Given a topologically mixing shift on a countable alphabet and a potential, we give criteria for the system to have exponential mixing.  That is, criteria for the potential to have an equilibrium state which also has exponential decay of correlations.  The first condition is that the potential should have Birkhoff averages on periodic points bounded away from its pressure.  The second is that we control the entropy at infinity.  Both conditions are sharp (in fact under the second condition, the first is both necessary and sufficient).  I will present this joint work with Boyuan Zhao using s

Using a deterministic perturbation result established by Galatolo and Lucena [1], we obtain quantitative estimates on the continuity of the invariant densities and entropies of the physical measures for a one parameter family of piecewise expanding maps. We apply these results to a family of two-dimensional tent maps.

[1] S. Galatolo, R. Lucena, Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps, Discrete and Continuous Dynamical Systems 40, 3 (2020), 1309--1360

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:

Romain Aimino 14:30-15:10 Title: Records for dynamical systems

Discussion period: 15:10-15:30

Coffee Break: 15:30-16:00

Jorge Freitas: 16:00 - 16:45

Discussion period: 16:45-17:00

In this talk, we will investigate the question of quantitative recurrence for ergodic dynamical systems. By fixing a set of small measure in phase space, we study the law of successive return times to this target as the measure tends to zero. When the invariant measure is finite and the system is sufficiently mixing, it is known that the limit law obtained for natural targets (typically balls or cylinders) is the Poisson process. In this presentation, we will focus on the case where the invariant measure is infinite.

The specification property, introduced in the 1970s by Bowen in the study of Axiom A diffeomorphisms, is a fundamental tool in dynamical systems and it is closely related to chaotic behavior and rich ergodic properties. Although important classes of systems have the specification property, several relevant classes of systems do not have it, and therefore, weaker notions we introduced, such as the weak and almost specification properties.

I will discuss the outcome  of  periodic perturbations of attracting cyclic dynamics.  
The system to be perturbed may be either a periodic orbit, a heteroclinic cycle or a flow-invariant torus. 
We look for frequency locked solutions that return after an integer multiple of the period of the perturbation.
The analysis consists in reducing to discrete-time dynamics on a cylinder and the golden number has a surprise participation.

This is joint work with Alexandre Rodrigues (ISEG, U. Lisboa).

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.