In this talk, we present an overview of modern Iterated Function System (IFS) theory, highlighting its geometric, topological, and dynamical aspects, along with several recent developments in the field. We then discuss applications to Thermodynamical Formalism, focusing on holonomic measures and results obtained in recent collaborations with R. R. Souza, A. O. Lopes, J. K. Mengue, among others.

The classical Livsic theorem is a simple and useful result for Anosov diffeomorphism (or flows) which shows that a smooth function is trivial (i.e., a coboundary) if the sum (or integral) of the function vanishes around every closed orbit. In 2024, Gogolev and F. Rodriguez Hertz gave a generalisation of the result to lifts of Anosov flows to manifolds whose covering group is an infinite abelian group.

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:

Alex Genaro 14:30-15:10 Title: A Counting Problem for a Symbolic System with Digit Density

Discussion period: 15:10-15:30

Coffee Break: 15:30-16:00

Gustavo Pessil: 16:00 - 16:45 Title: Dimensional complexity in non-smooth dynamics

Discussion period: 16:45-17:00

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