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Ergodic theory and dynamical systems

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Frühjahrssemester 2023

Datum / Zeit Referent:in Titel Ort
27. Februar 2023
13:30-14:30 		Prof. Dr. Stefano Marmi
Scuola Normale Superiore
Details

Ergodic theory and dynamical systems seminar

Titel Brjuno functions, Hölder continuity and modular forms
Referent:in, Affiliation Prof. Dr. Stefano Marmi, Scuola Normale Superiore
Datum, Zeit 27. Februar 2023, 13:30-14:30
Ort HG G 43
Abstract In 1988 Yoccoz proved that the size of the stability domain (Siegel disk) around an irrationally indifferent fixed point in the complex plane is given by a purely arithmetic function--called Brjuno's function--up to a more regular (L^\infty) correction. The Hölder interpolation conjecture (aka Marmi-Moussa-Yoccoz conjecture) states that for quadratic polynomials this correction is in fact 1/2-Hölder continuous. An analogous version of the conjecture stands also for other dynamical systems, including the standard family. Hölder continuity seems to be the relevant regularity for these problems also since it measures the difference between formulations of the arithmetical function corresponding to different continued fraction algorithms (Gauss, nearest integer, by-excess, ...) Surprisingly finally, very similar functions are used to study the convergence of trigonometric sums involving the divisor function, as discovered by Wilton almost a century ago, and also for the study of the differentiability properties of integrals of modular forms. The talk will include recent work in collaboration with Seul Bee Lee, Izabela Petrykiewicz and Tanja Schindler: https://arxiv.org/abs/2111.13553 and https://arxiv.org/abs/2111.10807
Brjuno functions, Hölder continuity and modular formsread_more
HG G 43
6. März 2023
13:30-14:30 		Prof. Dr. Klaus Schmidt
University of Vienna
Details

Ergodic theory and dynamical systems seminar

Titel Generators and symbolic representations of algebraic group actions (joint work with Hanfeng Li)
Referent:in, Affiliation Prof. Dr. Klaus Schmidt, University of Vienna
Datum, Zeit 6. März 2023, 13:30-14:30
Ort HG G 43
Abstract I will present natural symbolic representations of intrinsically ergodic, but not necessarily expansive, principal algebraic actions of countably infinite amenable groups and use these representations to find explicit generating partitions (up to null-sets) for such actions. This is joint work with Hanfeng Li.
Generators and symbolic representations of algebraic group actions (joint work with Hanfeng Li)read_more
HG G 43
13. März 2023
13:30-14:30 		Timothée Bénard
University of Cambridge
Details

Ergodic theory and dynamical systems seminar

Titel Limit theorems on nilpotent Lie groups
Referent:in, Affiliation Timothée Bénard, University of Cambridge
Datum, Zeit 13. März 2023, 13:30-14:30
Ort HG G 43
Abstract I will talk about my recent work with E. Breuillard establishing limit theorems for random walks on nilpotent Lie groups. Most previous works assumed the law of increment to be centered in the abelianization of the group. Our major contribution is to allow the law of increment to be non-centered. In this case, new phenomena appear: the large scale geometry of the walk depends on the increment average, and the limiting measure in the central limit theorem may not have full support in the group.
Limit theorems on nilpotent Lie groupsread_more
HG G 43
20. März 2023
13:30-14:30 		Dr. Jonguk Yang
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel A priori bounds for unimodal diffeomorphisms in dimension two
Referent:in, Affiliation Dr. Jonguk Yang, Universität Zürich
Datum, Zeit 20. März 2023, 13:30-14:30
Ort HG G 43
Abstract One of the most fundamental examples of non-linear dynamics is given by the class of unimodal interval maps. It is the simplest setting in which one can study the behavior of a critical orbit and the profound impact it has on the geometry of the system. By the works of Sullivan, McMullen and Lyubich, we have a complete renormalization theory for these maps, and as a result, their dynamics is now very well understood. In this talk, we discuss the extension of this theory to a higher dimensional setting--namely, to properly dissipative diffeomorphisms in dimension two. Using the notion of non-uniform partial hyperbolicity, we identify what it means for such maps to be "unimodal." Then we show that properly dissipative infinitely renormalizable unimodal diffeomorphisms have a priori bounds (a certain uniform control on their geometry that holds at arbitrarily small scales). This is based on a joint work with S. Crovisier, M . Lyubich and E. Pujals.
A priori bounds for unimodal diffeomorphisms in dimension tworead_more
HG G 43
27. März 2023
13:30-14:30 		Prof. Dr. Alex Eskin
University of Chicago
Details

Ergodic theory and dynamical systems seminar

Titel A homogeneous dynamics approach to Gibbs u-states
Referent:in, Affiliation Prof. Dr. Alex Eskin, University of Chicago
Datum, Zeit 27. März 2023, 13:30-14:30
Ort HG G 43
Abstract Gibbs u-states are a distinguished class of measures invariant by a partially hyperbolic diffeomorphism of a manifold. These measures and the related SRB measures have been studied extensively in various situations. There are some preliminary indications that homogeneous dynamics methods are relevant to this circle of questions. I will try to sketch this connection.
A homogeneous dynamics approach to Gibbs u-statesread_more
HG G 43
3. April 2023
13:30-14:30 		Dr. Charlene Kalle
Leiden University
Details

Ergodic theory and dynamical systems seminar

Titel Random intermittent dynamics
Referent:in, Affiliation Dr. Charlene Kalle, Leiden University
Datum, Zeit 3. April 2023, 13:30-14:30
Ort HG G 43
Abstract Intermittent dynamics, where systems irregularly alternate between long periods of different types of dynamical behaviour, has been studied since the work of Pomeau and Manneville in 1980. In random dynamical systems this phenomenon has only been well understood in a few specific cases. A random dynamical system consists of a family of deterministic systems, one of which is chosen to be applied at each time step according to some probabilistic rule. In this talk we will describe the intermittency of some families of random systems with a particular emphasis on how the intermittency of the random system depends on the intermittency of the underlying deterministic systems. This talk is based on joint works with Ale Jan Homburg, Tom Kempton, Valentin Matache, Marks Ruziboev, Masato Tsujii, Evgeny Verbitskiy and Benthen Zeegers.
Random intermittent dynamicsread_more
HG G 43
21. April 2023
13:30-14:30 		Dr. Frank Trujillo
Universität Zürich
Details

Ergodic theory and dynamical systems seminar

Titel On invariant measures of circle maps with breaks
Referent:in, Affiliation Dr. Frank Trujillo, Universität Zürich
Datum, Zeit 21. April 2023, 13:30-14:30
Ort HG G 43
Abstract By a classical theorem of Denjoy, any sufficiently regular piece-wise smooth circle homeomorphism with finitely many branches (often called a circle homeomorphism with breaks) and irrational rotation number is topologically conjugated to an irrational circle rotation. In particular, it admits a unique invariant probability measure. We will discuss dimensional properties of this measure and show that, generically, this unique invariant probability measure has zero Hausdorff dimension. To encode this generic condition, we consider piece-wise smooth homeomorphisms as generalized interval exchange transformations of the interval and rely on the notion of combinatorial rotation number.
On invariant measures of circle maps with breaksread_more
HG G 43
24. April 2023
13:30-14:30 		Dr. Frédéric Paulin
Université Paris-Saclay
Details

Ergodic theory and dynamical systems seminar

Titel Partial equidistribution of Farey rays in negative curvature
Referent:in, Affiliation Dr. Frédéric Paulin, Université Paris-Saclay
Datum, Zeit 24. April 2023, 13:30-14:30
Ort HG G 43
Abstract In the unit tangent bundle of a finite volume Riemannian manifold with negative curvature, a closed strong unstable leaf pushed by the geodesic flow equidistributes towards the maximal entropy measure. Fixing a family of discrete points with geometric origin (intersection with divergent orbits of the geodesic flow) on these unstable leaves, and having care of taking neither too many nor too few points (using a prescribed density), we prove that the family of points equidistributes towards a measure supported on a truncated weak stable leaf. We give arithmetic applications by varying arithmetic hyperbolic manifolds. This is a joint work with Jouni Parkkonen.
Partial equidistribution of Farey rays in negative curvatureread_more
HG G 43
8. Mai 2023
13:30-14:30 		Dr. Prasuna Bandi
IHES
Details

Ergodic theory and dynamical systems seminar

Titel Hausdorff dimension of exact sets in Ahlfors regular spaces
Referent:in, Affiliation Dr. Prasuna Bandi, IHES
Datum, Zeit 8. Mai 2023, 13:30-14:30
Ort HG G 43
Abstract In Diophantine approximation, it is a classical problem to determine the size of the sets related to \(\psi\) approximable set for a given non-increasing function \(\psi\). Bugeaud determined the Hausdorff dimension of the exact \(\psi\) approximable set answering a question posed by Beresnevich, Dickinson, and Velani. We compute the Hausdorff dimension of the exact set in the general setup of Ahlfors regular spaces. Our result applies to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces. This is joint work with Anish Ghosh and Debanjan Nandi.
Hausdorff dimension of exact sets in Ahlfors regular spacesread_more
HG G 43
15. Mai 2023
13:30-14:30 		Prof. Dr. Viviane Baladi
CNRS, Sorbonne Université & ITS-ETHZ
Details

Ergodic theory and dynamical systems seminar

Titel A parameter almost sure invariance principle (ASIP) for the quadratic family
Referent:in, Affiliation Prof. Dr. Viviane Baladi, CNRS, Sorbonne Université & ITS-ETHZ
Datum, Zeit 15. Mai 2023, 13:30-14:30
Ort HG G 43
Abstract After recalling what is an ASIP and how it can appear in dynamics, we will discuss and motivate the following joint result with Magnus Aspenberg and Tomas Persson: Consider the quadratic family \(T_a(x) = a x (1 - x)\), for \(x\) in \([0, 1]\) and parameters a in \((2,4)\). For any transversal Misiurewicz parameter b, we find a positive measure subset Omega of mixing Collet-Eckmann parameters such that for any Holder function f with nonvanishing autocorrelation for b, the functions \(f_a(T_a^{k}(1/2))\) (where \(f_a\) is a suitable normalisation of \(f\)) for the normalised Lebesgue measure on a positive measure subset of Omega (depending on \(f\)) satisfy an ASIP.
A parameter almost sure invariance principle (ASIP) for the quadratic familyread_more
HG G 43
22. Mai 2023
13:30-14:30 		Dr. Natalia Jurga
University of St Andrews
Details

Ergodic theory and dynamical systems seminar

Titel Hausdorff dimension of the Rauzy gasket
Referent:in, Affiliation Dr. Natalia Jurga, University of St Andrews
Datum, Zeit 22. Mai 2023, 13:30-14:30
Ort HG G 43
Abstract The Rauzy gasket is a fractal subset of the two dimensional simplex which is an important subset of parameter space in numerous dynamical and topological problems. Arnoux conjectured that the Hausdorff dimension of the Rauzy gasket is strictly less than 2, and since then there has been considerable interest in computing its Hausdorff dimension. In this talk we will also discuss a natural class of measures supported on the Rauzy gasket (the stationary measures for the projective action of the generators of the Rauzy gasket). We will show how recent developments from the theory of self-affine sets can be adapted to compute the dimension of these measures, which will allow us to establish an exact value for the Hausdorff dimension of the Rauzy gasket.
Hausdorff dimension of the Rauzy gasketread_more
HG G 43
30. Mai 2023
13:30-14:30 		Prof. Dr. Daniel Smania
Instituto de Ciências Matemáticas e de Computação USP
Details

Ergodic theory and dynamical systems seminar

Titel Deformations of one-dimensional dynamical systems
Referent:in, Affiliation Prof. Dr. Daniel Smania, Instituto de Ciências Matemáticas e de Computação USP
Datum, Zeit 30. Mai 2023, 13:30-14:30
Ort HG G 43
Abstract Perhaps one of the main features of one-dimensional dynamics (either real or complex) is that the theory of deformations is rich. By this we mean that the topological classes of such maps often are infinite dimensional manifolds, but with finite codimension. They are kind of "almost" structurally stable! Moreover for smooth families of maps inside a given topological class the associated family of conjugacies also moves in a smooth way. There are various applications in the study of renormalisation theory and linear response theory. There is a nice theory in complex dynamics but for real maps with finite smoothness on the interval our current understanding is far behind the complex setting. We will discuss recent developments obtained in joint work with Clodoaldo Ragazzo but also some results with Viviane Baladi and Amanda de Lima. Ergodic theory will be a crucial tool.
Deformations of one-dimensional dynamical systemsread_more
HG G 43
5. Juni 2023
13:30-14:30 		Dr. Thibault Lefeuvre
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Details

Ergodic theory and dynamical systems seminar

Titel Ergodicity of frame flows in low-rank
Referent:in, Affiliation Dr. Thibault Lefeuvre, Institut de Mathématiques de Jussieu-Paris Rive Gauche
Datum, Zeit 5. Juni 2023, 13:30-14:30
Ort HG G 43
Abstract The aim of this talk is to study the ergodicity of a class of partially hyperbolic flows constructed by taking unitary extensions to vector bundles of the geodesic flow over a closed negatively-curved manifold. We will show that, when the rank of the vector bundle is small compared to the dimension of the manifold, ergodicity can be completely characterized in geometric terms -- a certain connection admits no holonomy reduction. The proof combines classical hyperbolic dynamics, harmonic analysis, and, more surprisingly, real algebraic geometry (classification of polynomial maps between sphere). Joint work with Mihajlo Cekic.
Ergodicity of frame flows in low-rankread_more
HG G 43

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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