Ergodic theory and dynamical systems

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Spring Semester 2024

Date / Time

Speaker

Title

Location

28 February 2024
13:30-14:30

Prof. Dr. Krzysztof Krupinski
Uniwersytetu Wroclawskiego

Event Details

Ergodic theory and dynamical systems seminar

Title	On some applications of model theory and topological dynamics
Speaker, Affiliation	Prof. Dr. Krzysztof Krupinski, Uniwersytetu Wroclawskiego
Date, Time	28 February 2024, 13:30-14:30
Location	HG G 19.1
Abstract	Model theory is a fast growing branch of mathematical logic with deep interactions with algebra, algebraic geometry, combinatorics, and, more recently, topological dynamics. I will focus on a few interactions with topological dynamics and applications to additive combinatorics. I will discus type-definable components of definable groups, which lead to model-theoretic descriptions of Bohr compactificatios of groups and rings, and also to so-called locally compact models of approximate subgroups and subrings which in turn are crucial to get structural or even classification results about approximate subgroups and subrings. I will discuss my result that each approximate subring has a locally compact model, and mention some structural applications. In contrast to approximate subrings, not every approximate subgroup has a locally compact model. However, Ehud Hrushovski showed that instead it has such a model in a certain generalized sense (with morphisms replaced by quasi-homomorphisms). In order to do that, he introduced and developed local logics and definability patterns. In my recent paper with Anand Pillay, we gave a shorter and simpler construction of a generalized locally compact model, based on topological dynamics methods in a model-theoretic context. I will briefly discuss it, if time permits.

On some applications of model theory and topological dynamicsread_more

HG G 19.1

6 March 2024
13:30-14:30

Dr. Hao Wu
Universität Zürich

Event Details

Ergodic theory and dynamical systems seminar

Title	A central limit theorem for irrational rotations of bounded type
Speaker, Affiliation	Dr. Hao Wu, Universität Zürich
Date, Time	6 March 2024, 13:30-14:30
Location	HG G 19.1
Abstract	In hyperbolic dynamical systems, one can often prove the spatial central limit theorem (CLT), where the starting point is randomized with respect to the SRB measures. In zero-entropy systems such as irrational rotations, the spatial CLT often fails due to lack of mixing properties. However, using coding and Markov chains, Bromberg and Ulcigrai showed that a temporal CLT holds for bounded type irrational rotations with step functions whose jump point lies in a full Hausdorff dimension set. Here "temporal" means that we randomise time while fixing the starting point. In an ongoing joint work with Bromberg and Ulcigrai, we extend this result from full Hausdorff dimension to full Lebesgue measure.

A central limit theorem for irrational rotations of bounded typeread_more

HG G 19.1

13 March 2024
13:30-14:30

Prof. Dr. Martin Leguil
École polytechnique

Event Details

Ergodic theory and dynamical systems seminar

Title	Rigidity of u-Gibbs measures for certain Anosov diffeomorphisms of the 3-torus.
Speaker, Affiliation	Prof. Dr. Martin Leguil, École polytechnique
Date, Time	13 March 2024, 13:30-14:30
Location	HG G 19.1
Abstract	We consider Anosov diffeomorphisms of the 3-torus $\mathbb{T}^3$ which admit a partially hyperbolic splitting $\mathbb{T}^3 = E^s \oplus E^c \oplus E^u$ whose central direction $E^c$ is uniformly expanded. We may consider the 2-dimensional unstable foliation $W^{cu}$ tangent to $E^c \oplus E^u$, but also the 1-dimensional strong unstable foliation $W^u$ tangent to $E^u$. The behavior of $W^{cu}$ is reasonably well understood; in particular, such systems have a unique invariant measure whose disintegrations along the leaves of $W^{cu}$ are absolutely continuous: the SRB measure. The behavior of $W^u$ is less understood; we can similarly consider the class of measures whose disintegrations along the leaves of $W^u$ are absolutely continuous, the so-called u-Gibbs measures. It is well-known that the SRB measure is u-Gibbs; conversely, in a joint work with S. Alvarez, D. Obata and B. Santiago, we show that in a neighborhood of conservative systems, if the strong bundles $E^s$ and $E^u$ are not jointly integrable, then there exists a unique u-Gibbs measure, which is the SRB measure.

Rigidity of u-Gibbs measures for certain Anosov diffeomorphisms of the 3-torus.read_more

HG G 19.1

13 March 2024
14:45-15:45

Prof. Dr. Anna Florio
Université Paris Dauphine-PSL

Event Details

Ergodic theory and dynamical systems seminar

Title	Birkhoff attractor of dissipative billiards
Speaker, Affiliation	Prof. Dr. Anna Florio, Université Paris Dauphine-PSL
Date, Time	13 March 2024, 14:45-15:45
Location	HG G 19.1
Abstract	In a joint work with Olga Bernardi and Martin Leguil, we study the dynamics of dissipative convex billiards. In these billiards, the usual elastic reflection law is replaced with a new law where the angle bends towards the normal after each collision. For such billiard dynamics there exists a global attractor; we are interested in the topological and dynamical complexity of an invariant subset of this attractor, the so-called Birkhoff attractor, whose study goes back to Birkhoff, Charpentier, and, more recently, Le Calvez. We show that for a generic convex table, on one hand, the Birkhoff attractor is simple, i.e., a normally contracted submanifold, when the dissipation is strong; while, on the other hand, the Birkhoff attractor is topologically complicated and presents a rich dynamics when the dissipation is mild.

Birkhoff attractor of dissipative billiardsread_more

HG G 19.1

20 March 2024
13:30-14:30

Dr. Yi Pan
Institute of Science and Technology Austria

Event Details

Ergodic theory and dynamical systems seminar

Title	Reducibility of quasi-periodic cocycles valued in symplectic groups
Speaker, Affiliation	Dr. Yi Pan, Institute of Science and Technology Austria
Date, Time	20 March 2024, 13:30-14:30
Location	HG G 19.1
Abstract	Reducibility of quasi-periodic cocyles valued in symplectic groups is related to the spectrum of discrete Schrödinger operators on strips. We will talk about a global reducibility result: given one parameter family of such cocycles, for almost every parameter, either the maximal Lyapunov exponent is positive, or the cocycle is almost conjugate to some precise model. The techniques include Kotani thoery, KAM theory and in particular study of hyperbolicity of renormalization operator. This is a joint work with Artur Avila and Raphaël Krikorian.

Reducibility of quasi-periodic cocycles valued in symplectic groupsread_more

HG G 19.1

27 March 2024
13:30-14:30

Pedram Safaee
Universität Zürich

Event Details

Ergodic theory and dynamical systems seminar

Title	Nondegeneracy of the spectrum of the twisted Cocycle for interval exchange transformations
Speaker, Affiliation	Pedram Safaee, Universität Zürich
Date, Time	27 March 2024, 13:30-14:30
Location	HG G 19.1
Abstract	We will introduce the concept of twisted Birkhoff sum and the twisted Cocycle. We give some motivations for studying this Cocycle and discuss the positivity of the top Lyapunov exponent. This is based on joint work with Hesam Rajabzadeh.

Nondegeneracy of the spectrum of the twisted Cocycle for interval exchange transformationsread_more

HG G 19.1

10 April 2024
13:30-14:30

Prof. Dr. Carlos Matheus Silva Santos
CNRS

Event Details

Ergodic theory and dynamical systems seminar

Title	Non-conical strictly convex divisible sets are maximally anisotropic
Speaker, Affiliation	Prof. Dr. Carlos Matheus Silva Santos, CNRS
Date, Time	10 April 2024, 13:30-14:30
Location	HG G 19.1
Abstract	Let U be a non-conical strictly convex divisible set. Even though the boundary S of U is not C^2, Benoist showed that S is C^1+ and Crampon established that S has a sort of anisotropic Holder regularity -- described by a list L of real numbers -- at almost all of its points. In this talk, we discuss our joint work with P. Foulon and P. Hubert showing that S is maximally anisotropic in the sense that the list L contains no repetitions thanks to the features of the Hilbert flow.

Non-conical strictly convex divisible sets are maximally anisotropicread_more

HG G 19.1

17 April 2024
13:30-14:30

Prof. Dr. Sébastien Biebler
Université Paris Cité

Event Details

Ergodic theory and dynamical systems seminar

Title	Non-density of hyperbolicity in complex dynamics in several variables
Speaker, Affiliation	Prof. Dr. Sébastien Biebler, Université Paris Cité
Date, Time	17 April 2024, 13:30-14:30
Location	HG G 19.1
Abstract	One of the main goals in the theory of dynamical systems is to describe the dynamics of a "typical" map. For instance, in the case of diffeomorphisms of a given manifold, it was conjectured by Smale in the 60s that uniform hyperbolicity was generically satisfied. This hope was however fast discouraged by exhibiting dynamical systems displaying in a robust way dynamical configurations which are obstructions to hyperbolicity: robust homoclinic tangencies (this is the so-called Newhouse phenomenon) and robust heterodimensional cycles. In this talk, I will explain these phenomena and their extensions to the complex setting. In particular, I will show how to construct robust heterodimensional cycles in the family of polynomial automorphisms of C^3. The main tool is the notion of blender coming from real dynamics.

Non-density of hyperbolicity in complex dynamics in several variablesread_more

HG G 19.1

24 April 2024
13:30-14:30

Prof. Dr. Andreas Strömbergsson
Uppsala University

Event Details

Ergodic theory and dynamical systems seminar

Title	An effective equidistribution result in the space of 2-dimensional tori with k marked points
Speaker, Affiliation	Prof. Dr. Andreas Strömbergsson, Uppsala University
Date, Time	24 April 2024, 13:30-14:30
Location	HG G 19.1
Abstract	Let X be the homogeneous space Gamma \ G, where G is the semidirect product of SL(2,R) and a direct sum of k copies of R^2, and where Gamma is the subgroup of integer elements in G. I will present a result giving effective equidistribution of 1-dimensional unipotent orbits in the space X. The proof makes use of the delta method in the form developed by Heath-Brown. Joint work with Anders Södergren and Pankaj Vishe.

An effective equidistribution result in the space of 2-dimensional tori with k marked pointsread_more

HG G 19.1

8 May 2024
13:30-14:30

Prof. Dr. Weikun He
Chinese Academy of Sciences

Event Details

Ergodic theory and dynamical systems seminar

Title	Quantitative equidistribution of random walks on SL_2(R)/SL_2(Z)
Speaker, Affiliation	Prof. Dr. Weikun He, Chinese Academy of Sciences
Date, Time	8 May 2024, 13:30-14:30
Location	HG G 19.1
Abstract	We consider random walks on the homogeneous space SL_2(R)/SL_2(Z) induced by the action of a Zariski-dense subgroup consisting of matrices with algebraic entries. I will present a recent joint work with Timothée Bénard where we showed the following. The random walk equidistributes in law unless it is trapped in a finite orbit. Moreover, the convergence is fast unless the starting point is close to a finite orbit or high in the cusp.