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.