Seminario de Sistemas Dinámicos

Consider a Hamiltonian H in n degrees of freedom, admitting functionally independent Poisson commuting first integrals J_1,..,J_k. When k=n, this system is called Arnold Liouville integrable, and on compact common levels, the dynamic is quasi periodic. When k<n, let us further assume that the dynamical system restricted to the smooth common zero level S of these first integrals admit additional first integrals K_1,..,K_{n-k}. We will show that the property ``Poisson commuting'' can be properly defined for these additional first integrals. What can be said about the dynamic on S?

We will prove that the system is always integrable by quadrature. For k=1, the compact common levels are always tori, and under a Diophantine condition, the motion is quasi periodic. For k=2, n=3, we will prove that a compact common level N is an orientable torus bundle over a circle. It is either a 3-torus, a nilmanifold, or there exists an additional first integral whose level are 2-tori. In the 3 torus case, under a Diophantine condition, the motion is quasi periodic, in the 2 torus case, the dynamic is quasi periodic but with non global action angles coordinates over N. In the nilmanifold case, the system has a parabolic dynamic on it. We will then present the construction explicit of examples and obstructions.

Magnetic systems are a well-known class of classical Hamiltonian systems, serving as a fundamental model for the motion of a charged particle on a Riemannian manifold under the influence of a magnetic force. They were first formalized by V. Arnold in the 1980s, and in recent decades,  they have been actively investigated by numerous mathematical communities. Through many examples, this talk aims to provide a straightforward overview of both classical and recent developments in the theory. A particular focus will be placed on the new concept of magnetic curvature, and, if time permits, I will also present some new applications that can be derived from it. The presentation is designed to be accessible and requires no prior background, making it suitable for audiences of all types.

The aim of this talk is to provide an overview of some of the main results concerning scaling limits of random compact metric spaces, with a particular emphasis on random trees and random surfaces. We will focus especially on the geometric aspects of the theory. In this context, we will outline the main ideas behind the convergence of large random planar maps with large faces, obtained in recent joint work with Nicolas Curien (Orsay) and Grégory Miermont (ENS Lyon).

No specific prerequisites are required for this talk.

Given a topological dynamical system (X,T), where X is a compact metric space and T is a continuous self-map on X, the Ellis semigroup E(X,T) is the pointwise closure of {T^i : i ≥ 0}. In this talk, I will give a gentle introduction to the Ellis semigroup. My aim is to convince everyone of why it matters and why it’s interesting.

There is one aspect of the Ellis semigroup that doesn’t need much convincing: the fact that it is not particularly tractable. We will discuss a situation in which this is especially the case; specifically, we will talk about non-tameness. I will present a very hands-on new criterion for non-tameness obtained in [1]. If time permits, we will look at an application of this criterion in the context of so-called Floyd-Auslander systems.

In any case, the talk will be accessible and no prior knowledge of the Ellis semigroup or Floyd-Auslander systems is expected.

[1] G. Fuhrmann and C. Liu, Idempotents in the Ellis semigroup of Floyd–
Auslander systems, preprint, arXiv:2512.13341 [math.DS], 2025.

When the injectivity radius of the surface is finite, it is known that horocycle trajectories are closed or have non minimal closure, except if the surface is "convex-cocompact". 

If we add the condition that the injectivity radius is >0, then all ergodic measure m, invariant by the horocycle flow are quasi-invariant ( i-e the image of m by any time of the geodesic flow is absolutely continuous with respect to m).

In this talk, I will explain how to construct a hyperbolic surface admitting a non trivial minimal set for the horocyclic flow and a conservative and ergodic invariant measure which is not quasi-invariant.