Questions about how "random" or "deterministic" a sequence is appear in many areas of mathematics.  For real-valued sequences, this can be formulated in terms of how well distributed points are in the real line, or in an interval.  More detailed information can be found if we look at the gaps between pairs of points in finite segments of such sequences.  If these are sufficiently random then the mean gap data converges in a natural way, and the sequence has "Poisson pair correlations" (PPC) introduced by Rudnick and Sarnak.  There is a rich literature on such gaps of pairs in sequences stud

For simple fractal sets (such as the Sierpinski triangle) the dimension is explicitly known. For more general limit sets of a conformal iterated function scheme (a family of conformal contracting maps satisfying the open set condition) the dimension is known implicitly via the pressure function and Bowen's formula. Unfortunately, this breaks down when the contractions are not strict or are not conformal. However, some of the thermodynamic ideas can still be adapted to get bounds in some key cases.

Consider the space of two-dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such cocycles and establish a Furstenberg type formula characterizing the Lyapunov exponent. Using the spectral properties of the corresponding Markov operator and a parameter elimination argument, we prove that Lebesgue almost every cocycle in this space satisfies large deviations estimates and a central limit theorem.

Françoise Pène has proven functional limit theorems for Birkhoff sums of observables of the Lorentz gas in the infinite-horizon setting. However, the limiting processes are constant in time. In this talk, I will explain how these results can be revisited using a non-linear time normalization in order to obtain non-trivial limiting processes. I will first consider the case of an integrable observable, and then the case of a centered observable.

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There will be a coffee break after the seminar.

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.