This talk concerns entropy-dimensional concepts recently developed in order to estimate the complexity of systems with infinite topological entropy. I will motivate the search for variational principles and explain why we can understand this problem more clearly, and solve it, by considering the quantization of measures. This is joint work with Maria Carvalho.

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There will be a coffee break after the seminar.

We introduce a class of linear bounded invertible operators on Banach spaces, called shift operators, which comprises weighted backward shifts and models finite products of weighted backward shifts and dissipative composition operators. We classify vast families of these shift operators, including the ones generated by orthogonal, diagonalizable, rotation or hyperbolic matrices. Our classification yields verifiable conditions which we use to construct concrete examples of shift operators with a variety of dynamical properties.

In this talk, we explore a martingale approximation framework that provides quantitative estimates for large deviations in invertible dynamical systems. Assuming suitable decay of correlations, we derive these estimates and, as an application, construct Young structures with matching recurrence tails for partially hyperbolic diffeomorphisms with mostly expanding central directions.

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There will be a coffee break after the seminar.

Ideas from dynamical systems and thermodynamic formalism can be useful in rigorously estimating the numerical value of the (Hausdorff) dimension of some sets.  We will discuss applications of this approach to a variety of different problems (e.g., Bounded continued fractions, Feigenbaum attractors, Circle packings, Diophantine Approximation).  No prior knowledge is required.

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There will be a coffee break after the seminar.

One of the most important notions in Dynamical Systems is the topological entropy. It is a topological invariant and, roughly speaking, measures how chaotic the system is. More precisely, it quantifies the rate of dispersal of points in the future by the action of the system on the phase space. In particular, it is an effective tool to decide whether two systems can be conjugated.

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