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Ergodic theory and dynamical systems

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Frühjahrssemester 2019

Datum / Zeit Referent:in Titel Ort
18. Februar 2019
14:00-15:00 		Barbara Schapira
Details

Ergodic theory and dynamical systems seminar

Titel Regularity of entropy on non compact negatively curved manifolds
Referent:in, Affiliation Barbara Schapira,
Datum, Zeit 18. Februar 2019, 14:00-15:00
Ort HG G 43
Abstract In this work, we proved that the entropy of the geodesic flow of non compact manifolds varies in a C1-way along a C1-variation of metrics, under an assumption, called Strong positive recurrence, which is satisfied on a wide class of non compact manifolds. It generalizes an old result of Katok-Knieper-Weiss in the compact case. This assumption means that a certain entropy at infinity is strictly smaller than the topological entropy. I will explain the relevant concepts and strategy of the proof. Joint work with Samuel Tapie.
Regularity of entropy on non compact negatively curved manifoldsread_more
HG G 43
25. Februar 2019
14:00-15:00 		Dr. Nicolas de Saxcé
University Paris-Nord
Details

Ergodic theory and dynamical systems seminar

Titel Diophantine approximation and product of linear forms
Referent:in, Affiliation Dr. Nicolas de Saxcé, University Paris-Nord
Datum, Zeit 25. Februar 2019, 14:00-15:00
Ort HG G 43
Abstract Fix a one-parameter diagonal flow a_t on the space of unimodular lattices in R^d, and consider its conjugates L^{-1}a_tL, where L is chosen randomly on a subvariety M of SL(d,R). Under some algebraicity condition on M, we shall give a formula for the rate of escape of the orbit L^{-1}a_tL.Z^d and describe the shape of the lattice, for t large and almost every L. Finally, we shall give some applications to diophantine approximation on manifolds.
Diophantine approximation and product of linear formsread_more
HG G 43
4. März 2019
14:00-15:00 		Dr. Ian Morris
University of Surrey
Details

Ergodic theory and dynamical systems seminar

Titel Computability of some matrix growth invariants arising in fractal geometry
Referent:in, Affiliation Dr. Ian Morris, University of Surrey
Datum, Zeit 4. März 2019, 14:00-15:00
Ort HG G 43
Abstract The dimension theory of self-similar sets without overlaps (which includes familiar examples of fractals such as the von Koch curve and Sierpinski gasket) was resolved essentially completely in the early 1980s. The dimension theory of their non-conformal relatives, self-affine sets, has by contrast been a source of stubborn open problems for over three decades. In this talk I will discuss a dimension formula for self-affine sets introduced by Falconer in the 1980s and describe a proof that the dimension value predicted by this formula is theoretically computable in a precise sense. As a by-product this argument also yields a new proof of the continuity of the affinity dimension, a result previously established by Feng and Shmerkin in 2014.
Computability of some matrix growth invariants arising in fractal geometryread_more
HG G 43
11. März 2019
14:00-15:00 		Prof. Dr. Anke Pohl
University of Bremen
Details

Ergodic theory and dynamical systems seminar

Titel Automorphic functions twisted with non-expanding cusp monodromies, and dynamics
Referent:in, Affiliation Prof. Dr. Anke Pohl, University of Bremen
Datum, Zeit 11. März 2019, 14:00-15:00
Ort HG G 43
Abstract Automorphic functions play an important role in several subareas of mathematics and mathematical physics. The correspondence principle between quantum und classical mechanics suggests that automorphic functions are closely related to geometric and dynamical entities of the underlying locally symmetric spaces. Despite intensive research, the extent of this relation is not yet fully understood. A seminal and very influencial result in this direction has been provided by Selberg, showing that for any hyperbolic surface X of finite area, the Selberg zeta function (a generating function for the geodesic length spectrum of X) encodes the spectral parameters of the untwisted automorphic forms for X among its zeros. Subsequently, this type of relation was generalized to automorphic forms twisted by unitary representations, to resonances and to spaces of infinite area by various researchers. Recently, a deeper relation could be established by means of transfer operator techniques. For the so-called Maass cusp forms, these methods allow us to provide a purely dynamical characterization of these automorphic functions themselves, not only of their spectral parameters. The structure of this approach indicates that an extension to automorphic forms with well-behaved, also non-unitary twists should be expected. While such a generalization seems to be a long-term goal, we could already show first steps in this direction on the level of the Selberg zeta functions. After surveying the transfer operator techniques, we discuss the current state of art regarding dynamical approaches towards automorphic forms with non-unitary twists.
Automorphic functions twisted with non-expanding cusp monodromies, and dynamicsread_more
HG G 43
18. März 2019
14:00-15:00 		Dr. Sasha Skripchenko
National Research University Higher School of Economics
Details

Ergodic theory and dynamical systems seminar

Titel Cohomological equations for linear involutions
Referent:in, Affiliation Dr. Sasha Skripchenko, National Research University Higher School of Economics
Datum, Zeit 18. März 2019, 14:00-15:00
Ort HG G 43
Abstract Famous Roth theorem about diophantine approximations states that a given algebraic number may not have too many rational number approximations, that are "very good". More precisely, Roth first defined a class of numbers that are not very easy to approximate by rationals (they are called Roth numbers) and then showed that almost all algebraic irrationals are of Roth type, and that they form a set of a full measure which is invariant under the natural action of the modular group SL(2,Z).
In addition to their interesting arithmetical properties, Roth type irrationals appear in a study of the cohomological equation associated with a rotation R_a : R_a(x) = x+a of the circle T=R/Z: a is of Roth type if and only iff for all r,s : r>s+1>1 and for all functions g of class C^r on T with zero mean there exists a unique function f in C^s(T) with zero mean such that
f-f R_a = f
In 2005 Marmi, Moussa and Yoccoz established an analogue of Roth theorem for interval exchange transformations (IETs). In particular, they defined the notion of Roth type IETs and proved existence of the solution of cohomological equation for this class; they also showed that IET of Roth type form a full measure set in the parameter space of IETs.
In a fresh joint work with Erwan Lanneau and Stefano Marmi we get a certain generalization of this result for linear involutions that can be considered as a natural extension of IETs to non-orientable case.
Cohomological equations for linear involutionsread_more
HG G 43
* 8. April 2019
13:15-14:15 		Prof. Dr. Pascal Hubert
Aix Marseille Université
Details

Ergodic theory and dynamical systems seminar

Titel Tiling billiards
Referent:in, Affiliation Prof. Dr. Pascal Hubert, Aix Marseille Université
Datum, Zeit 8. April 2019, 13:15-14:15
Ort HG G 43
Abstract In this joint work with Olga Romaskevich, I will discuss tiling billiards, a family of dynamical systems introduced by Diana Davis and her students. I will reduce this problem to the study of some interval exchange transformations with flips (or flows on non orientable surfaces). I will study these systems in low complexity and address open questions about the dynamics of interval exchange transformations with flips.
Tiling billiardsread_more
HG G 43
15. April 2019
14:00-15:00 		Prof. Dr. Federico Rodriguez-Hertz
Penn State University
Details

Ergodic theory and dynamical systems seminar

Titel Rigidity in hyperbolic systems
Referent:in, Affiliation Prof. Dr. Federico Rodriguez-Hertz, Penn State University
Datum, Zeit 15. April 2019, 14:00-15:00
Ort HG G 43
Abstract In the 80's de work of de la LLave, Marco and Moriyon showed that for two dimensional Anosov diffeomorphisms, the marked Lyapunov spectrum determines the smooth isomorphism type of the system. Also in the 80's Otal and Croke showed that for negatively curved surface the marked length spectrum determines the isometry type of the surface. In this talk I will discuss new developments along this line of problems, discussing a more general framework where these theory can be developed. This project is joint with A. Gogolev.
Rigidity in hyperbolic systemsread_more
HG G 43
29. April 2019
14:00-15:00 		Dr. Davoud Cheragi
Imperial College London
Details

Ergodic theory and dynamical systems seminar

Titel Quasi-periodic dynamics in complex dimension one
Referent:in, Affiliation Dr. Davoud Cheragi, Imperial College London
Datum, Zeit 29. April 2019, 14:00-15:00
Ort HG G 43
Abstract Quasi-periodic dynamics in one complex variable reveals fascinating interplay between complex analysis and Diophantine approximations. The question of whether a nonlinear perturbation of a linear rotation is conjugate to a linear rotation (linearisation) dates back to more than a century ago, with remarkable contributions by C. Siegel, A. Brjuno, and J.-C. Yoccoz. The behaviour of non-linearisable maps is very complicated. Indeed, there is not a single example of a non-linearisable map whose local behaviour is completely understood. There is major recent advances on this problem which has lead to a complete description of the topological behaviour of typical orbits. This is an introductory talk to demonstrate some of these results.
Quasi-periodic dynamics in complex dimension oneread_more
HG G 43
6. Mai 2019
14:00-15:00 		Prof. Dr. Klaus Schmidt
University of Vienna
Details

Ergodic theory and dynamical systems seminar

Titel Homoclinic points of algebraic dynamical systems
Referent:in, Affiliation Prof. Dr. Klaus Schmidt, University of Vienna
Datum, Zeit 6. Mai 2019, 14:00-15:00
Ort HG G 43
Abstract An algebraic action of a countable discrete group $\Gamma $ is an action of $\Gamma $ by automorphisms of a compact abelian group $X$. Classical examples are $Z^d$-actions by (commuting) toral automorphisms, Ledrappier's Example, or analogous examples of shift-actions of $Z^2$ on closed, shift-invariant subgroup of $(R/Z)^{Z^2}$ (which may or may not have positive entropy). For groups bigger than $Z$, algebraic actions offer one of the key playgrounds for the study of positive entropy ergodic actions of such groups. For an algebraic action $lpha lon \gamma o lpha ^\gamma $ of a group $\Gamma $ on a compact abelian group $X$, a extit{homoclinic point} is a nonzero point $x\in X$ for which $lpha ^\gamma x$ converges to $0$ as $\gamma o\infty $. Such a point is 'summable' if $lpha ^\gamma x --> 0$ sufficiently fast as $\gamma --> \infty $. The existence of a summable homoclinic point implies positive entropy of the action, and for expansive actions the reverse implication holds under quite general conditions. For nonexpansive actions, the existence of summable homoclinic points is much more mysterious, but implies many nice dynamical properties such actions. This talk is based on results and examples by Martin G"{o}ll, Hanfeng Li, Doug Lind, Evgeny Verbitskiy, and KS.
Homoclinic points of algebraic dynamical systemsread_more
HG G 43
13. Mai 2019
14:00-15:00 		Dr. Jialun Li
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel Decrease of Fourier coefficients of Furstenberg measures and renewal theory
Referent:in, Affiliation Dr. Jialun Li, Universität Zürich
Datum, Zeit 13. Mai 2019, 14:00-15:00
Ort HG G 43
Abstract Let ? be a Borel probability measure on SL2(R) with a finite exponential moment, such that the support of ? generates a Zariski dense subgroup in SL2(R). We can define a unique probability measure on the circle, which is called the ? stationary measure or Furstenberg measure. We will prove, using Bourgain's discretized sum-product estimate, that the Fourier coefficients of this measure go to zero with a polynomial speed. Starting from this result, we can obtain a spectral gap of the transfer operator, whose properties enable us to get an exponential error term in the renewal theorem in the context of random products of matrices.
Decrease of Fourier coefficients of Furstenberg measures and renewal theoryread_more
HG G 43
20. Mai 2019
14:00-15:00 		Dr. Feng De-Jun
The Chinese University of Hong Kong
Details

Ergodic theory and dynamical systems seminar

Titel Dimension of invariant measures for affine iterated function systems
Referent:in, Affiliation Dr. Feng De-Jun, The Chinese University of Hong Kong
Datum, Zeit 20. Mai 2019, 14:00-15:00
Ort HG G 43
Abstract Iterated function system (IFS) is a broad scheme for generating fractal sets and measures. In this talk, we discuss the dimensional properties of certain fractal measures associated with affine IFS. We prove the exact dimensionality of ergodic stationary measures for any average-contractive affine IFS. Applications are given to the dimensions of self-affine sets, as well as their projections and slices.
Dimension of invariant measures for affine iterated function systemsread_more
HG G 43
24. Juni 2019
14:00-15:00 		Dr. Paolo Giulietti
Centro de Giorgi
Details

Ergodic theory and dynamical systems seminar

Titel On Non-Linear Pseudo Anosov map and parabolic flows
Referent:in, Affiliation Dr. Paolo Giulietti, Centro de Giorgi
Datum, Zeit 24. Juni 2019, 14:00-15:00
Ort HG G 19.1
Abstract I will present a work-in-progress which allows to study ergodic averages and cohomological equations related to parabolic dynamics by means of hyperbolic renormalizations, exploiting transfer operator techniques on anisotropic Banach spaces. I will focus on parabolic flows on surfaces with singularities that can be renormalized by nonlinear pseudo Anosov transformations. Time permitting, I will show how to apply our results to study averages of the Teichmuller flow. Joint work with M. Artigiani.
On Non-Linear Pseudo Anosov map and parabolic flowsread_more
HG G 19.1
15. August 2019
10:30-11:30 		Dr. Richard Aoun
American University of Beirut
Details

Ergodic theory and dynamical systems seminar

Titel Random matrix products without irreducibility
Referent:in, Affiliation Dr. Richard Aoun, American University of Beirut
Datum, Zeit 15. August 2019, 10:30-11:30
Ort HG G 43
Abstract Random matrix products theory has known a large spectrum of applications in the last few years: homogeneous dynamics, expander graphs, diophantine approximation... Roughly speak- ing, its goal is to study the behavior of a random walk on the general linear group GL(V ), where V is a finite dimensional vector space defined on any local field. In this talk, we present new results in this theory obtained in a joint work with Yves Guiv- arc’h. We consider a probability measure μ on the general linear group GL(V ) whose top Lyapunov exponent is simple and assume no other algebraic condition on its support. We will show that there exists a unique stationary probability measure on the projective space P(V ) corresponding to the top Lyapunov exponent. We also describe the new dynamics behind this setting and determine the support of the stationary measure in terms of μ.
Random matrix products without irreducibilityread_more
HG G 43

Hinweise: mit einem Stern gekennzeichnete Ereignisse (*) zeigen an, dass die Zeit und/oder der Ort von der üblichen Zeit und/oder dem üblichen Ort abweichen.

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