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Ergodic theory and dynamical systems

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Frühjahrssemester 2024

Datum / Zeit Referent:in Titel Ort
28. Februar 2024
13:30-14:30 		Prof. Dr. Krzysztof Krupinski
Uniwersytetu Wroclawskiego
Details

Ergodic theory and dynamical systems seminar

Titel On some applications of model theory and topological dynamics
Referent:in, Affiliation Prof. Dr. Krzysztof Krupinski, Uniwersytetu Wroclawskiego
Datum, Zeit 28. Februar 2024, 13:30-14:30
Ort 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. März 2024
13:30-14:30 		Dr. Hao Wu
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel A central limit theorem for irrational rotations of bounded type
Referent:in, Affiliation Dr. Hao Wu, Universität Zürich
Datum, Zeit 6. März 2024, 13:30-14:30
Ort 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. März 2024
13:30-14:30 		Prof. Dr. Martin Leguil
École polytechnique
Details

Ergodic theory and dynamical systems seminar

Titel Rigidity of u-Gibbs measures for certain Anosov diffeomorphisms of the 3-torus.
Referent:in, Affiliation Prof. Dr. Martin Leguil, École polytechnique
Datum, Zeit 13. März 2024, 13:30-14:30
Ort 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. März 2024
14:45-15:45 		Prof. Dr. Anna Florio
Université Paris Dauphine-PSL
Details

Ergodic theory and dynamical systems seminar

Titel Birkhoff attractor of dissipative billiards
Referent:in, Affiliation Prof. Dr. Anna Florio, Université Paris Dauphine-PSL
Datum, Zeit 13. März 2024, 14:45-15:45
Ort 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. März 2024
13:30-14:30 		Dr. Yi Pan
Institute of Science and Technology Austria
Details

Ergodic theory and dynamical systems seminar

Titel Reducibility of quasi-periodic cocycles valued in symplectic groups
Referent:in, Affiliation Dr. Yi Pan, Institute of Science and Technology Austria
Datum, Zeit 20. März 2024, 13:30-14:30
Ort 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. März 2024
13:30-14:30 		Pedram Safaee
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel Nondegeneracy of the spectrum of the twisted Cocycle for interval exchange transformations
Referent:in, Affiliation Pedram Safaee, Universität Zürich
Datum, Zeit 27. März 2024, 13:30-14:30
Ort 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
Details

Ergodic theory and dynamical systems seminar

Titel Non-conical strictly convex divisible sets are maximally anisotropic
Referent:in, Affiliation Prof. Dr. Carlos Matheus Silva Santos, CNRS
Datum, Zeit 10. April 2024, 13:30-14:30
Ort 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
Details

Ergodic theory and dynamical systems seminar

Titel Non-density of hyperbolicity in complex dynamics in several variables
Referent:in, Affiliation Prof. Dr. Sébastien Biebler, Université Paris Cité
Datum, Zeit 17. April 2024, 13:30-14:30
Ort 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
Details

Ergodic theory and dynamical systems seminar

Titel An effective equidistribution result in the space of 2-dimensional tori with k marked points
Referent:in, Affiliation Prof. Dr. Andreas Strömbergsson, Uppsala University
Datum, Zeit 24. April 2024, 13:30-14:30
Ort 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. Mai 2024
13:30-14:30 		Prof. Dr. Weikun He
Chinese Academy of Sciences
Details

Ergodic theory and dynamical systems seminar

Titel Quantitative equidistribution of random walks on SL_2(R)/SL_2(Z)
Referent:in, Affiliation Prof. Dr. Weikun He, Chinese Academy of Sciences
Datum, Zeit 8. Mai 2024, 13:30-14:30
Ort 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.
Quantitative equidistribution of random walks on SL_2(R)/SL_2(Z)read_more
HG G 19.1
15. Mai 2024
13:30-14:30 		Dr. Homin Lee
Northwestern University
Details

Ergodic theory and dynamical systems seminar

Titel Absolute continuity of stationary measure
Referent:in, Affiliation Dr. Homin Lee, Northwestern University
Datum, Zeit 15. Mai 2024, 13:30-14:30
Ort HG G 19.1
Abstract In this talk, we discuss about the smooth random dynamical systems on surfaces. Based on the measure rigidity work by Aaron Brown and Federico Rodriguez Hertz, we know that a stationary measure has SRB property unless there is a certain obstruction. Here, SRB property implies that, morally, the measure is 'absolutely continuous' along 1 dimensional piece. In this talk, we consider about a different mechanism to get 'measure rigidity' which promote SRB to absolute continuity using 'transversality' motivated by Tsujii's works on partially hyperbolic endomorphisms and Bernoulli convolution. This is a work in progress with Aaron Brown, Davi Obata, and Yuping Ruan.
Absolute continuity of stationary measureread_more
HG G 19.1
22. Mai 2024
13:30-14:30 		Dr. Mikolaj Fraczyk
Jagiellonian University
Details

Ergodic theory and dynamical systems seminar

Titel Higher rank rigidity and discrete subgroups of semisimple Lie groups
Referent:in, Affiliation Dr. Mikolaj Fraczyk, Jagiellonian University
Datum, Zeit 22. Mai 2024, 13:30-14:30
Ort HG G 19.1
Abstract I will talk about a new strategy of studying discrete subgroup of a higher rank semisimple Lie group using the high and low entropy methods for actions of higher rank abelian groups. They were developed by Einsiedler-Katok-Lindenstrauss to study A-invariant measures on (typically finite volume) quotients of semisimple Lie groups. The properties of such measures can reveal a lot about the geometry of the quotient, wich in turn makes them a useful tool in studying (a priori) infinite covolume subgroups of semisimple Lie groups. Based on a joint work with Minju Lee.
Higher rank rigidity and discrete subgroups of semisimple Lie groupsread_more
HG G 19.1
22. Mai 2024
13:30-14:30 		Prof. Dr. Harald Helfgott
CNRS
Details

Ergodic theory and dynamical systems seminar

Titel Expansion, divisibility and parity
Referent:in, Affiliation Prof. Dr. Harald Helfgott, CNRS
Datum, Zeit 22. Mai 2024, 13:30-14:30
Ort HG G 19.1
Abstract We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining the main result with Matomaki-Radziwill. (This is joint work with M. Radziwill.) For instance: for lambda the Liouville function (that is, the completely multiplicative function with lambda(p) = -1 for every prime), (1/\log x) \sum_{n\leq x} lambda(n) \lambda(n+1)/n = O(1/sqrt(log log x)), which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that lambda(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Omega(n)=k, for any "popular" value of k (that is, k = log log N + O(sqrt(log log N)) for n<=N). We shall also discuss a recent generalization by C. Pilatte, who has succeeded in proving that a graph with edges that are rough integers, rather than primes, also has a strong local expander property almost everywhere, following the same strategy. As a result, he has obtained a bound with O(1/(log x)^c) instead of O(1/sqrt(log log x)) in the above, as well as other improvements in consequences across the board.
Expansion, divisibility and parityread_more
HG G 19.1
29. Mai 2024
13:30-14:30 		Magali Jay
Aix-Marseille Université
Details

Ergodic theory and dynamical systems seminar

Titel Tiling billiard in the wind-tree model
Referent:in, Affiliation Magali Jay, Aix-Marseille Université
Datum, Zeit 29. Mai 2024, 13:30-14:30
Ort HG G 19.1
Abstract In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and Eaton lenses. The three of them are motivated by physics. The wind-tree model was instoduced by Paul and Tatyana Ehrenfest to study a gaz: a particule is moving in a plane where obstacles are periodically placed, on which the particule bounces. The Eaton lenses are a periodic array of lenses in the plane, in which we consider a light ray that is reflected each time it crosses a lens. In the beginning of the 2000's, physicists have conceived metamaterials with negative index of refraction. Tilling billiards' trajectories consist of light rays moving in a arrangement of metamaterials with opposite indew of refraction. After having introduced these dynamical systems, I will consider a mix of them: an arrangement of rectangles in the plane, like in the wind-tree model, but made of metamaterials, like for tiling billiards. I study the trajectories of light in this plane. They are refracted each time they cross a rectangle. I show that these trajectories are traped in a strip, for almost every parameter. This behavior is similar to the one of Eaton lenses.
Tiling billiard in the wind-tree modelread_more
HG G 19.1

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