Seminar 09.08.22 Dilsavor – Virtually

Title: Statistics of periodic points and a positive proportion Livsic theorem

Speaker:  Caleb Dilsavor – Ohio State University

Abstract: The connection between the Ruelle-Perron-Frobenius operator and the statistics of a Hölder observable g with respect to an equilibrium state has a rich history, tracing back to an exercise in Ruelle’s book. A somewhat lesser known, but related, statistical theorem studied first by Lalley, and later by Sharp using the RPF operator, states that the periods of g grow approximately linearly with respect to length, with square rootoscillations chosen according to a normal distribution whose variance is equal to the (dynamical) variance of g. This result is known for aperiodic shifts of finite type, but surprisingly it is still notknown in full generality for their Hölder suspensions. I will describe a tentative result that fills in this gap, along with joint work with James Marshall Reber which uses this result to deduce a strengthening of Livsic’s theorem not previously considered: if a positive-upper-density proportion of the periods of g are zero, then g is in fact a coboundary.

Zoom link: https://osu.zoom.us/j/91943812487?pwd=K1lhTU02UTdMelBFTzhDdXRNcm80QT09

Meeting ID: 919 4381 2487

Password: Mixing

Link of recorded talk: https://osu.zoom.us/rec/share/ZiOZu_LJaCIMt0oBPGmFrenNVehsf2ZxaM8Myw1DiBNJ9cyVzrdFZHaqTIOoP3vO.ap18_rehC7ecOOgQ

Seminar 09.01.22 Bersudsky – In person

Title: On the image in the torus of sparse points on expanding analytic curves

Speaker: Michael Bersudsky (OSU)

Abstract: It is known that the projection to the 2-torus of the normalised parameter measure on a circle of radius R in the plane becomes uniformly distributed as R grows to infinity. I will discuss the following natural discrete analogue for this problem. Starting from an angle and a sequence of radii {Rn} which diverges to infinity, I will consider the projection to the 2-torus of the n’th roots of unity rotated by this angle and dilated by a factor of Rn. The interesting regime in this problem is when Rn is much larger than n so that the dilated roots of unity appear sparsely on the dilated circle.

Seminar program for Fall 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 co-organizer, and responsible for the virtual component of the seminar, Andreas Koutsogianis.

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

August 25: In peron – Dmitri Scheglov

September 1: In person – Michael Bersudsky

September 8: Virtually – Caleb Dilsavor

September 15: In person – Andrey Gogolev

September 22: In person – Tomasz Downarowicz

September 29: No talk

October 6: Virtually – Yunied Puig de Dios

October 13: No seminar, Fall break

October 20: Virtually – Jiajie Zheng

October 27: In person –  Andreas Koutsogiannis

November 3: In person – Michał Misiurewicz

November 10: Virtually – Borys Kuca

November 17: TBA

November 24: No seminar, Thanksgiving break

December 1: Virtually – Mariusz Mirek

December 8: In person – Martin Leguil

Seminar 04.21.22 Zelada Cifuentes

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.

Zoom link: https://osu.zoom.us/j/93885989739?pwd=bUNWdjgzMS93NHRUcmVZRkljTDBHZz09

Meeting ID: 938 8598 9739

Password: Mixing

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.

Zoom link: https://osu.zoom.us/j/93885989739?pwd=bUNWdjgzMS93NHRUcmVZRkljTDBHZz09

Meeting ID: 938 8598 9739

Password: Mixing

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.

Zoom link: https://osu.zoom.us/j/93885989739?pwd=bUNWdjgzMS93NHRUcmVZRkljTDBHZz09

Meeting ID: 938 8598 9739

Password: Mixing

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 03.10.22 Le

Title: Interpolation sets for nilsequences

Speaker: Anh N. Le – Ohio State University

Abstract: Interpolation sets are classical objects in harmonic analysis whichhave a natural generalization to ergodic theory regardingnilsequences. A set $E$ of natural numbers is an interpolation set fornilsequences if every bounded function on E can be extended to anilsequence on $\mathbb{N}$. By a result of Strzelecki, lacunary setsare interpolation sets for nilsequences. In this talk, I show that nosub-lacunary sets are interpolation sets for nilsequences and theclass of interpolation sets for nilsequences is closed under unionwith finite sets.

Zoom link: https://osu.zoom.us/j/93885989739?pwd=bUNWdjgzMS93NHRUcmVZRkljTDBHZz09

Meeting ID: 938 8598 9739

Password: Mixing

Recorded Talk: https://osu.zoom.us/rec/share/nIo2Tnfv7PMRIP3U_EG7FWw7N1YhFRL4BeJa_gqE0voCXN3enu_jnHuH-tW1H5q2.84ac0THimUVpKQfW

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 $0<\mu(A)<1$.

Zoom link: https://osu.zoom.us/j/93885989739?pwd=bUNWdjgzMS93NHRUcmVZRkljTDBHZz09

Meeting ID: 938 8598 9739

Password: Mixing

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.)

Zoom link: https://osu.zoom.us/j/94136097274

Meeting ID: 941 3609 7274

Password: Mixing

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.)

Zoom link: https://osu.zoom.us/j/94136097274

Meeting ID: 941 3609 7274

Password: Mixing

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.

Zoom link: https://osu.zoom.us/j/94136097274

Meeting ID: 941 3609 7274

Password: Mixing

Recorded Talk: https://osu.zoom.us/rec/share/gPPVM_NKnpK4WSPjcFje6GgZEKpqDKVYr_m6Fim7oC03n9uwA4ktmAd7yuO6BjsI.308Dh7WhXQPgI7jm

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.

Zoom link: https://osu.zoom.us/j/94136097274

Meeting ID: 941 3609 7274

Password: Mixing

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)

Seminar 12.09.21 Tiozzo

Title: Hitting measures for random walks on cocompact Fuchsian groups

Speaker: Giulio Tiozzo – University of Toronto

Abstract: A recurring question in the theory of random walks on hyperbolic spaces asks whether the hitting (harmonic) measures can coincide with measures of geometric origin, such as the Lebesgue measure. This is also related to the inequality between entropy and drift.  For finitely-supported random walks on cocompact Fuchsian groups with symmetric fundamental domain, we prove that the hitting measure is singular with respect to Lebesgue measure; moreover, its Hausdorff dimension is strictly less than 1. Along the way, we prove a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.  Joint with P. Kosenko.

Zoom link: https://osu.zoom.us/j/91638927725

Meeting ID: 916 3892 7725

Password: Mixing

Recorded Talk: https://osu.zoom.us/rec/share/dcqrRfiTn8n_stVOUUooY9q8cCpzFwreq0L3ih3lUrgoR-rdbQefsdxYwfUJu-Y.p0RYOqWCdsYDR1iA

Seminar 10.28.21 Sun

Title: Joint ergodicity conjecture for systems with commuting transformations

Speaker: Wenbo Sun – Virginia Tech

Abstract: It is well know by the Mean Ergodic Theorem that for any measure preserving system $(X,\mathcal{B},\mu,T)$ and $L^{\infty}$ function f, the time average of $T^{n}f$ converges to the integral of f if and only if T is ergodic. It is a natural question to ask when the average of products of polynomial iterates of  $L^{\infty}$ functions (known as multiple ergodic averages) converges to the product of the integrals of the functions. This question is called the Joint Ergodicity Problem. In this talk, I will introduce some recent advances in this problem. This talk is based on joint works with Sebasti\’an Donoso, Andreu Ferr\’e Moragues and Andreas Koutsogiannis.

Zoom link: https://osu.zoom.us/j/91638927725

Meeting ID: 916 3892 7725

Password: Mixing

Recorded Talk: https://osu.zoom.us/rec/play/dk–MjWHHK8ex0sdF0ILXOXT338U71LQ1awWnexrtUuyYiEtC-noT76YCLpX4bnCvTAT2mU-xxTPv9d1.OUHqT_H7CAgbLfFM?continueMode=true

Seminar 10.21.21 Maass

Title: Spectral analysis of topological finite rank systems

Speaker: Alejandro Maass – University of Chile

Abstract: Finite topological rank Cantor minimal systems represent a broad class of sub shifts of zero entropy or odometers [Downarowiz-Maass], it contains well studied systems like substitution sub shifts or linearly recurrent systems. In this talk we will present the study of measure-theoretical and topological eigenvalues for such class of systems, given formulas characterizing them. This work is motivated by the seminal work of Bernard Host where it is proved that measure-theoretical and topological eigenvalues of substitutions systems coincide. This is a joint work with Fabien Durand and Alexander Frank.

Zoom link: https://osu.zoom.us/j/91638927725

Meeting ID: 916 3892 7725

Password: Mixing

Recorded Talk: https://osu.zoom.us/rec/share/6x1GYGHuPkjR4KFdpi4usJMP1ert17FX_RHSF_MVaxIkD6PrLXjLO83fXN_-CR8u.0rZGFYbboMqO4Ps7

Seminar 10.08.21 Tanaka

Title: The Manhattan curve and rough similarity rigidity

Speaker: Ryokichi Tanaka – Kyoto University

Abstract: For every non-elementary hyperbolic group, we consider the Manhattan curve, which was originally introduced by M. Burger (1993),
associated with any pair of (say) word metrics. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, that is, they are within bounded distance after multiplying by a positive constant. I would like to explain how it is related to the central limit theorem for uniform counting measures on spheres, to ergodic theory of topological flows built on general hyperbolic groups, and to the multifractal structure of Patterson-Sullivan measures. Furthermore, I will present some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for a pair of word metrics by using automatic structures of the group.
Joint work with Stephen Cantrell (University of Chicago).

Zoom link: https://osu.zoom.us/j/91638927725

Meeting ID: 916 3892 7725

Password: Mixing

Recorded Talk: https://osu.zoom.us/rec/share/LPAREOkf7DJ5zT6tKMPY1dZP_Vs824-zAEFnAJ7CTzXJKS-pMk_Bx8o2eD6QXg-y.lFp9Na68FCmloePw