Ergodic theory and dynamical systems

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Herbstsemester 2023

Datum / Zeit Referent:in Titel Ort
25. September 2023
15:00-16:00
Prof. Dr. Jayadev Athreya
University of Washington
Details

Ergodic theory and dynamical systems seminar

Titel Counting Pairs of Saddle Connections
Referent:in, Affiliation Prof. Dr. Jayadev Athreya, University of Washington
Datum, Zeit 25. September 2023, 15:00-16:00
Ort Y27 H 25
Abstract Motivated by the comparison of sets of holonomies of saddle connections on translation surfaces to Poisson point processes, we show that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has asymptotic quadratic growth. The proof techniques combine ergodic methods for counting saddle connections with the fact that the Siegel--Veech transform is in L2. The talk will not assume prior knowledge of translation surfaces, and we will attempt to keep it broadly accessible. This is joint work with Sam Fairchild and Howard Masur.
Counting Pairs of Saddle Connectionsread_more
Y27 H 25
2. Oktober 2023
15:00-16:00
Dr. Gabriela Estevez
Universidade Federal Fluminense
Details

Ergodic theory and dynamical systems seminar

Titel Some recent results on multicritical circle maps
Referent:in, Affiliation Dr. Gabriela Estevez, Universidade Federal Fluminense
Datum, Zeit 2. Oktober 2023, 15:00-16:00
Ort Y27 H 25
Abstract In 1984 Yoccoz proved that any two C^3 orientation-preserving circle homeomorphisms, with the same quantity of non-flat critical points and the same irrational rotation number, are topologically conjugated. For maps with only one critical point, it has been shown that the conjugacy is, in fact, a C^1-diffeomorphism. Moreover, in a total Lebesgue measure set of irrational rotation numbers the conjugacy can be improved to be C^{1+\alpha}. In this talk, we will discuss some recent results concerning the smoothness of the conjugacy for maps with more than one critical point, and how these results are obtained using renormalization.
Some recent results on multicritical circle mapsread_more
Y27 H 25
9. Oktober 2023
15:00-16:00
Nihar Gargava
EPFL
Details

Ergodic theory and dynamical systems seminar

Titel Random Arithmetic Lattices as Sphere Packings
Referent:in, Affiliation Nihar Gargava, EPFL
Datum, Zeit 9. Oktober 2023, 15:00-16:00
Ort Y27 H 25
Abstract In 1945, Siegel showed that the expected value of the lattice-sums of a function over all the lattices of unit covolume in an n-dimensional real vector space is equal to the integral of the function. In 2012, Venkatesh restricted the lattice-sum function to a collection of lattices that had a cyclic group of symmetries and proved a similar mean value theorem. Using this approach, new lower bounds on the most optimal sphere packing density in n dimensions were established for infinitely many n. In the talk, we will outline some analogues of Siegel's mean value theorem over lattices. This approach has modestly improved some of the best known lattice packing bounds in many dimensions. We will speak of some variations and related ideas. (Joint work with V. Serban, M. Viazovska)
Random Arithmetic Lattices as Sphere Packingsread_more
Y27 H 25
16. Oktober 2023
15:00-16:00
Christopher Enno Hardy Lutsko
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel Effective Counting of Sphere Packings
Referent:in, Affiliation Christopher Enno Hardy Lutsko, Universität Zürich
Datum, Zeit 16. Oktober 2023, 15:00-16:00
Ort Y27 H 25
Abstract In this talk I'll present a method, using abstract spectral theory, for counting points in a group orbit. That is, given a discrete group of isometries acting on the (n+1) dimensional upper half-space, our method allows one to count the number of points in an orbit a distance T from an observer. In particular, the method does not rely on an explicit pre-trace formula, and thus can be used in infinite volume. Further, I will present how to extend the method to obtain effective error rates when counting Apollonian (and more generally Kleinian) sphere packings. This is based on joint work with Alex Kontorovich, and joint work with Dubi Kelmer and Alex Kontorovich.
Effective Counting of Sphere Packingsread_more
Y27 H 25
23. Oktober 2023
15:00-16:00
Dr. Daniele Galli
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel A cohomological approach to Ruelle-Pollicott resonances of Anosov diffeomorphisms
Referent:in, Affiliation Dr. Daniele Galli, Universität Zürich
Datum, Zeit 23. Oktober 2023, 15:00-16:00
Ort Y27 H 25
Abstract Given a transitive Anosov diffeomorphism on a closed connected manifold, it is known that, for enough smooth observables, the system is mixing w.r.t. the measure of maximal entropy. Accordingly, it makes sense to investigate the speed of decay of correlations and to look for the so-called Ruelle-Pollicott resonances, in order to determine an asymptotic for the correlation limit. In this talk I will describe some recent ideas to tackle these questions. In particular, I will point out some surprising connections between the spectrum of a particular transfer operator acting on suitable Anisotropic Banach spaces of currents and the spectrum of the action induced by the Anosov map on the De Rham cohomology. As a corollary, we obtain an upper bound for the speed of mixing. This talk is based on my PhD thesis that I defended last June at the University of Bologna.
A cohomological approach to Ruelle-Pollicott resonances of Anosov diffeomorphismsread_more
Y27 H 25
30. Oktober 2023
15:00-16:00
Dr. Dominik Kwietniak
Jagiellonian University
Details

Ergodic theory and dynamical systems seminar

Titel An anti-classification theorem for the topological conjugacy of Cantor minimal systems
Referent:in, Affiliation Dr. Dominik Kwietniak, Jagiellonian University
Datum, Zeit 30. Oktober 2023, 15:00-16:00
Ort Y27 H 25
Abstract

The isomorphism problem in dynamics  dates back to a question of von Neumann from 1932: Is it possible to classify (in some reasonable sense) the ergodic measure-preserving diffeomorphisms of a compact manifold up to isomorphism? We want to study a similar problem: Let C be the Cantor set and let Min(C) stand for the space of all minimal homeomorphisms of the Cantor set. Recall that a homeomorphism f is in Min(C) if every orbit of f is dense in C.  We say that f and g in Min(C) are topologically conjugate if there is a Cantor set homeomorphism h such that f o h = h o g. We prove an anti-classification result showing that even for very liberal interpretations of what a "reasonable'' classification scheme might be, a classification of minimal Cantor set homeomorphism up to topological conjugacy is impossible. We see it as a consequence of the following: we prove that the topological conjugacy relation of Cantor minimal systems TopConj treated as a subset of Min(C)xMin(C) is complete analytic. In particular, TopConj is a non-Borel subset of Min(C)xMin(C). Roughly speaking, it is impossible to tell if two minimal Cantor set homeomorphisms are topologically conjugate using only a countable amount of information and computation.

Our result is proved by applying a Foreman-Rudolph-Weiss-type construction used for an anti-classification theorem for ergodic automorphisms of the Lebesgue space. We find a continuous map F from the space of all trees over non-negative integers with arbitrarily long branches into the class of minimal homeomorphisms of the Cantor set. Furthermore, F is a reduction, which means that a tree T is ill-founded if and only if F(T) is topologically conjugate to its inverse. Since the set of ill-founded trees is a well-known example of a complete analytic set, it is impossible to classify which minimal Cantor set homeomorphisms are  topologically conjugate to their inverses.

This is joint work with Konrad Deka, Felipe García-Ramos, Kosma Kasprzak, Philipp Kunde (all from the Jagiellonian University in Kraków).

An anti-classification theorem for the topological conjugacy of Cantor minimal systemsread_more
Y27 H 25
6. November 2023
15:00-16:00
Yuval Yifrach
Technion
Details

Ergodic theory and dynamical systems seminar

Titel Non Algebraic Versions of the p-Adic Littlewood Conjecture and of Duke's Theorem for Subcollections
Referent:in, Affiliation Yuval Yifrach, Technion
Datum, Zeit 6. November 2023, 15:00-16:00
Ort Y27 H 25
Abstract

Various algebraic phenomenons in homogeneous dynamics have non algebraic counterparts. For example, the equidistribution of Hecke neighbors can be seen as a non-algebraic counterpart of Duke's Theorem. In this talk, we consider non-algebraic counterparts of the p-Adic Littlewood Conjecture and of Duke's Theorem for subcollections.
One of the non-algebraic counterparts of the p-Adic Littlewood Conjecture involves unboundedness of the A-orbits of arbitrary choices of p-Hecke neighbors of a lattice as p goes to infinity along the primes.
We prove, using expanders, a bootstrap argument and the equidistribution of Hecke neighbors, that the set of exceptions for this conjecture has Hausdorff dimension strictly smaller than 1 in [0,1] (where we assign lattices to points in [0,1]). Moreover, we discuss evidence for the conjecture in some cases using GRH.

This talk is based on a joint ongoing work with Erez Nesharim from the Technion.

Non Algebraic Versions of the p-Adic Littlewood Conjecture and of Duke's Theorem for Subcollectionsread_more
Y27 H 25
20. November 2023
15:00-16:00
Prof. Dr. Françoise Pène
Université de Brest
Details

Ergodic theory and dynamical systems seminar

Titel Probabilistic limit theorems for the periodic Lorentz gas and for the geodesic flow on a Z^d-cover of a negatively curved compact surface
Referent:in, Affiliation Prof. Dr. Françoise Pène, Université de Brest
Datum, Zeit 20. November 2023, 15:00-16:00
Ort Y27 H 25
Abstract The two models mentioned in the title are natural examples of dynamical systems preserving an infinite measure. Because of their periodicity, they can be represented by a Z^d-extension over a chaotic probability preserving dynamical system (resp. Sinai billiard, geodesic flow on a compact surface). Thus, their ergodic properties are closely related to those of the underlying probability preserving chaotic system (studied namely by Sinai, Bunimovich, Chernov, Young, Ratner, Pesin, etc.) and in particular with the local limit theorem established by resp. Domokos Szász and Tamás Varjú and Yves Guivarc'h and J. Hardy. When the horizon is finite, the free flight is bounded, and powerful tools can be used to establish many strong results, such as quantitative recurrence results, expansions in mixing, limit theorems for Birkhoff sums, for pin-ball, for non-stationary Birkhoff sums and for solutions of perturbed differential equations (results in collaboration with Benoît Saussol, with Dima Dolgopyat and Péter Nándori, with Damien Thomine, results by Nasab Yassine and Maxence Phalempin). Finally we will also state results in the more difficult case of the Lorentz gas in infinite horizon (results in collaboration with Dalia Terhesiu, and also with Ian Melbourne).
Probabilistic limit theorems for the periodic Lorentz gas and for the geodesic flow on a Z^d-cover of a negatively curved compact surfaceread_more
Y27 H 25
27. November 2023
15:00-16:00
Prof. Dr. Sophie Grivaux
CNRS, Lille
Details

Ergodic theory and dynamical systems seminar

Titel Some new results regarding convergence under xqxp-invariant measures on the circle
Referent:in, Affiliation Prof. Dr. Sophie Grivaux, CNRS, Lille
Datum, Zeit 27. November 2023, 15:00-16:00
Ort Y27 H 25
Abstract For each integer n \geq 1, denote by T_n the map x \mapsto nx mod 1 from the circle group T=R/Z into itself. Let p,q \geq 2 be two multiplicatively independent integers. Using Baire Category arguments, we will show that generically, a continuous T_p-invariant probability measure \mu on T is such that (T_q^{n}\mu)_{n\geq 0} does not weak-star converge to the Lebesgue measure on T. This disproves Conjecture (C3) from a 1988 paper by R. Lyons, which is a stronger version of Furstenberg's rigidity conjecture on xp and xq invariant measures on T, and complements previous results by Johnson and Rudolph. If time permits, I will also present some generalizations of this result concerning convergence to the Lebesgue measure of sequences of the form (T_{c_{n}}\mu)_{n\geq 0}, as well as some extensions to the multidimensional setting. The talk will be based on a joint work with Catalin Badea (Lille).
Some new results regarding convergence under xqxp-invariant measures on the circleread_more
Y27 H 25
4. Dezember 2023
15:00-16:00
Prof. Dr. Samuel Tapie
Université de Lorraine
Details

Ergodic theory and dynamical systems seminar

Titel Entropy at infinity and applications in negative curvature
Referent:in, Affiliation Prof. Dr. Samuel Tapie, Université de Lorraine
Datum, Zeit 4. Dezember 2023, 15:00-16:00
Ort Y27 H 25
Abstract In this talk, I will focus on some relationships between topology, analysis and geometry which are provided by studying the geodesic flow on non-compact manifolds with negative curvature. I will first recall some classical notions of entropy and then present entropies at infinity and entropy gap property. I will sketch various applications to counting closed orbits, Laplace spectrum and mixing properties of the geodesic flow.
Entropy at infinity and applications in negative curvatureread_more
Y27 H 25
11. Dezember 2023
15:00-16:00
Prof. Dr. Tim Austin
University of Warwick
Details

Ergodic theory and dynamical systems seminar

Titel A dynamical proof of the Shmerkin--Wu theorem
Referent:in, Affiliation Prof. Dr. Tim Austin, University of Warwick
Datum, Zeit 11. Dezember 2023, 15:00-16:00
Ort Y27 H 25
Abstract Let a < b be multiplicatively independent integers, both at least 2. Let A,B be closed subsets of [0,1] that are forward invariant under multiplication by a, b respectively, and let C be A x B. An old conjecture of Furstenberg asserted that any line not parallel to either axis must intersect C in Hausdorff dimension at most max(dim C,1)-1. He was able to prove a partial result in this direction using a new class of measure-valued processes, now referred to as "CP chains". A few years ago, Shmerkin and Wu independently gave two different proofs of Furstenberg's conjecture. In this talk I will sketch a more recent third proof that builds on some of Furstenberg's original results. In addition to those, the main ingredients are a version of the Shannon--McMillan--Breiman theorem relative to a factor and some standard calculations with entropy and Hausdorff dimension.
A dynamical proof of the Shmerkin--Wu theoremread_more
Y27 H 25
11. Dezember 2023
16:15-17:15
Prof. Dr. Konstantin Khanin
University of Toronto
Details

Ergodic theory and dynamical systems seminar

Titel Typical rotation numbers for families of circle maps with singularities
Referent:in, Affiliation Prof. Dr. Konstantin Khanin, University of Toronto
Datum, Zeit 11. Dezember 2023, 16:15-17:15
Ort Y27 H 25
Abstract I shall discuss how one can define in a natural way the notion of typical rotation numbers for families of circle maps with singularities. This problem is related to a well known fact that in the case of maps with singularities the set of parameters, corresponding to irrational rotation numbers, has zero Lebesgue measure. Our approach is based on the hyperbolicity of renormalizations. I shall also discuss a natural setting for the Kesten theorem in the case of maps with singularities.
Typical rotation numbers for families of circle maps with singularitiesread_more
Y27 H 25
18. Dezember 2023
15:00-16:00
Dr. Nicolas de Saxcé
University Paris-Nord
Details

Ergodic theory and dynamical systems seminar

Titel Lattices, subspaces and diophantine approximation
Referent:in, Affiliation Dr. Nicolas de Saxcé, University Paris-Nord
Datum, Zeit 18. Dezember 2023, 15:00-16:00
Ort Y27 H 25
Abstract Since the work of Minkowski in the early twentieth century, the space of lattices has been a fundamental tool in the study of natural or rational numbers. Then, Margulis and his followers, in particular Dani, showed that methods from ergodic theory could be used very efficiently in that setting. More recently, Schmidt and Summerer started the "parametric geometry of numbers", which is a way to describe diagonal orbits in the space of lattices, using a simple combinatorial coding. The goal of this mini-course is to introduce the main concepts of parametric geometry of numbers, and to use them to study two problems going back to Jarník and Schmidt: - (Jarník) Given r>(n+1)/n, does there exist x in R^n such that the inequality |x-p/q| < q^(-r) has infinitely many solutions p/q in Q^n, but for all c<1, the inequality |x-p/q| < c q^(-r) has only finitely many solutions p/q in Q^n? (And what is the Hausdorff dimension of the set of such points?) - (Schmidt) Given an l-dimensional subspace x in R^d, for what values of r can one always find an l-dimensional rational subspace v in Q^d arbitrarily close to x and such that the distance to x satisfies d(v,x) < H(v)^(-r)? (Height and distance on Grassmann varieties will be defined in the first lecture -- note that this talk will last 1h30 and it will be introductory lecture for the ensuing mini-course that will take place on Tuesday 19th and Wednesday 20th of December. Please contact organizers for more info.)
Lattices, subspaces and diophantine approximationread_more
Y27 H 25
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