# 2021-22

##### The seminars are held on Tuesdays at 14:00 in B3.02 (unless stated otherwise)

##### Term 3

- May 3, 2022

Benthen Zeegers (Leiden University)**Title:**Statistical properties of random intermittent systems**Abstract:**For nonuniformly hyperbolic dynamical systems where the evolution map is deterministic there exists a fair amount of literature on statistical properties like decay of correlations and limit laws such as CLT. Much less is known for nonuniformly hyperbolic dynamical systems where the evolution map is random. In this talk we consider a class of random interval maps that are composed of two types of maps: regular and chaotic maps. These random models exhibit dynamical features of both types of maps by alternating between periods of either chaotic behaviour or being in a seemingly steady state, a phenomenon that is referred to as intermittency. We discuss results on the existence of a finite absolutely continuous invariant measure for these random models as well as results on decay of correlations. - May 10, 2022

Joel Moreira (University of Warwick)**Title:**Iterated infinite sumsets in sets with positive density**Abstract:**A set $A\subset{\mathbb{N}}$ has positive upper density if $\limsup_{N\to\infty}|A\cap\{1,\dots,N\}|/N>0$. I will talk about a recent result stating that if $A\subset{\mathbb{N}}$ has positive upper density then for any $k\in{\mathbb{N}}$ there are infinite sets $B_1,\dots,B_k\subset{\mathbb{N}}$ such that $B_1+\cdots+B_k:=\{b_1+\cdots+b_k:b_1\in B_1,\dots,b_k\in B_k\}\subset A$. The case $k=2$, which was originally obtained in joint work with Richter and Robertson in 2018, settled a conjecture of Erdos, but the proof does not generalize to higher $k$. The first step of the proof is to turn the combinatorial problem into a dynamical statement, using a modern version of Furstenberg's correspondence principle. The proof of the resulting dynamical statement involves several classical (and modern) tools from ergodic theory.

Based on joint work with Kra, Richter and Robertson. - May 17, 2022

Jamie Walton (University of Nottingham)**Title:**Aperiodic Order and Linear Repetitivity of Polytopal Cut and Project Sets**Abstract:**A cut and project set is a point pattern in Euclidean space given by*cutting*an irrational slice of a higher dimensional lattice and then*projecting*it to the so-called physical space. The vertices of the Penrose Tilings are famous examples of cut and project sets. These patterns inherit some of the order of the original lattice – sometimes even rotational symmetry – but the irrationality of the slice removes any global translational symmetry, making them examples of*Aperiodic Order*. In this talk I will introduce the study of Aperiodic Order through some of its main examples, constructions and properties of interest. I will then present recent work, joint with Henna Koivusalo, which gives a characterisation of linear repetitivity (a signifier of high structural order) of a large class of cut and project sets in terms of Diophantine approximation properties of the original cut and project scheme. - May 24, 2022

François Ledrappier (Paris)**Title:**Exact dimension of Furstenberg measures**Abstract:**We consider a random walk on a group of matrices with finite support and all exponents distinct. Then, we show that the distribution of the Oseledets unstable flag is exact-dimensional (both these notions will be defined). This is a joint work with Pablo Lessa (Montevideo). - May 31, 2022

David Parmenter (Warwick)**Title:**Gibbs measures for hyperbolic attractors defined by densities**Abstract:**In this talk I will describe a new construction for Gibbs measures for hyperbolic attractors generalizing the original construction of Sinai, Bowen and Ruelle of SRB measures. The classical construction of the SRB measure is based on pushing forward the normalized volume on a piece of unstable manifold. By modifying the density at each step appropriately we show that the resulting measure is a prescribed Gibbs measure. To prove this, we require a property on the growth rate of unstable manifolds. This is joint work with my supervisor Mark Pollicott. - June 14, 2022

Caroline Wormell (Paris)**Title:**Diffusion maps and Sinkhorn balancing**Abstract:**TBA

##### Term 2

- January 25, 2022

Victor Kleptsyn (University of Rennes) (**online)**

**Title:**From the percolation theory to Fuchsian equations and Riemann-Hilbert problem**Abstract:**Consider the critical percolation problem on the hexagonal lattice: each of (tiny) hexagons is independently declared « open » or « closed » with probability (1/2) — by fair coin tossing. Assume that on the boundary of a simply connected domain four points A,B,C,D are marked. Then either there exists an « open » path, joining AB and CD, or there is a « closed » path, joining AD and BC (one can recall the famous « Hex » game here). Cardy’s formula, rigorously proved by S. Smirnov, gives an explicit value of the limit of such percolation probability, when the same smooth domain is put onto lattices with smaller and smaller mesh. Though, a next natural question is: what if more than four points are marked? And thus that there are more possible configurations of open/closed paths joining the arcs? In our joint work with M. Khristoforov we obtain the answer as an explicit integral for the case of six marked points on the boundary, passing through Fuchsian differential equations, Riemann surfaces, and Riemann-Hilbert problem. We also obtain a generalisation of this answer to the case when one of the marked points is inside the domain (and not on the boundary). - February 1, 2022

Alexey Korepanov

**Title:**Probabilistic properties of the measure of maximal entropy for Sinai billiards.**Abstract:**We know much about the physical measure for dispersing billiards, results like exponential mixing and Central Limit Theorem (CLT) are basic and standard, with a variety of different proofs. Much less is known about the measure of maximal entropy (MME): we have only very recently learned that it exists and is unique (Baladi-Demers 2020). I'll talk about (surprizing!) mixing rates and CLT-type theorems for the MME. This is a work in progress, joint with M.Demers. - February 8, 2022

Alexey Okunev

Solutions passing through a resonance exhibit a small quasi-random jump, this is called scattering on a resonance. Some solutions passing through a resonance can also be captured into resonance, remaining near the resonance for a long time, but this only happens for small measure of initial data. Far from separatrices there are only finitely many resonances such that capture is possible, however, such resonances can accumulate on separatrices. We estimate how the amplitude of scattering on resonances and the measure of initial data captured into resonances decrease for resonances near separatrices. We also show that the infinite number of resonances near separatrices such that capture is possible can be split into a finite number of series such that dynamics near all resonances in the same series is close to each other and can be written in terms of the Melnikov function. We obtain realistic estimates on the accuracy of averaging method and on the measure of initial data badly described by averaging method (such as initial data captured into resonances) for solutions crossing separatrices. We also prove formulas for probability of capture into different domains after separatrix crossing. Our results can also be applied to perturbations of generic two-frequency integrable systems near separatrices, as they can be reduced to periodic perturbations of one-frequency systems.**Title:**Averaging and passage through resonances in two-frequency systems near separatrices**Abstract:**There are two major obstacles to applying the averaging method, resonances and separatrices. We study averaging method for the simplest situation where both these obstacles are present at the same time, time-periodic perturbations of one-frequency Hamiltonian systems with separatrices. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (so slow-fast Hamiltonian systems with two and a half degrees of freedom are included in our class). - February 15, 2022

Carlangelo Liverani

**Title:**Fast-slow systems and partially hyperbolic dynamics**Abstract:**I will describe some motivations and some recent results pertaining partially hyperbolic dynamics of the fast-slow type. In particular, I will concentrate on the study of their quantitative statistical properties. - February 22, 2022

Valerie Berthe (**online)**

**Title:**On the dynamics of Ostrowski’s numeration**Abstract:**Ostrowski's numeration allows the representation of integers and real numbers with respect to the convergents in the continued fraction expansion of a given real real number. This numeration has many applications from inhomogeneous Diophantine approximation to word combinatorics. We consider here this numeration according to a dynamical and probabilistic approach. We establish the existence of normal laws for the statistical properties of digits and we discuss estimates on the Hausdorff dimension of sets of points whose Ostrowski expansions have restricted digits. These results are deduced from the spectral study of the associated transfer operator. These results have been obtained in collaboration with Jungwon Lee. - March 1, 2022

Jungwon Lee

**Title:**Another view of Ferrero--Washington Theorem**Abstract:**Ferrero and Washington observed the joint equidistribution of digits of p-adic integers, which describes the arithmetic invariants of a tower of number fields. We reprove the main equidistribution instance in their proof of the vanishing of cyclotomic Iwasawa \mu-invariant, based on the ergodicity of a certain p-adic skew-extension dynamical system that can be identified with Bernoulli shift (joint with Bharathwaj Palvannan). - March 8, 2022

Michael Magee

**Title:**The maximal spectral gap of a hyperbolic surface**Abstract:**A hyperbolic surface is a surface with metric of constant curvature -1. The spectral gap between the first two eigenvalues of the Laplacian on a closed hyperbolic surface contains a good deal of information about the surface, including its connectivity, dynamical properties of its geodesic flow, and error terms in geodesic counting problems. For arithmetic hyperbolic surfaces the spectral gap is also the subject of one of the biggest open problems in automorphic forms: Selberg’s eigenvalue conjecture. It was an open problem from the 1970s whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and spectral gap tending to 1/4. (The value 1/4 here is the asymptotically optimal one.) Recently we proved that this is indeed possible. I’ll discuss the very interesting background of this problem in detail as well as some ideas of the proof. This is joint work with Will Hide. - March 15, 2022

Viveka Erlandsson

**Title:**Non-orientable surfaces and mapping class group orbit closures**Abstract:**The space of measured laminations on a surface—which can be viewed as the closure of the set of weighted simple closed geodesics—is a fundamental tool when studying hyperbolic surfaces. When the surface is orientable, this space and the action of the mapping class group on it is by now well-understood. However, the situation is pretty different in the non-orientable case and I will talk about some recent results in this direction. In particularly we will describe the orbit closures of the mapping class group action on the space of measured laminations, which in the orientable setting is a result of Lindenstrauss-Mirzakhani. This is joint work with Gendulphe, Pasquinelli, and Souto.

##### 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 -term arithmetic progressions. In the case of -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 actions (where the average is taken over a Flner 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 . 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 structure theorem that is new even in the case of -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:**Real-analytic FBI transform and Anosov flows.**Abstract:**Anosov flows form an extensively studied class of chaotic dynamical systems. In this talk, I will explain how PDE techniques developed by Helffer and Sjöstrand in the 80s-90s can be used to study the statistical properties of real-analytic Anosov flows, and the complex-analytic properties of associated zeta functions. This is a joint work with Yannick Guedes Bonthonneau.