# Seminar series in Dynamical Systems and Number Theory

The seminars are held every other Friday at 10:15 to 11:15 in room 64119, unless otherwise stated.

## Previous seminars

### 2022

Date: 112522

• Lecturer: Michal Rams (IMPAN)
• Title: Cocycles of circle diffeomorphisms

Abstract: I will present an introduction to the world of cocycles of circle diffeomorphisms. The system is simple: given a finite family $\{f_i\}_{i=1}^n$ of $C^1$ diffeomorphisms of a circle, we consider the dynamical system $F:S^1 \times \Sigma, F(x,\xi) = (f_{\xi_0}(x), \sigma\xi)$ (where $\Sigma = \{1,\ldots,n\}^{\mathbb Z}$). In simple words, we apply the maps $f_i$ in any order we want.

Such systems can be of many types, including hyperbolic ones -- but that is the boring case that I'll skip. I will concentrate on the really interesting case of robustly nonuniformly hyperbolic cocycles (that is, the set of possible Lyapunov exponents contains 0 as its interior point, and this property is preserved under small perturbations). I'll try to explain where that kind of systems comes from, what are they useful for, and what we know about them. The results presented will be from joint works with Lorenzo Diaz and Katrin Gelfert.

Date: 111122

• Lecturer: Charlene Kalle (Leiden University)
• Title: Random intermittent dynamics

Abstract: Intermittent dynamics, where systems irregularly alternate between long periods of different types of dynamical behaviour, has been studied since the work of Pomeau and Manneville in 1980. In random dynamical systems this phenomenon has only been well understood in a few specific cases. A random dynamical system consists of a family of deterministic systems, one of which is chosen to be applied at each time step according to some probabilistic rule. In this talk we will describe the intermittency of some families of random systems with a particular emphasis on how the intermittency of the random system depends on the intermittency of the underlying deterministic systems. This talk is based on joint works with Ale Jan Homburg, Tom Kempton, Valentin Matache, Marks Ruziboev, Masato Tsujii, Evgeny Verbitskiy and Benthen Zeegers.

Date: 102822

• Lecturer: Cagri Sert (Universität Zürich)
• Title: Stationary measures for $\SL_2(\R)$-actions on homogeneous bundles over flag varieties

Abstract: Let $X_{k,d}$ denote the space of rank-$k$ lattices in $\R^d$. Topological and statistical properties of the dynamics of discrete subgroups of $G=\SL_d(\R)$ on $X_{d,d}$ were described in the seminal works of Benoist--Quint. A key step/result in this study is the classification of stationary measures on $X_{d,d}$. Later, Sargent--Shapira initiated the study of dynamics on the spaces $X_{k,d}$. When $k \neq d$, the space $X_{k,d}$ is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure $\mu$ Zariski-dense in a copy of $\SL_2(\R)$ in $G$, we give a classification of $\mu$-stationary measures on $X_{k,d}$ and prove corresponding equidistribution results. In contrast to the results of Benoist--Quint, the type of stationary measures that $\mu$ admits depends strongly on the position of $\SL_2(\R)$ relative to parabolic subgroups of $G$. I will start by reviewing preceding major works and ideas. Joint work with Alexander Gorodnik and Jialun Li.

Date 102022

• Lecturer: Marc Graczyk (Uppsala University)
• Title: On regularity of integrable families

Abstract: In the eighties, M. Misiurewicz posed the problem if there exists non-trivial integrable circle maps. The problem is directly related to a result of M. Herman on mode-locking in families of diffeomorphisms of the circle. Using techniques coming from one-parameter families, we prove that in the class of C^3 integrable circle maps the conjugation must be C^2 independently of the rotation number. This is based on a joint work with Professor Jarek Kwapisz from Montana State University.

Date: 100322

• Lecturer: Min Lee (University of Bristol)
• Title: An extension of converse theorems to the Selberg class

Abstract: The converse theorem for automorphic forms has a long history beginning with the work of Hecke (1936) and a work of Weil (1967): relating the automorphy relations satisfied by classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. The classical converse theorems were reformulated and generalised in the setting of automorphic representations for GL(2) by Jacquet and Langlands (1970). Since then, the converse theorem has been a cornerstone of the theory of automorphic representations.
Venkatesh (2002), in his thesis, gave new proof of the classical converse theorem for modular forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this talk, we extend Venkatesh’s proof of the converse theorem to forms of arbitrary levels and characters with the gamma factors of the Selberg class type.
This is joint work with Andrew R. Booker and Michael Farmer.

Date: 093022

• Lecturer: Chiara Caracciolo (Uppsala University)
• Title: On the applicability of KAM theory to realistic physical problems: effective stability and computer-assisted proofs

Abstract: Many physical problems can be described by nearly-integrable Hamiltonian systems and KAM theorem is one of the main results in this field. Since it ensures the existence of invariant solutions, it can be used to prove stability. However, the strict hypotheses (in particular the smallness of the perturbation parameter) make this instrument most of the time unusable in realistic models. In this seminar, I want to show how the problem of the applicability can be overcome with the use of
computer-assisted techniques. I am going to present a simple example about how to rigorously compute a lower bound for the time of stability around an elliptic equilibrium point (joint work with U. Locatelli). The outcome is the so-called effective stability, that is a stability for times longer than the characteristic time of the system, in the spirit of Nekhoroshev theorem (where the stability time is exponentially long w.r.t. the inverse of the small parameter). Similar estimates can be used to bound the time of stability around KAM tori or lower dimensional elliptic tori.

Date: 092322

Seminar held by Daniel El-Baz (TU Graz) with the title "Primitive rational points on expanding horospheres: effective joint equidistribution".

Abstract: I will report on ongoing work with Min Lee and Andreas Strömbergsson. Using techniques from analytic number theory, spectral theory, geometry of numbers as well as a healthy dose of linear algebra and building on a previous work by Bingrong Huang, Min Lee and myself, we furnish a new proof of a 2016 theorem by Einsiedler, Mozes, Shah and Shapira. That theorem concerns the equidistribution of primitive rational points on certain manifolds and our proof has the added benefit of yielding a rate of convergence. It turns out to have several (perhaps surprising) applications to number theory and combinatorics, which I shall also discuss.

Date: 091622

Mahbub Alam (Uppsala University) holds a seminar with the title "Rational approximation on spheres".

Abstract: Classical diophantine approximation intends to quantify the density of rational numbers in Euclidean spaces. In recent years there have been some interest in intrinsic diophantine approximation, where one considers an algebraic variety with a dense set of rational points and aims to prove analogues of the classical results, such as Dirichlet’s theorem, Khinchin-Groshev theorem, Dani correspondence etc. A general theory in this case still eludes mathematicians. In this talk we will focus on spheres, compare and contrast it with the classical case and mention some recent developments.

Date: 060322

Speaker:  Martin Leguil (Université de Picardie Jules Verne)

Title: U-Gibbs and SRB measures for certain Anosov diffeomorphisms of the 3-torus

Abstract: In this talk, we will consider Anosov diffeomorphisms of the 3-torus T3 which admit a partially hyperbolic splitting T3 = Es ⊕ Ec ⊕ Eu, s.t. Ec is a weak unsta- ble direction. We may consider the 2-dimensional unstable foliation Wcu tangent to Ec ⊕ Eu, but also the 1-dimensional strong unstable foliation Wu tangent to Eu. Such systems have a unique invariant measure whose disintegrations along the leaves of Wcu are absolutely continuous, namely, the SRB measure. We can also consider another class of measures, namely, u-Gibbs measures, whose disintegra- tions along the leaves of Wu are absolutely continuous. It is well-known that the SRB measure is u-Gibbs. Conversely, in a joint work with Sebastien Alvarez, Davi Obata and Bruno Santiago, we show that in a neighborhood of conservative sys- tems, if the strong bundles Es and Eu are not jointly integrable, then any u-Gibbs measure is SRB; in particular, there exists a unique u-Gibbs measure. Our proof borrows techniques from different works on measure rigidity, in particular, a ver- sion of the exponential drift argument present in a paper of Eskin-Lindenstrauss, but also from the work of Brown and Rodriguez-Hertz on random iteration of surface diffeomorphisms.

Date: 050622

Speaker: Jordi-Lluís Figueras (Uppsala University)

Title: Quantitative KAM theory in Hamiltonian Systems

Abstract:  In this talk I want to explain how to prove the existence of quasiperiodic solutions, invariant tori, in hamiltonian systems that are far from integrable. The techniques we use are the parameterization method combined with a taylored version of Quantitative KAM Theory.

I plan to not overwhelm the audience with technicalities. My wish here is that you get the following 3 (concatenated) ideas:

•  Proving the existence of quasiperiodic solutions is a fundamental question.
• There is an efficient way of proving the existence of quasiperiodic solutions for "good" hamiltonians.
•  One needs to combine Analysis, Geometry and Computer for getting qualitative and quantitative estimates of these proven quasiperiodic solutions.

If time permits I will talk about:

• Hamiltonian Systems, Integrability and Action-Angle Coordinates.
•  Brief presentation of the results of classical KAM (Kolmogorov-Arnold-Moser) Theory and one main tool used: normal forms.
• Justification of our point of view: forget about Normal Forms and try to just work the invariant torus you are interested in (parameterization method).
•  Newton scheme in parameterization method.
• Cohomological equations and Diophantine conditions.

PS: I never published anything of the above for Hamiltonian Systems but for Symplectic Maps, so if you try to look for papers on this with my name and don't find them, it is not your fault. However, I can assure you that what I did and what I will talk are epsilon close.

PS2: The field of KAM is huge and it will be impossible for me to make justice to all the great mathematicians that have work on it, so I take the decision of not start listing contributions of others since it will be overwhelming to the audience and ashaming for me since I will miss 100 $\pm \epsilon$ of the references.

This is a joint work with Alex Haro and Alejandro Luque.

Date: 042222

Speaker: Michael Björklund (Chalmers University)

Title:  Higher-order equidistribution – What is it and what can it do for you?

Abstract:  I will survey some recent applications of (quantitative) higher order equidistribution (and higher order mixing) in metric Diophantine approximation. Most of these applications are concerned with counting integer vectors which satisfy multiplicative Diophantine conditions. Some questions and open problems will be posed. Based on joint works with Alexander Gorodnik and Reynold Fregoli.

Date: 040822

Speaker: Michael Bersudsky (Technion-Israel Institute of Technology)

Title: A non-Euclidean "Linnik type" problem and statistics of orthogonal lattices

Abstract: A classical work of Linnik shows that the directions of primitive integer vectors lying on a large sphere are equidistributed. I will discuss a joint work with Uri Shapira in which we consider a variant of Linnik’s problem for a certain type of homogeneous varieties inside the special linear group. The main motivation behind the result is an extension of a previous work of Uri Shapira, Menny Aka and Manfred Einsiedler concerning the statistics of orthogonal lattices of primitive integral vectors.

The main idea in the proof is to recast the problem into a measure classification problem of a sequence of periodic orbits of S-arithemetic groups. I will present our results and sketch some proof ideas.

Date: 032522

Speaker: Philippe Thieullen (University of Bordeaux)

Title: Zero temperature convergence of Gibbs measures for locally a finite potential in a 2-dimensional lattice

Abstract: We consider the space of all configurations over a 2-dimensional lattice and over a finite alphabet. A Gibbs measure is a spatially invariant measure on the space of configurations associated to a given potential at a certain temperature. A potential is locally
finite if it depends on a finite index set. We want to understand the limit of these Gibbs measures when the temperature goes to zero. It may happen that the limit does not exist and several accumulation measures appear. For 1-dimensional lattices and locally finite potentials, the limit is unique. We show that it is not any more true in 2-dimensional lattice. This result is a joint work with S. Barbieri, R. Bissacot, and Gregorio Dalle Vedove.

Date: 031122

Speaker: Davit Karagulyan (KTH)

Title: Exponential Fermi accelerator

Abstract: We consider a resonant Fermi accelerator, which is realized as a square billiard with a periodically oscillating platform. We show that it has an infinite measure set of exponentially escaping orbits. We use normal forms to describe how the energy changes in a period and employ techniques for hyperbolic systems with singularities to show the exponential drift of these normal forms on a divided time-energy phase. This is joint work with Jing Zhou.

Date: 022522

Title: Recurrence

Abstract: Recurrence is a classical topic in ergodic theory and dynamical systems, which goes back to Poincaré's recurrence theorem. I will talk about old, less old, and new results on recurrence. In particular I will talk about how to obtain asymptotic results on the number of times a typical point returns to a shrinking neighbourhood around itself.

Date: 021122

Speaker:  Kiho Park (Korea Institute for Advanced Study)

Title: Limit laws in Dynamical Systems

Abstract: For a class of dynamical systems f : X → X that are considered chaotic, we will look at the validity of various statistical limit laws for additive observables and matrix-valued observables under suitable assumptions (with respect to certain invariant measures on X, including the measure of maximal entropy). Such limit laws include the Law of Large Numbers, Central Limit Theorem, and Large Deviation Principle, and the transfer operator and its spectral properties play key roles in establishing these limit laws. This is joint work with Mark Pi- raino.

Speaker: Kristian Holm (Chalmers University)

Title:  A Central Limit Theorem for Symplectic Lattice Point Counting Functions

Abstract: We study a sequence of counting functions on the space of 2d-dimensional symplectic lattices where d is at least 4. Using a combinatorial device introduced by Björklund and Gorodnik in order to estimate cumulants (alternating sums of moments), we prove that our sequence satisfies a central limit theorem.

Date: 012822

Speaker: Matthew Palmer (Uppsala University)

Title: Vaaler's theorem in number fields

Abstract: Classical Diophantine approximation concerns the approximation of real numbers by rational numbers: in particular, with how well real numbers can be approximated by rational numbers. A long-standing conjecture in Diophantine approximation was the Duffin-Schaeffer conjecture, stated in 1941; this result was recently (2020) proven by Dimitris Koukoulopoulos and James Maynard.

An area which is gaining more attention recently concerns generalisations of the notions of Diophantine approximation to other spaces in which we have a dense subfield with some sort of fractional structure. In this talk, I will discuss Diophantine approximation in number fields, where we use elements of an algebraic number field to simultaneously approximate in all of the number field's embeddings; I will state a version of a result known as Vaaler's theorem (a precursor to Koukoulopoulos and Maynard's theorem), give the rough structure of the proof in the classical case, and discuss how these elements generalise to the number fields case.