## 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.