Departement Mathematik

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Ergodic theory and dynamical systems

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Herbstsemester 2021

Datum / Zeit Referent:in Titel Ort
27. September 2021
15:00-16:00 		Prof. Dr. Mikhael Lyubich
Stony Brook University
Details

Ergodic theory and dynamical systems seminar

Titel Wandering around the Mandelbrot set
Referent:in, Affiliation Prof. Dr. Mikhael Lyubich, Stony Brook University
Datum, Zeit 27. September 2021, 15:00-16:00
Ort Y27 H 28
Abstract The Mandelbrot set M encodes in one picture the whole dynamical story of the complex quadratic family f_c(z)= z^2+c. This story is quite remarkable, beginning with a simple map z^2 and then developing, through a sequence of bifurcations, intricate chaotic and fractal structures. In some places it is self-similar, while in others is completely unrecognizable. How to make sense of these patterns? There are powerful geometric and dynamical ideas that lead to a comprehensive understanding of the set. But there is also one open problem on the way: ``MLC Conjecture" asserting that M is locally connected.
Wandering around the Mandelbrot setread_more
Y27 H 28
4. Oktober 2021
15:00-16:00 		Davide Ravotti - Emilio Corso
University of Vienna - ETH Zurich
Details

Ergodic theory and dynamical systems seminar

Titel Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner
Referent:in, Affiliation Davide Ravotti - Emilio Corso, University of Vienna - ETH Zurich
Datum, Zeit 4. Oktober 2021, 15:00-16:00
Ort Y27 H 28
Abstract In the first part of this talk, we will present a new method to study ergodic integrals for horocycle flows which does not rely on the study of the cohomological equation. The approach is inspired by Ratner's work on quantitative mixing for the geodesic flow. We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni, and we strengthen it by showing that the coefficients in the expansion are Hölder continuous with respect to the base point. The second part will focus on the distributional limit theorems that can be deduced from this result, in particular we will present streamlined proofs of Bufetov and Forni's spatial limit theorem and Dolgopyat and Sarig's temporal limit theorem.
Asymptotics and limit theorems for horocycle ergodic integrals à la Ratnerread_more
Y27 H 28
11. Oktober 2021
15:00-16:00 		Michael Bersudsky
Technion
Details

Ergodic theory and dynamical systems seminar

Titel Linnik's problem and statistics of orthogonal lattices of primitive integral vectors
Referent:in, Affiliation Michael Bersudsky, Technion
Datum, Zeit 11. Oktober 2021, 15:00-16:00
Ort Y27 H 28
Abstract I will discuss my joint work with Uri Shapira in which we generalize Aka, Einsiedler and Shapira's results on the statistics of shapes of orthogonal lattices of primitive integral vectors lying on large spheres (in dimensions larger than 4). Part of my talk will explain how the result obtained by Aka, Einsiedler and Shapira can be deduced from a result that can be thought of as a variant in a non-Euclidean setting of Linnik's classical equidistribution theorem about the directions of primitive integral vectors on a large sphere.
Linnik's problem and statistics of orthogonal lattices of primitive integral vectorsread_more
Y27 H 28
18. Oktober 2021
15:00-16:00 		Prof. Dr. Alex Kontorovich
Rutgers University
Details

Ergodic theory and dynamical systems seminar

Titel Asymptotic Length Saturation for Zariski Dense Surfaces
Referent:in, Affiliation Prof. Dr. Alex Kontorovich, Rutgers University
Datum, Zeit 18. Oktober 2021, 15:00-16:00
Ort Online Seminar
Abstract The lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set "saturates" (resp. "asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding one-half (below which saturation is impossible).
Asymptotic Length Saturation for Zariski Dense Surfacesread_more
Online Seminar
25. Oktober 2021
15:00-16:00 		Simon Machado
University of Cambridge
Details

Ergodic theory and dynamical systems seminar

Titel Approximate lattices in higher rank simple Lie groups
Referent:in, Affiliation Simon Machado, University of Cambridge
Datum, Zeit 25. Oktober 2021, 15:00-16:00
Ort Y27 H 28
Abstract Strong approximate lattices are defined as discrete approximate subgroups of finite co-volume in locally compact groups. To make sense of the notion of "finite co-volume" for a subset that is not a subgroup we need to introduce the notion of invariant hull. Given a closed subset X of a locally compact group G, the invariant hull of X is defined as the set of those subsets of G that cannot be distinguished locally from a translate of X. I will discuss certain properties of the invariant hull, and of approximate lattices in general. I will then explain how a careful analysis of Borel cocycles on the invariant hull, combined with cocycle superrigidity results, enables us to characterise the strong approximate lattices of SL_n(R) with n at least 3. They correspond to sets of matrices with coefficients in the set of Pisot-Vijayaraghavan numbers of some real number field K.
Approximate lattices in higher rank simple Lie groupsread_more
Y27 H 28
1. November 2021
15:00-16:00 		Tsviqa Lakrec
Institut für Mathematik, Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel Equidistribution of affine random walks on some nilmanifolds
Referent:in, Affiliation Tsviqa Lakrec, Institut für Mathematik, Universität Zürich
Datum, Zeit 1. November 2021, 15:00-16:00
Ort Y27 H 28
Abstract We consider the action of the group of affine transformations on a nilmanifold. Given a probability measure on this group and a starting point, a random walk on the nilmanifold is defined. We study quantitative equidistribution in law of such affine random walks on nilmanifolds. Under certain assumptions, we show that a failure to have fast equidistribution on a nilmanifold is due to a failure on some factor nilmanifold. Combined with equidistribution results on the torus, this leads to an equidistribution statement on some nilmanifolds, such as Heisenberg nilmanifolds. This talk is based on joint works with Weikun He and Elon Lindenstrauss.
Equidistribution of affine random walks on some nilmanifoldsread_more
Y27 H 28
8. November 2021
15:00-16:00 		Dr. Manuel Lüthi
EPFL
Details

Ergodic theory and dynamical systems seminar

Titel Random walks on homogeneous spaces, Spectral Gaps, and Khintchine's theorem on fractals
Referent:in, Affiliation Dr. Manuel Lüthi, EPFL
Datum, Zeit 8. November 2021, 15:00-16:00
Ort Y27 H 28
Abstract Khintchine's theorem in Diophantine approximation gives a zero one law describing the approximability of typical points by rational points. In 1984, Mahler asked how well points on the middle third Cantor set can be approximated. His question fits into an attempt to determine conditions under which subsets of Euclidean space inherit the Diophantine properties of the ambient space. I will discuss a complete analogue of the theorem of Khintchine for certain fractal measures which was recently obtained in collaboration with Osama Khalil. Our results hold for fractals generated by rational similarities of Euclidean space that have sufficiently small Hausdorff co-dimension. The main ingredient to the proof is an effective equidistribution theorem for associated fractal measures on the space of unimodular lattices. The latter is established using a spectral gap property of a type of Markov operators associated with an S-arithmetic random walk related to the generating similarities.
Random walks on homogeneous spaces, Spectral Gaps, and Khintchine's theorem on fractalsread_more
Y27 H 28
15. November 2021
15:00-16:00 		Prof. Dr. Romain Dujardin
Université Sorbonne Paris Nord
Details

Ergodic theory and dynamical systems seminar

Titel Dynamics of groups of automorphisms of compact complex surfaces
Referent:in, Affiliation Prof. Dr. Romain Dujardin, Université Sorbonne Paris Nord
Datum, Zeit 15. November 2021, 15:00-16:00
Ort Y27 H 28
Abstract I will review some results obtained in the past few years in collaboration with Serge Cantat on the dynamics of groups of algebraic automorphisms of real and complex projective surfaces: classification of stationary and invariant measures, orbit closures, and finite orbits. This relies on a variety of techniques from complex and algebraic geometry as well as random dynamics.
Dynamics of groups of automorphisms of compact complex surfacesread_more
Y27 H 28
22. November 2021
15:00-16:00 		Dr. Mihajlo Cekic
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel Ergodicity of frame flows on even-dimensional manifolds
Referent:in, Affiliation Dr. Mihajlo Cekic, Universität Zürich
Datum, Zeit 22. November 2021, 15:00-16:00
Ort Y27 H 28
Abstract Flows of frames over negatively curved Riemannian manifolds (M, g) are one of the oldest examples of partially hyperbolic dynamics. It is well known that frame flows of hyperbolic manifolds are ergodic, while Kahler manifolds never have ergodic frame flows; Brin conjectured in the 70's that all manifolds with sectional curvature between -1 and -0.25 (i.e. curvature is 0.25-pinched) have ergodic frame flows. In this talk I will explain recent progress on this conjecture: we show that in dimensions 4k+2 the frame flow is ergodic if (M, g) is ~0.27 pinched, and in dimensions 4k if it is ~0.55 pinched. Our new method uses techniques in hyperbolic dynamics (transitivity group, Parry's representation), topology of structure groups of spheres, and Fourier analysis in the vertical fibre of the unit sphere bundle (based on Pestov identity). This is joint work with Lefeuvre, Moroianu, and Semmelmann.
Ergodicity of frame flows on even-dimensional manifoldsread_more
Y27 H 28
29. November 2021
15:00-16:00 		Dr. Nicolas Matte Bon
Université Lyon 1
Details

Ergodic theory and dynamical systems seminar

Titel Locally moving groups acting on intervals
Referent:in, Affiliation Dr. Nicolas Matte Bon, Université Lyon 1
Datum, Zeit 29. November 2021, 15:00-16:00
Ort Y27 H 28
Abstract Given a group, we are interested in understanding and classifying its actions on one-dimensional manifolds, that its representations into the groups of homeomorphisms or diffeomorphisms of an interval or the circle. In this talk we will address this problem for a class of groups arising via an action on intervals of a special type, called locally moving. A well studied example in this class is the Thompson group. I will explain that if G is a locally moving group of homeomorphisms of a real interval, then every action of G on an interval by diffeomorphisms (of class C^1) is semiconjugate to the natural defining action of G. In contrast such a group can admit a much richer space of actions on intervals by homeomorphisms, and for a class of locally moving groups I will present a structure theorem for such actions. This is joint work with Joaquín Brum, Cristóbal Rivas and Michele Triestino.
Locally moving groups acting on intervalsread_more
Y27 H 28
6. Dezember 2021
15:00-16:00 		Prof. Dr. Valérie Berthé
IRIF, Université de Paris
Details

Ergodic theory and dynamical systems seminar

Titel Symbolic discrepancy and Pisot dynamics
Referent:in, Affiliation Prof. Dr. Valérie Berthé, IRIF, Université de Paris
Datum, Zeit 6. Dezember 2021, 15:00-16:00
Ort Online Seminar
Abstract Discrepancy is a measure of equidistribution for sequences of points. A bounded remainder set is a set with bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. We discuss dynamical, symbolic and spectral approaches to the study of bounded remainder sets for Kronecker sequences. We also consider discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts. We focus on the case of Pisot parameters for toral translations and then show how to construct symbolic codings in terms of multidimensional continued fraction algorithms which lead to renormalization schemes. This is joint work with W. Steiner and J. Thuswaldner.
Symbolic discrepancy and Pisot dynamicsread_more
Online Seminar

Archiv: HS 24 FS 24 HS 23 FS 23 HS 22 FS 22 HS 21 FS 21 HS 20 FS 20 HS 19 FS 19 HS 18

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