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Spring Semester 2020

Date / Time Speaker Title Location
24 February 2020
13:45-14:45 		Prof. Dr. Alex Eskin
University of Chicago
Event Details

Ergodic theory and dynamical systems seminar

Title Exponential drift and character varieties
Speaker, Affiliation Prof. Dr. Alex Eskin, University of Chicago
Date, Time 24 February 2020, 13:45-14:45
Location Y27 H 28
Abstract Let G be a compact Lie group, and let S be a surface group. The character variety is the set of homomorphisms from S to G, modulo conjugation. Since the mapping class group is the outer automorphism group of S, it acts on the character variety by precomposition. It is natural to ask about stationary measures and orbit closures for this action. The breakthrough work of Benoist and Quint and subsequent developments gives a new set of tools for attacking this problem. I will give a progress report and describe the difficulties in the general case.
Exponential drift and character varietiesread_more
Y27 H 28
2 March 2020
13:45-14:45 		Dr. Sebastian Hurtado
University of Chicago
Event Details

Ergodic theory and dynamical systems seminar

Title Random walks by homeomorphisms on the line and orderability of lattices
Speaker, Affiliation Dr. Sebastian Hurtado, University of Chicago
Date, Time 2 March 2020, 13:45-14:45
Location Y27 H 28
Abstract The standard random walk in the integers is known to be recurrent, it passes through any integer infinitely many times. We will discuss a generalization of this theorem for random walks given by homeomorphisms of the line due to Deroin-Navas-Kleptsyn-Parwani and discuss some applications to the theory of left-orderable groups. Our main result is that cocompact lattices in simple Lie groups of higher rank are not left-orderable groups (as a consequence, do not act by homeomorphisms in the line or the circle), a conjecture due to Witte-Morris and Ghys. The proof makes use of of higher-rank hyperbolic dynamics and some basic notions in infinite ergodic theory. (Joint work with Bertrand Deroin)
Random walks by homeomorphisms on the line and orderability of latticesread_more
Y27 H 28
10 March 2020
15:00-16:00 		Prof. Dr. Jon Chaika
University of Utah
Event Details

Ergodic theory and dynamical systems seminar

Title There exists a weakly mixing billiard in a polygon (Video seminar)
Speaker, Affiliation Prof. Dr. Jon Chaika, University of Utah
Date, Time 10 March 2020, 15:00-16:00
Location Y27 H 28
Abstract This main result of this talk is that there exists a billiard flow in a polygon that is weakly mixing with respect to Liouville measure (on the unit tangent bundle to the billiard). This strengthens Kerckhoff, Masur and Smillie's result that there exists ergodic billiard flows in polygons. The existence of a weakly mixing billiard follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues. This talk will explain the problem, related results, and approach. Time permitting it will present a bit of the argument to rule out shared eigenvalues. The talk will not assume familiarity with translation surfaces. This is joint work with Giovanni Forni.
There exists a weakly mixing billiard in a polygon (Video seminar)read_more
Y27 H 28
16 March 2020
13:45-14:45 		Prof. Dr. Alexander Bufetov
Institut de Mathématiques de Marseille
Event Details

Ergodic theory and dynamical systems seminar

Title e-Seminar: Spectral cocycle for substitutions
Speaker, Affiliation Prof. Dr. Alexander Bufetov, Institut de Mathématiques de Marseille
Date, Time 16 March 2020, 13:45-14:45
Location Y27 H 28
e-Seminar: Spectral cocycle for substitutions
Y27 H 28
6 April 2020
13:45-14:45 		Dr. Selim Ghazouani
University of Warwick
Event Details

Ergodic theory and dynamical systems seminar

Title e-Seminar: Renormalisation and rigidity of generalised interval exchange maps
Speaker, Affiliation Dr. Selim Ghazouani, University of Warwick
Date, Time 6 April 2020, 13:45-14:45
Location Y27 H 28
Abstract A celebrated result of Herman asserts that a circle diffeomorphism whose rotation number satisfies a mild arithmetic condition is smoothly conjugate to its linear model. One can wonder the extent to which the theory of circle diffeomorphisms extends to smooth, non-linear (generalised) interval exchange transformations (GIET). In particular, is a smooth GIET with irrational "rotation number" always smoothly conjugate to its linear model? Building upon the development of Teichmüller dynamics, Forni and later on Marmi-Moussa-Yoccoz brought to light a finite dimensional set of obstructions for this problem. These obstructions are of purely ergodic-theoretic nature. In this talk, I will present a linearisation result establishing that these obstructions are indeed the only ones, in the case where the rotation number satisfies a strong arithmetic condition. If time permits, I will discuss elements of the proof which centres around the dynamics of a renormalisation operator.
e-Seminar: Renormalisation and rigidity of generalised interval exchange mapsread_more
Y27 H 28
27 April 2020
13:45-14:45 		Dr. Rhiannon Dougall
University of Bristol
Event Details

Ergodic theory and dynamical systems seminar

Title e-Seminar: Critical exponents via twisted Patterson-Sullivan measures
Speaker, Affiliation Dr. Rhiannon Dougall, University of Bristol
Date, Time 27 April 2020, 13:45-14:45
Location Y27 H 28
Abstract This is joint work with R. Coulon, B. Schapira and S. Tapie. One of the first things we learn about a (proper) Gromov hyperbolic geodesic space X is the construction of the visual boundary of X. An ergodic theorist then learns that for a non-elementary discrete group of isometries G acting properly on X, there is an interesting family of \delta_G-quasi-conformal measures on the boundary, called Patterson-Sullivan measures. The parameter \delta_G is called the critical exponent of G, and is equal to the exponential growth rate of the orbit Gx in X. Given a subgroup H of G, how do we compare \delta_H and \delta_G? For G with "good dynamics", we expect that \delta_G = \delta_H if and only if H is co-amenable in G. (As in the case that X is a rank 1 symmetric space, some results were known using Brooks' characterisation of amenability in terms of the bottom of the spectrum of the Laplacian.) We obtain the theorem: if the action of G is SPR, then we have \delta_G = \delta_H if and only if H is co-amenable in G. What is particularly appealing about our method is the construction of a "twisted Patterson-Sullivan measure".
e-Seminar: Critical exponents via twisted Patterson-Sullivan measuresread_more
Y27 H 28
11 May 2020
13:45-14:45 		Prof. Dr. Anders Karlsson
Section de Mathématiques, Université de Genève
Event Details

Ergodic theory and dynamical systems seminar

Title e-Seminar: Noncommuting random products and metric functionals
Speaker, Affiliation Prof. Dr. Anders Karlsson, Section de Mathématiques, Université de Genève
Date, Time 11 May 2020, 13:45-14:45
Location Y27 H 28
Abstract The study of the composition of random maps that do not commute started in the 1950s with the need to solve linear difference equations with random coefficients, giving rise to products of random matrices (today appearing in the study of quasi-periodic Schrödinger operators). It also arises in differentiable dynamics, with the derivative cocycle. In both these settings a basic fundamental theorem is the multiplicative ergodic theorem of Oseledets asserting the existence of Lyapunov exponents. But there are several non-linear settings of interests: random walks on groups, surface homeomorphisms, deep learning, etc. In joint works with Ledrappier and with Gouëzel we established a general extension of the multiplicative ergodic theorem, that in particular provided an answer to a question raised in a seminal paper by Furstenberg from 1963. It is given in terms of metric functionals (which include horofunctions) and suggests a (non-linear) metric functional analysis / spectral theory.
e-Seminar: Noncommuting random products and metric functionalsread_more
Y27 H 28
18 May 2020
13:45-14:45 		Dr. Michael Magee
Durham University
Event Details

Ergodic theory and dynamical systems seminar

Title e-Seminar: The spectral gap of a random hyperbolic surface
Speaker, Affiliation Dr. Michael Magee, Durham University
Date, Time 18 May 2020, 13:45-14:45
Location Y27 H 28
Abstract On a compact hyperbolic surface, the Laplacian has a spectral gap between 0 and the next smallest eigenvalue if and only if the surface is connected. The size of the spectral gap measures both how highly connected the surface is, and the rate of exponential mixing of the geodesic flow on the surface. There is an analogous concept of spectral gap for graphs, with analogous connections to connectivity and dynamics. Motivated by theorems about the spectral gap of random regular graphs, we proved that for any r>0, a random cover of a fixed compact connected hyperbolic surface has no new eigenvalues below 3/16 - r, with probability tending to 1 as the covering degree tends to infinity. The number 3/16 is, mysteriously, the same spectral gap that Selberg obtained for congruence modular curves. The talk is intended to be accessible to graduate students and is based on joint works with Frédéric Naud and Doron Puder.
e-Seminar: The spectral gap of a random hyperbolic surfaceread_more
Y27 H 28
25 May 2020
13:45-14:45 		Prof. Dr. Benoît Kloeckner
Université Paris-Est
Event Details

Ergodic theory and dynamical systems seminar

Title e-Seminar: Extensions with shrinking fibers
Speaker, Affiliation Prof. Dr. Benoît Kloeckner, Université Paris-Est
Date, Time 25 May 2020, 13:45-14:45
Location Y27 H 28
Abstract Let T and S be two discrete-time dynamical systems, the first being an extension of the second with a contracting dynamics along the fibers in a very general meaning. The main goal of the talk will be to show the simple result, yet not known in this generality before, that every S-invariant measure has a unique T-invariant lift. Then, many properties of invariant measures can easily be lifted as well (ergodicity, mixing, physicality, entropy) possibly under additional mild assumptions. The main tool is a variant of the Wasserstein distance issued from optimal transportation. If time permits, we shall evoke the lighting of more subtle properties (statistical properties, speed of decay of correlations).
e-Seminar: Extensions with shrinking fibersread_more
Y27 H 28
2 June 2020
13:45-14:45 		Prof. Dr. Pablo Shmerkin
Universidad Torcuato Di Tella
Event Details

Ergodic theory and dynamical systems seminar

Title e-Seminar: Arithmetic and geometric properties of planar self-similar sets
Speaker, Affiliation Prof. Dr. Pablo Shmerkin, Universidad Torcuato Di Tella
Date, Time 2 June 2020, 13:45-14:45
Location Y27 H 28
Abstract Furstenberg's conjecture on the dimension of the intersection of x2,x3-invariant Cantor sets can be restated as a bound on the dimension of linear slices of the product of x2,x3-Cantor sets, which is a self-affine set in the plane. I will discuss some older and newer variants of this, where the self-affine set is replaced by a self-similar set such as the Sierpinski triangle, Sierpinski carpet or (support of) a complex Bernoulli convolution. Among other things, I will show that the intersection of the Sierpinski carpet with circles has small dimension, but on the other hand it can be covered very efficiently by linear tubes (neighborhoods of lines). The latter fact is a recent result joint with A. Pyörälä, V. Suomala and M. Wu.
e-Seminar: Arithmetic and geometric properties of planar self-similar setsread_more
Y27 H 28

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