This a scientific meeting gathering researchers, PhD students, master students and undergraduate students about the recent advances in quantitative recurrence for dynamical systems. There will be two talks with plenty of discussion.

Schedule:

Mubarak Muhammad 14:00-14:30 Title: Trimmed sums for slowly mixing systems

Discussion period: 14:30-15:00

Coffee Break: 15:00-15:30

Duarte Sá Pinho: 15:30 - 16:00 Title: Extremal index for bidimensional systems

Discussion period: 16:30-16:30

In this talk I will talk about some new results for the escape and
hitting statistics for various open dynamical systems. This includes
1. Poisson limit laws for arbitrary slow mixing hyperbolic billiards,
a connection to RH will be presented too
2. polynomial and exponential escape rates, and
3. where orbits prefer going in the phase space of nonuniformly
expanding dynamical systems. I will outline the idea of the proof for
it, which uses operator renewal theory and generalized Keller-Liverani
perturbation theory.

The game of Rock-Paper-Scissors is an instructive example of cyclic competition between competing populations or strategies in evolutionary biology and game theory, where no single species is an overall winner. Mathematically, this cyclic behaviour can be modelled by ordinary differential equations containing heteroclinic cycles: sequences of equilibria along with trajectories that connect them in a cyclic manner. In simple examples, the equilibria are all similar to each other, in that they all have the same number of non-zero components.

Oscillator networks, including neuronal ensembles, can exhibit multiple cooperative rhythms such as chimera and cluster states. However, understanding which rhythm prevails remains challenging. In this talk, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. We show that three-cluster splay states with two distinct, coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units.

The unipotent flow on the unit tangent bundle of the modular surface is a classic example of a homogeneous flow when understood through the identification with P SL2(R)/P SL2(Z). The ergodicity of the flow implies that almost every orbit is dense in the space and hence must eventually make excursions deeper and deeper into the cusp. We are interested in understanding the nature of these excursions.

We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild
assumptions we prove the existence of a unique invariant mixing absolutely continuous
probability measure, study its rate of decay of correlation and prove a number of limit
theorems.