Title: Polynomial Ergodic Theorems for Strongly Mixing Commuting Transformations

Speaker:  Rigo Zelada Cifuentes – University of Maryland

Abstract: We present new polynomial ergodic theorems dealing with probability measure preserving $\mathbb Z^L$-actions having at least one strongly mixing element. We prove that, under different conditions, the set of $n\in\mathbb Z$ for which the multi-correlation expressions $$\mu(A_0\cap T_{\vec v_1(n)}A_1\cap \cdots\cap T_{\vec v_L(n)}A_L)$$ are $\epsilon$-independent, must be $\Sigma_m^*$. Here $\vec v_1,…,\vec v_L$ are $\mathbb Z^L$-valued polynomials in one variable and $\Sigma_m^*$, $m\in\N$, is one of a family of notions of largeness intrinsically connected with strongly mixing. We will also present two examples showing the limitations of our results. The existence of these examples suggests further questions dealing with the weakly, mildly, and strongly mixing properties of a multi-correlation sequence along a polynomial path.  This talk is based in joint work with Vitaly Bergelson.

Meeting ID: 938 8598 9739

Recorded Talk:

Seminar 04.14.22 Yang

Title: Entropy rigidity for 3D Anosov flows

Speaker:  Yun Yang – Virginia Tech

Abstract: Anosov systems are among the most well-understood dynamical systems. Special among them are the algebraic systems. In the diffeomorphism case, these are automorphisms of tori and nilmanifolds. In the flow case, the algebraic models are suspensions of such diffeomorphisms and geodesic flows on negatively curved rank one symmetric spaces. In this talk, we will show that given an integer k ≥ 5, and a C^k Anosov flow Φ on some compact connected 3-manifold preserving a smooth volume, the measure of maximal entropy is the volume measure if and only if Φ is C^{k−ε}-conjugate to an algebraic flow, for ε > 0 arbitrarily small. This is a joint work with Jacopo De Simoi, Martin Leguil and Kurt Vinhage.

Meeting ID: 938 8598 9739

Recorded Talk:

Seminar 04.07.22 Tsinas

Title:Multiple ergodic theorems for sequences of polynomial growth

Speaker:  Konstantinos Tsinas – University of Crete (Greece)

Abstract: Following the classical results of Host-Kra and Leibman on the polynomial ergodic theorem, it is natural to ask whether we can establish mean convergence of multiple ergodic averages along several other sequences, which arise from functions that have polynomial growth. In 1994, Boshernitzan proved that for a function f, which belongs to a large class of smooth functions (called a Hardy field) and which has polynomial growth, its “distance” from rational polynomials is crucial in determining whether or not the sequence of the fractional parts of f(n) is equidistributed on [0,1]. This, also, implies a corresponding mean convergence theorem in the case of single ergodic averages along the sequence ⌊f(n)⌋ of integer parts. In the case of multiple averages, it was conjectured by Frantzikinakis that a similar condition on the linear combinations of the involved functions should imply mean convergence. We verify this conjecture and show that in all ergodic systems we have convergence to the “expected limit”, namely, the product of the integrals. We rely mainly on the recent joint ergodicity results of Frantzikinakis, as well as some seminorm estimates for functions belonging to a Hardy field. We will also briefly discuss the “non-independent” case, where the L^2-limit of the averages exists but is not equal to the product of the integrals.

Meeting ID: 938 8598 9739

Recorded Talk: https://osu.zoom.us/rec/share/Gf98gFbI9Itd1STAukYTGjTHeePNXMHIsdoCITVDNs0cCpKQbNDEjUaYfEEVHbms.BBHTyrGjdrrvmvPr

Seminar 03.31.22 Chen – In person

Title: Marked boundary rigidity and Anosov extension

Speaker: Dong Chen – Penn State University

Abstract: In this talk we will show how a sufficiently small geodesic ball in any Riemannian manifold can be embedded into an Anosov manifold with the same dimension. Furthermore, such embedding exists for a larger family of domains even with hyperbolic trapped sets. We will also present some applications to boundary rigidity and related open questions. This is a joint work with Alena Erchenko and Andrey Gogolev.

Seminar 03.24.22 Koutsogiannis – In person

Title: Convergence of polynomial multiple ergodic averages for totally ergodic systems

Speaker: Andreas Koutsogiannis – Aristotle University of Thessaloniki (Greece)

Abstract: A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge to “the expected limit” in the mean, i.e., the product of the integrals. Exploiting a recent approach of Frantzikinakis, which allows one to avoid deep tools from ergodic theory that were previously used to establish similar results, we study joint ergodicity in totally ergodic systems for integer parts of real polynomial iterates. More specifically, our main results in this direction are a sufficient condition for k terms, and a characterization in the k=2 case. Joint work with Wenbo Sun.

Seminar 03.03.22 Griesmer

Title: Rigidity sequences for measure preserving transformations

Speaker: John Griesmer – Colorado School of Mines

Abstract:Let $(X,\mu,T)$ be a probability measure preserving system.  An increasing sequence $(n_k)$ of natural numbers is a rigidity sequence for $(X,\mu,T)$ if $\lim_{k\to\infty} \mu(A\triangle T^{-n_k}A)=0$ for every measurable $A\subset X$.  A classical result says that a generic measure preserving transformation is weak mixing and has a rigidity sequence, and it is natural to wonder which sequences are rigidity sequences for some weak mixing system.  Bergelson, del Junco, Lemańczyk, and Rosenblatt (2012) popularized many problems inspired by this question, and interesting constructions have since been provided by T. Adams; Fayad and Thouvenot; Fayad and Kanigowski; Griesmer; Badea, Grivaux, and Matheron; and Ackelsberg, among others.   This talk will summarize the relevant foundations and survey some recent results. We also consider two variations: union rigidity, where $\lim_{K\to\infty} \mu\Bigl(A\triangle \bigcup_{k>K}T^{-n_k}A\Bigr)=0$ for some  $A$ with $0<\mu(A)<1$, and summable rigidity, where $\sum_{k=1}^\infty \mu(A\triangle T^{-n_k}A)$ converges for some $A$ with $0<\mu(A)<1$.

Meeting ID: 938 8598 9739

Recorded Talk: https://osu.zoom.us/rec/share/LhoRfB_gvaAVAFyou-BQhRojLm0dQ0sk4uFbeQuWVXu1g5ytspNTkgGS25Li1a8Z.anEF2zLVmvlunkCo

Seminar 02.24.22 Ackelsberg

Title: Large intersections for multiple recurrence in abelian groups

Speaker: Ethan Ackelsberg – Ohio State University

Abstract: With the goal of a common extension of Khintchine’s recurrence theorem and Furstenberg’s multiple recurrence theorem in mind, Bergelson, Host, and Kra showed that, for any ergodic measure-preserving system (X, ℬ, μ, T), any measurable set A ∈ ℬ, and any ε > 0, there exist (syndetically many) n ∈ ℕ such that μ(A ∩ TnA ∩ … ∩ TknA) > μ(A)k+1 – ε if k ≤ 3, while the result fails for k ≥ 4. The phenomenon of large intersections for multiple recurrence was later extended to the context of ⊕𝔽p-actions by Bergelson, Tao, and Ziegler. In this talk, we will address and give a partial answer to the following question about large intersections for multiple recurrence in general abelian groups: given a countable abelian group G, what are necessary and sufficient conditions for a family of homomorphisms φ1, …, φk : G → G so that for any ergodic measure-preserving G-system (X, ℬ, μ, (Tg)gG), any A ∈ ℬ, and any ε > 0, there is a syndetic set of g ∈ G such that μ(A ∩ Tφ1(g)A ∩ … ∩ Tφk(g)A) > μ(A)k+1 – ε? We will also discuss combinatorial applications in ℤd and (ℕ, ·). (Based on joint work with Vitaly Bergelson and Andrew Best and with Vitaly Bergelson and Or Shalom.)

Meeting ID: 941 3609 7274

Recorded Talk: https://osu.zoom.us/rec/share/TY64JIVXsqzNP_i1eNUIiwC0LriToGI6PVmOqPdJGnNuvNFRKkSLVvXiRP27RPU-.lyS_YtUQpBEuOhpC

Seminar 02.17.22 Sharp

Title: Helicity and linking for 3-dimensional Anosov flows

Speaker: Richard Sharp – University of Warwick, UK

Abstract: Given a volume-preserving flow on a closed 3-manifold, one can, under certain conditions, define an invariant called the helicity. This was introduced as a topological invariant in fluid dynamics by Moffatt and measures the total amount of linking of orbits. When the manifold is a real homology 3-sphere, Arnold and Vogel identified this with the so-called asymptotic Hopf invariant, obtained by taking the limit of the normalised linking number of two typical long orbits. We obtain a similar result for null-homologous volume preserving Anosov flows, in terms of weighted averages of periodic orbits. (This is joint work with Solly Coles.)

Meeting ID: 941 3609 7274

Recorded Talk: https://osu.zoom.us/rec/share/cYO8hmX37fCGqR5DfrnHAnCNK04udNHsLvehiztiGOKAOEiByu-F2FpNPl7GDCGZ.WyVU6UdNkxaxw0fQ

Seminar 02.10.22 Quas

Title: Lyapunov Exponents for Transfer Operators

Speaker: Anthony Quas – University of Victoria, Canada

Abstract: Transfer operators are used, amongst other ways, to study rates of decay of correlation in dynamical systems. Keller and Liverani established a remarkable result, giving conditions in which the (non-essential) part of the spectrum of a transfer operator changes continuously under small perturbations to the operator. This talk is about an ongoing project with Cecilia Gonzalez-Tokman in which we aim to develop non-autonomous versions of this theory.

Meeting ID: 941 3609 7274

Seminar 01.27.22 Call

Title: Uniqueness and the K-property of equilibrium states for the geodesic flow on translation surfaces

Speaker: Benjamin Call – Ohio State University

Abstract: In the general setting of CAT(0) spaces, Ricks has provided necessary and sufficient conditions for uniqueness and mixing of the measure of maximal entropy for the geodesic flow. I will discuss recent work establishing uniqueness and the K-property of a class of equilibrium states for the geodesic flow on translation surfaces, a subclass of CAT(0) spaces. This result builds on the orbit-decomposition machinery developed by Climenhaga and Thompson, and is joint work with Dave Constantine, Alena Erchenko, Noelle Sawyer, and Grace Work.

Meeting ID: 941 3609 7274

Recorded Talk: https://osu.zoom.us/rec/share/G56ox7c5B3dudA9ZO303EvX8VW4Fa_z7sCM4S4tdBDsfe4LfolcLJ8p4TGgVgY-X.nY-n-jPdltDSMOCX

Seminar program for Spring 2022

Our seminar continues with a mixture of in person and virtual talks. As usual, we meet on (most) Thursdays at 3.00pm EST unless otherwise noted. In person talks will be in MW154.

For virtual talks, the Zoom link can be obtained from the organizers, Andreas Koutsogianis and Dan Thompson. For most virtual talks, video will be posted afterwards, and will remain viewable on Zoom for 120 days after the talk.

The following is our current schedule; more talks might be announced soon.

Jan 27: Virtual –  Ben Call (OSU)

Feb 10: Virtual – Anthony Quas (University of Victoria, Canada)

Feb 17: Virtual – Richard Sharp (University of Warwick, UK)

Feb 24: Virtual – Ethan Ackelsberg (OSU)

Mar 3: Virtual – John Griesmer (Colorado School of Mines)

Mar 10: Virtual – Anh N. Le (OSU)

Mar 24: In Person – Andreas Koutsogiannis (Aristotle University of Thessaloniki, Greece)

Mar 31: In Person – Dong Chen (PennState)

April 7: Virtual – Konstantinos Tsinas (University of Crete, Greece)

Apr 14: Virtual – Yun Yang (Virginia Tech)

Apr 21: Virtual – Rigo Zelada Cifuentes (University of Maryland)