# 2021-22

##### Term 1
• October 5, 2021
Julia Slipantschuk (University of Warwick)

Title: Toral Anosov diffemorphisms with computable Pollicott-Ruelle resonances.
Abstract: In the one-dimensional setting, Blaschke products give rise to analytic expanding circle maps for which the entire spectrum of the associated (compact) transfer operator is computable. Inspired by these examples, in this talk I will present a class of Anosov diffeomorphisms on the torus, constructed using Blaschke factors, for which the spectrum of transfer operators defined on suitable anisotropic Hilbert spaces can be determined explicitly and related to the dynamical features of the underlying maps.

• October 12, 2021
Richard Sharp (University of Warwick)

Title: Helicity and linking for 3-dimensional Anosov flows.
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.)

• October 19, 2021
Nicolo Paviato (University of Warwick)

Title: Rates of convergence for a functional CLT.
Abstract: It is well known that chaotic systems give rise to interesting statistical properties, such as the strong law of large numbers and the central limit theorem. In this talk we will glimpse into the world of smooth ergodic theory, stating similar limit theorems for random variables which are generated by a dynamical system. Moreover, we will see a new result on the speed of convergence to Brownian motion for nonuniformly expanding semiflows.

• October 26, 2021
Selim Ghazouani

Title: Rigidity for smooth flows on surfaces
Abstract:I will consider the general problem of the regularity of conjugacies between smooth dynamical systems. After recalling the history of the problem and reviewing the classical cases of expanding maps of the circle and circle diffeomorphisms, I will report on recent work (joint with Corinna Ulcigrai) about flows and foliations on closed surfaces. Our approach is based on the study of a renormalisation operator acting upon a Banach space of dynamical systems and yields a proof of a conjecture by Marmi-Moussa-Yoccoz on the rigidity of most generalised interval exchange transformations in genus 2.

• November 2, 2021
Caroline Wormell (Sorbonne Université, Paris)

Title: Linear response for the Lozi map and mixing of SRB measure cross-sections.
Abstract: Dynamical systems of sufficiently high dimension are expected, regardless of their hyperbolicity properties, to have a linear response, but this property does not follow from the existing theory. Ruelle conjectured that linear response arises from generic average dynamics of singularities in the SRB measure: we pursue this idea in the tractable setting of 2D piecewise hyperbolic maps, whose 1D equivalents fail to have linear response.
We show, rigorously, that linear response formally obtains for these maps if the SRB measure has a strong mixing property: that its conditional measure on the singularity set–a measure of dimension strictly less than one–converges exponentially quickly back to the SRB measure under the action of the map. We conjecture that this property holds generically, presenting strong numerical evidence and some parallels with some recent one-dimensional results. This unexpected phenomenon may furnish a general mechanism for linear response in higher dimensions, as well as being of its own interest.

• November 9, 2021 (Note that there are two seminars!)
• 2pm
Oleg Karpenkov (University of Liverpool, UK)

Title: Geometrisation of Markov Numbers
Abstract: In this talk we link discrete Markov spectrum to geometry of continued fractions. As a result of that we get a natural generalisation of classical Markov tree which leads to an efficient computation of Markov minima for all elements in generalized Markov trees.

• 5pm, online only
Seonhee Lim (Seoul National University and Univ. of California at San Diego)

Title: Inhomogeneous Diophantine approximation and Hausdorff dimension, a quantitative result
Abstract: We consider the Diophantine approximation of a target vector $b$ by integral multiples $Aq$ of an $m$ by $n$ matrix $A$ modulo integral vectors. By an inhomogeneous version of Khintchine-Groshev theorem, the liminf of |q|^n <Aq-b>|^m is zero, where $<.>$ denotes the norm from the nearest integral vector. We call a pair $(A,b)$ epsilon-badly approximable if the liminf above is bounded below by epsilon. We show that the Hausdorff dimension of the set of epsilon-badly approximable pair is bounded above by $mn+ m - c\epsilon^M$ for some $c, M$ not depending on $\epsilon$ nor $b$. We also show a quantitative upper bound of the Hausdorff dimension of the set of matrices $A$ above for a fixed $b$. This is joint work with Wooyeon Kim and Taehyeong Kim.

• November 16, 2021 (online)
Or Shalom (HUJI, Jerusalem)

Title: Structure theory for the Gowers-Host-Kra seminorms in countable abelian groups and Khintchine type recurrence.
Abstract: Furstenberg's famous proof of Szemeredi's theorem leads to a natural question about the convergence and limit of some multiple ergodic averages related to $k$-term arithmetic progressions. In the case of $\mathbb{Z}$-actions these averages were studied by Host-Kra and independently by Ziegler. They show that the limiting behavior of such multiple ergodic average is determined on a certain factor that can be given the structure of an inverse limit of nilsystems (i.e. rotations on nilmanifolds). This structure result can be generalized to $\mathbb{Z}^d$ actions (where the average is taken over a F$\o$lner sequence), but the non-finitely generated case is still open. The only progress prior to our work is due to Bergelson Tao and Ziegler, who studied actions of the infinite direct sum of $\mathbb{Z}/p\mathbb{Z}$. In this talk we will discuss generalizations of these results to arbitrary countable abelian groups. In particular, we will define a generalized version of nilsystems and deduce an $\textit{exact}$ structure theorem that is new even in the case of $\mathbb{Z}$-actions. Moreover, we will show that the structure theorem for non-finitely generated groups is related to a multiplicative version of the famous Bergelson-Host-Kra multiple Khintchine-type recurrence.

• November 23, 2021
Huadi Qu (South University of Science and Technology, China)

Title: A C^∞ Closing lemma on tori
Abstract: Closing lemma is one of the fundamental problems in dynamical systems. In this talk, we introduce some recent progress on this problem. Recently a C^∞ closing lemmas for Hamiltonian diffeomorphisms of closed surfaces is proved by some application of Embedded Contact Homology(ECH). We reformulated their techniques into a more general perturbation lemma for area-preserving diffeomorphism and proved a C^∞ closing lemma for area-preserving diffeomorphisms on a torus T^2 that is isotopic to identity. i.e., we show that the set of periodic orbits is dense for a generic diffeo- morphism isotopic to identity area-preserving diffeomorphism on T2. The main tool is the flux vector of area-preserving diffeomorphisms which is, different from Hamiltonian cases, non-zero in general.

• November 30, 2021
Peter Varju (University of Cambridge, UK)

Title: Exponential mixing of commuting nilmanifold automorphisms
Abstract: Gorodnik and Spatzier proved that Z^l actions of ergodic nilmanifold automorphisms are mixing of all orders. They proved an exponential rate of mixing of order at most 3 for Holder test functions. I will discuss a result extending the exponential rate to mixing of all orders. This problem is intimately related to a problem in Diophantine approximation, which is solved using Schmidt's subspace theorem. Work in progress, joint with Timothée Bénard.

• December 7, 2021 (online)
Malo Jézéquel (MIT, Boston)

Title: (tbc).
Abstract: (tbc)