speaker: Asaf Katz – University of Chicago

Abstract: Using the factorization method, a method pioneered by Eskin and Mirzkhani in their groundbreaking work about measure classification for P-invariant measures over the moduli space of translation surfaces, we show that generalized u-Gibbs states over quantitatively non-integrable Anosov systems are absolutely continuous with respect to the whole unstable manifold.

Seminar 10.31.19 Downarowicz

Title: Asymptotic Pairs Versus Positivity of Entropy

Speaker: Tomasz Downarowicz – Wroclaw University of Technology

Abstract: Consider a dynamical system (X,T) consisting of a compact metric space X and iterates of a self-homeomorphism T of this space. Topological entropy of the system depends on the speed of growth of complexity of orbits. Zero entropy means that this growth is subexponential and positive entropy corresponds to exponential growth. All this seems quite sophisticated and subtle. On the other hand, there is a very simple-minded notion of an asymptotic pair: two points x, y in X are asymptotic pair if their orbits come closer and closer together as time advances. It might seem surprising but this simple concept SUFFICES to distinguish between positive and zero entropy systems. During my talk I will try to familiarize the audience with the main ideas behind this result (which is due to Blanchard-Host-Ruette in one direction and D. and Lacroix in the other). Moreover, in recent time D., Oprocha and Zhang have obtained a similar criterion for positive entropy in actions of any countable abelian group. If time permits. I will briefly say why this case is very different from the Z-case.

Seminar 10.24.19 Lindsey

Title: Thurston’s Master Teapot

Speaker: Kathryn Lindsey – Boston College

Abstract: When a multimodal self-map of an interval is postcritically finite (PCF), its growth rate (the exponential of its topological entropy) is a special type of algebraic number called a weak Perron number. W. Thurston plotted the set of all Galois conjugates of growth rates of PCF unimodal maps; this visually stunning image revealed that this set has a rich and mysterious geometric structure. Thurston’s Master Teapot is a closely related 3D set. This talk will present some of the basic topological and geometrical properties of these sets. Based on joint work with C. Wu. H. Bray, D. Davis.

Seminar 10.17.19 Velozo

Title: Pressure in Symbolic Dynamics

Speaker: Anibal Velozo – Yale

Abstract: Symbolic dynamics is a particularly useful tool to understand smooth systems with certain hyperbolicity; this is typically done via a “coding”, i.e. a Markov partition/section or inducing schemes. Moreover, the thermodynamic formalism of finite state and countable Markov shifts is well understood, and many results can be pushed to the smooth setting via such codings. The plan of the talk is to overview some known results about the thermodynamic formalism of finite state and countable Markov shifts, and to discuss some recent works about the limiting behavior of the pressure of invariant measures in the non-compact setting. If time permits we will also discuss analogous results for flows.

Seminar 10.3.19 Siddiqi

Title: Decay of correlations for isometric extensions of Anosov flows

Speaker: Salman Siddiqi, Michigan

Abstract: I will briefly provide some historical context discussing known results on exponential correlation decay (or exponential mixing) for Anosov flows. I’ll summarize some classical techniques, and sketch a proof that locally accessible isometric extensions of Anosov flows are exponentially mixing under certain conditions – this includes, for example, some classes of frame flows and flows on principal bundles.