Departement Mathematik

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Ergodic theory and dynamical systems

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Frühjahrssemester 2021

Datum / Zeit Referent:in Titel Ort
18. Januar 2021
15:00-16:15 		Dr. Charles Fougeron
Université de Paris
Details

Ergodic theory and dynamical systems seminar

Titel Dynamics of simplicial systems and multi-dimensional continued fraction algorithms
Referent:in, Affiliation Dr. Charles Fougeron, Université de Paris
Datum, Zeit 18. Januar 2021, 15:00-16:15
Ort Y27 H 28
Abstract Motivated by the richness of the Gauss algorithm which allows to efficiently compute the best approximations of a real number by rationals, many mathematicians have proposed generalisations of these algorithms to approximate vectors of dimension greater than 1. Examples include Poincaré's algorithm introduced at the end of the 19th century or those of Brun and Selmer in the middle of the 20th century. Since the beginning of the 1990s to the present day, there has been a number of works studying the convergence of these algorithms. In particular, Schweiger and Broise have shown that the algorithms of Selmer and Brun are convergent and ergodic. Perhaps more surprisingly, Nogueira demonstrated that the algorithm proposed by Poincaré almost never converged. Starting from the classical case of Farey's algorithm, which is an "additive" version of Gauss's algorithm, I will present a combinatorial point of view on these algorithms which allows the passage from a deterministic view to a probabilistic approach. Indeed, in this model, taking a random vector for the Lebesgue measurement will correspond to following a random walk with memory in a labelled graph called symplicial system. The laws of probability for this random walk are elementary and we can thus develop probabilistic techniques to study their generic dynamical behaviour. This will lead us to describe a purely graph theory criterion to demonstrate the convergence or not of a continuous fraction algorithm.
Dynamics of simplicial systems and multi-dimensional continued fraction algorithmsread_more
Y27 H 28
25. Januar 2021
15:00-16:15 		Paul Frigge
Stony Brook University
Details

Ergodic theory and dynamical systems seminar

Titel Smoothness of stable manifolds of circle renormalization
Referent:in, Affiliation Paul Frigge, Stony Brook University
Datum, Zeit 25. Januar 2021, 15:00-16:15
Ort Y27 H 28
Abstract In the setting of finitely smooth circle diffeomorphisms, the renormalization operator is not differentiable. In this lecture, I will discuss generalizable methods and techniques which allow us to prove that the topological classes of C^{4+ epsilon} circle diffeomorphisms with irrational rotation number of bounded combinatorial type are C^1 global smooth stable manifolds of renormalization. First I will explain how to extend renormalization to decomposition spaces, and construct a normed space of Mobius-Schwarzian decompositions on which the renormalization operator is jump-out hyperbolic. Using this notion, I will explain the construction of the local smooth stable manifolds in a neighborhood of the rigid rotations. This structure will then be globalized with the help of monotone families which transversely intersect the topological classes.
Smoothness of stable manifolds of circle renormalizationread_more
Y27 H 28
1. März 2021
15:00-16:15 		Prof. Dr. Alan Haynes
University of Houston
Details

Ergodic theory and dynamical systems seminar

Titel Gap theorems for linear forms and for rotations on higher dimensional tori
Referent:in, Affiliation Prof. Dr. Alan Haynes, University of Houston
Datum, Zeit 1. März 2021, 15:00-16:15
Ort HG G 19.2
Abstract This talk is based on joint work with Jens Marklof, and with Roland Roeder. The three distance theorem states that, if x is any real number and N is any positive integer, the points x, 2x, ... , Nx modulo 1 partition the unit interval into component intervals having at most 3 distinct lengths. We will present two higher dimensional analogues of this problem. In the first we consider points of the form mx+ny modulo 1, where x and y are real numbers and m and n are integers taken from an expanding set in the plane. This version of the problem was previously studied by Geelen and Simpson, Chevallier, Erdos, and many other people, and it is closely related to the Littlewood conjecture in Diophantine approximation. The second version of the problem is a straightforward generalization to rotations on higher dimensional tori which, surprisingly, has been mostly overlooked in the literature. For the two dimensional torus, we are able to prove a five distance theorem, which is best possible. In higher dimensions we also have bounds, but establishing optimal bounds is an open problem.
Gap theorems for linear forms and for rotations on higher dimensional toriread_more
HG G 19.2
8. März 2021
15:00-16:15 		Prof. Dr. Sébastien Gouezel
CNRS
Details

Ergodic theory and dynamical systems seminar

Titel Ruelle resonances for geodesic flows on noncompact manifolds
Referent:in, Affiliation Prof. Dr. Sébastien Gouezel, CNRS
Datum, Zeit 8. März 2021, 15:00-16:15
Ort HG G 19.2
Abstract Ruelle resonances are complex numbers associated to a dynamical system that describe the precise asymptotics of the correlations for large times. It is well known that this notion makes sense for smooth uniformly hyperbolic dynamics on compact manifolds. In this talk, I will consider the case of the geodesic flow on some noncompact manifolds. In a class of such manifolds (called SPR), I will explain that one can define Ruelle resonances in a half-plane delimited by a critical exponent at infinity. Joint with Barbara Schapira and Samuel Tapie.
Ruelle resonances for geodesic flows on noncompact manifoldsread_more
HG G 19.2
15. März 2021
15:00-16:15 		Laura Monk
Université Strasbourg
Details

Ergodic theory and dynamical systems seminar

Titel Geometry and spectrum of random hyperbolic surfaces
Referent:in, Affiliation Laura Monk, Université Strasbourg
Datum, Zeit 15. März 2021, 15:00-16:15
Ort HG G 19.2
Abstract The main aim of this talk is to present geometric and spectral properties of typical hyperbolic surfaces. More precisely, I will: - introduce a probabilistic model, first studied by Mirzakhani, which is a natural and convenient way to sample random hyperbolic surfaces - describe the geometric properties of these random surfaces - explain how one can deduce from this geometric information estimates on the number of eigenvalues of the Laplacian in an interval [a,b], using the Selberg trace formula.
Geometry and spectrum of random hyperbolic surfacesread_more
HG G 19.2
22. März 2021
15:00-16:15 		Francisco Arana-Herrera
Stanford University
Details

Ergodic theory and dynamical systems seminar

Titel Effective mapping class group dynamics
Referent:in, Affiliation Francisco Arana-Herrera, Stanford University
Datum, Zeit 22. März 2021, 15:00-16:15
Ort HG G 19.2
Abstract Much is known about the dynamics of the mapping class group on different spaces: Teichmüller space, the space of singular measured foliations, the space of geodesic currents. However, very little is known about its effective dynamics. In this talk I will discuss work in progress that aims at clearing up this picture. Applications to counting problems on surfaces, including a partial solution to an open problem of Wright, will also be discussed. No previous knowledge of any of these topics will be assumed.
Effective mapping class group dynamicsread_more
HG G 19.2
29. März 2021
15:00-16:15 		Yiftach Dayan
Tel Aviv University
Details

Ergodic theory and dynamical systems seminar

Titel Random walks on tori and an application to normality of numbers in self-similar sets
Referent:in, Affiliation Yiftach Dayan, Tel Aviv University
Datum, Zeit 29. März 2021, 15:00-16:15
Ort HG G 19.2
Abstract We show that under certain conditions, random walks on a d-dim torus by affine expanding maps have a unique stationary measure. We then use this result to show that given an IFS of contracting similarity maps of the real line with a uniform contraction ratio 1/D, where D is some integer > 1, under some suitable condition, almost every point in the attractor of the given IFS (w.r.t. a natural measure) is normal to base D. (Joint work with Arijit Ganguly and Barak Weiss.)
Random walks on tori and an application to normality of numbers in self-similar setsread_more
HG G 19.2
12. April 2021
16:00-17:15 		Prof. Dr. Bryna Kra
Northwestern University
Details

Ergodic theory and dynamical systems seminar

Titel Symbolic systems, automorphism groups, and invariant measures
Referent:in, Affiliation Prof. Dr. Bryna Kra, Northwestern University
Datum, Zeit 12. April 2021, 16:00-17:15
Ort HG G 19.2
Abstract The automorphism group of a shift system can be quite complicated: for example, for a topologically mixing shift of finite type, the automorphism group contains isomorphic copies of all finite groups and the free group on two generators and such behavior is common for shifts of high complexity. In the opposite setting of low complexity, there are numerous restrictions on the automorphism group, and for many classes of shift systems, it is known to be virtually abelian. In both of these extreme cases, there exists a Borel probability measure supported on the shift that is invariant under the full automorphism group. However, it unknown if every shift system supports such a measure. I will discuss progress on this question, including its solution for a new class of shift systems, which we have named the language stable shifts. This is joint work with Van Cyr.
Symbolic systems, automorphism groups, and invariant measuresread_more
HG G 19.2
14. April 2021
16:00-17:15 		Prof. Dr. Alireza Golsefidy
University of California, San Diego
Details

Ergodic theory and dynamical systems seminar

Titel Local randomness and spectral independence
Referent:in, Affiliation Prof. Dr. Alireza Golsefidy, University of California, San Diego
Datum, Zeit 14. April 2021, 16:00-17:15
Ort HG G 19.2
Abstract I will talk about two new concepts related to random walks in compact groups. Roughly locally random compact groups are groups that do not have low complexity models. I will indicate how such a property gives us a mixing inequality. Two compact groups G and H are called spectrally independent if the following holds: suppose (X,Y) is a random variable with values in GxH. If X and Y have spectral gap property in G and H, respectively, then (X,Y) has spectral gap property in GxH. I will explain why compact open subgroups of two non-locally isomorphic almost simple analytic groups are spectrally independent. Along the way we show that a local approximate homomorphism between two compact open subgroups of almost simple analytic groups that has large image is close to an isogeny. Among other things this result extends a work of Kazhdan on approximate homomorphisms. (Joint with K. Mallahi-Karai and A. Mohammadi)
Local randomness and spectral independenceread_more
HG G 19.2
26. April 2021
15:00-16:15 		Dr. Shahriar Mirzadeh
Michigan State University
Details

Ergodic theory and dynamical systems seminar

Titel On the dimension drop conjecture for diagonal flows on the space of lattices
Referent:in, Affiliation Dr. Shahriar Mirzadeh, Michigan State University
Datum, Zeit 26. April 2021, 15:00-16:15
Ort HG G 19.2
Abstract Consider the set of points in a homogeneous space X = G/\Gamma whose g_t-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when has real rank 1. In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: G=SL_{m+n}(R), \Gamma= \SL_{m+n}(Z), and g_t=\diag (e^{t/m},..., e^{t/m},e^{-t/n},..., e^{-t/n}). We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.
On the dimension drop conjecture for diagonal flows on the space of latticesread_more
HG G 19.2
3. Mai 2021
15:00-16:15 		Prof. Dr. Dmitry Kleinbock
Brandeis University
Details

Ergodic theory and dynamical systems seminar

Titel Some remarks on the `eventually always hitting' property
Referent:in, Affiliation Prof. Dr. Dmitry Kleinbock, Brandeis University
Datum, Zeit 3. Mai 2021, 15:00-16:15
Ort HG G 19.2
Abstract Eventually always hitting (EAH) points are those whose long orbit segments eventually hit the corresponding shrinking targets for all future times. This is a uniform version of the classical hitting property in ergodic theory with shrinking targets; the terminology is due to Dubi Kelmer. Unlike its classical counterpart, not much is known about conditions on the targets for which almost all vs. almost no points are EAH. I will talk about systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes. For such systems tight conditions on the shrinking rate of the targets can be stated so that the set of eventually always hitting points is null or co-null. This is a joint work with Ioannis Konstantoulas and Florian Richter.
Some remarks on the `eventually always hitting' propertyread_more
HG G 19.2
10. Mai 2021
15:00-16:15 		Prof. Dr. Nimish Shah
Ohio State University
Details

Ergodic theory and dynamical systems seminar

Titel Limiting distributions in the space of 3-Lattices and non-improvability of Dirichlet's approximation
Referent:in, Affiliation Prof. Dr. Nimish Shah, Ohio State University
Datum, Zeit 10. Mai 2021, 15:00-16:15
Ort HG G 19.2
Abstract Earlier a question of Davenport and Schmidt about non-improvability of Dirichlet's approximation was answered by proving equidistribution of translates of non-degenerate regular curves in SL(n,R)/SL(n,Z) under the action of certain diagonal subgroups. I will speak about a joint work with D. Kleinbock, N. de Saxcé and P. Yang, where we show that equidistribution results of various flavors for the translates of lines, rather than curves, in the space of unimodular 3-lattices hold under certain precise Diophantine conditions on the lines. In particular, one can show that for almost all points on an irrational line, the Dirichlet's approximation cannot be improved. Our proof uses Ratner's theorem and the linearization technique combined with ideas from geometric invariant theory.
Limiting distributions in the space of 3-Lattices and non-improvability of Dirichlet's approximationread_more
HG G 19.2
17. Mai 2021
15:00-16:15 		Prof. Dr. Tatiana Smirnova-Nagnibeda
Université de Genève
Details

Ergodic theory and dynamical systems seminar

Titel Various types of spectra and spectral measures on Schreier and Cayley graphs
Referent:in, Affiliation Prof. Dr. Tatiana Smirnova-Nagnibeda, Université de Genève
Datum, Zeit 17. Mai 2021, 15:00-16:15
Ort HG F 3
Abstract We will be interested in the Laplacian on graphs associated with finitely generated groups: Cayley graphs and more generally Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general. We will discuss various techniques that allow to construct examples with different types of spectra: connected, union of two intervals, totally disconnected..., and how this depends on the choice of the generating set in the group. Types of spectral measures that can arise in these examples will also be discussed.
Various types of spectra and spectral measures on Schreier and Cayley graphsread_more
HG F 3
31. Mai 2021
15:00-16:15 		Dr. Tal Horesh
IST Austria
Details

Ergodic theory and dynamical systems seminar

Titel Distribution of primitive lattices and their flags
Referent:in, Affiliation Dr. Tal Horesh, IST Austria
Datum, Zeit 31. Mai 2021, 15:00-16:15
Ort
Abstract Integral (or primitive) lattices in \(Z^n\) are the higher dimensional analog of integral (or primitive) vectors in \(Z^n\). Therefore, many counting and equidistribution problems regarding integral/primitive vectors can be generalized to integral/primitive lattices. For example, Linnik type questions concern the distribution of projections of integral points on the unit sphere; these can be generalized to questions about the distribution of the projections of rank \(d\) lattices in \(Z^n\) to the Grassmannian \(Gr(d,n)\). Such counting and equidistribution results will be the topic of the talk, as well as their generalizations to flags of lattices. This is joint work with Yakov Karasik. (---> The talk is taking place at ETH ML H 37.1)
Distribution of primitive lattices and their flagsread_more
7. Juni 2021
15:00-16:15 		Dr. Przemyslaw Berk
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel Some properties of automorphisms disjoint with all ergodic systems
Referent:in, Affiliation Dr. Przemyslaw Berk, Universität Zürich
Datum, Zeit 7. Juni 2021, 15:00-16:15
Ort HG F 3
Abstract The talk is based on a joint work with Tim Austin, Martyna Gorska and Mariusz Lemanczyk. It is a classical fact that identity is disjoint from every ergodic system. It turns out that the family of systems with this property is far richer. First I will give some examples and a general construction of elements of this class. Then I will talk about certain properties of systems disjoint with ergodic automorphisms, with a particular emphasis on joinings of such systems. Time permitting, I will give a sketch of the proof that the product of two systems disjoint with ergodic systems also has this property
Some properties of automorphisms disjoint with all ergodic systemsread_more
HG F 3

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