Seminario de Sistemas Dinámicos

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.