Tag - Dynamical systems

Slawomir Solecki: The dynamics and structure of transformation groups

17th January, 2023

This is a 24-lecture course, with each lecture being 75 minutes, given by Slawomir Solecki. Note that the 2nd lecture was not recorded. The other lectures might still be of significant interest, but this needs to be known.

This course focuses on the interaction between set theory, geometry, group theory, and dynamics. It will present parts of Rosendal’s Coarse Geometry of Topological Groups, Kechris-Pestov-Todorcevic’s Fraïssé Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups, as well as theory of Borel and measurable combinatorics.

Christian Lange: Orbifolds and Systolic Inequalities

13th January, 2023

In this talk, I will first discuss some instances in which orbifolds occur in geometry and dynamics, in particular, in the context of billiards and systolic inequalities. Then I will present topological conditions for an orbifold to be a manifold together with applications to foliations and to Besse geodesic and Reeb flows (joint work with Manuel Amann, Marc Kegel and Marco Radeschi). Here a flow is called Besse if all its orbits are periodic. Such flows are related to systolic inequalities. Namely, I will explain a characterization of contact forms on 3-manifolds whose Reeb flow is Besse as local maximizers of certain 'higher' systolic ratios, and mention other related systolic-like inequalities (joint work with Alberto Abbondandolo, Marco Mazzucchelli and Tobias Soethe).

Laura DeMarco: Lattès maps, bifurcations, and arithmetic

8th December, 2022

In the field of holomorphic dynamics, we learn that the Lattès maps - the rational functions on ℙ1 that are quotients of maps on elliptic curves - are rather boring. We can understand their dynamics completely. But viewed arithmetically, there are still unanswered questions. I'll begin the talk with some history of these maps. Then I'll describe one of the recent questions and how it has led to interesting complex-dynamical questions about other families of maps on ℙ1 and, in turn, new perspectives on the arithmetic side. The new material is a joint project with Myrto Mavraki.

Shreyasi Datta: S-arithmetic Diophantine Approximation

2nd December, 2022

Diophantine approximation deals with quantitative and qualitative aspects of approximating numbers by rationals. A major breakthrough by Kleinbock and Margulis in 1998 was to study Diophantine approximations for manifolds using homogeneous dynamics. After giving an overview of recent developments in this subject, I will talk about Diophantine approximation in the S-arithmetic set-up, where S is a finite set of valuations of

Krzysztof Fraczek: Deviation Spectrum of Ergodic Integrals for Locally Hamiltonian Flows on Surfaces

8th November, 2022

The talk will consist of a long historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration with Corinna Ulcigrai and Minsung Kim. New developments include a better understanding of the asymptotic of so-called error term (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime).

Camila Sehnem: Equilibrium on Toeplitz extensions of higher-dimensional non-commutative tori

1st April, 2022

The C∗-algebra generated by the left-regular representation of ℕn twisted by a 2-cocycle is a Toeplitz extension of an n-dimensional non-commutative torus, on which each vector r ∈ [0,∞)n determines a one-parameter subgroup of the gauge action. I will report on joint work with Z. Afsar, J. Ramagge and M. Laca, in which we show that the equilibrium states of the resulting C∗-dynamical system are parametrized by tracial states of the non-commutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of r. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Our results generalize the phase transition on the Toeplitz non-commutative tori used as building blocks in work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims.