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Date & Time: 10/07/2025 — 11:00 AM (Brazil time)
Speaker: Neige Paulet (Queen's University, Canada)
Title: Finiteness of the gluing procedure of Anosov flows in dimension 3
Abstract: In this talk, I will first introduce Anosov flows and their significance in the study of hyperbolic dynamical systems and the topology of 3-manifolds. A fundamental question is whether a given 3-manifold supports only finitely many Anosov flows up to orbit equivalence. A powerful construction method involves gluing "building blocks" of flows. I will present a finiteness result for Anosov flows obtained through this procedure. I will also explain how this result, combined with classification results, contributes to an ongoing joint work with Thomas Barthelmé on the finiteness problem for Anosov flows on 3-manifolds.
30/06/2025
Speaker: Ana Cristina Barreto de Araújo, (IMPA, Brasil)
Title: Hölder Continuity of the Lyapunov Exponents of Products of Invertible Random Matrices
Abstract: The cocycle associated with a random product of invertible 2×2 matrices, under a probability distribution with compact support, has two Lyapunov exponents. When these exponents are distinct, Tall and Viana proved that the exponents depend pointwise Hölder-continuously on the probability measure. A natural question arises: can this result be extended to any dimension D≥2? In this seminar, we will explore this possibility. This is a joint work with Adriana Sanchéz, El Hadji Yaya Tall, and Marcelo Viana.
06/02/2025
Speaker: Lamartine Medeiros (Universidade Federal do Rio de Janeiro, Brasil)
Title: Thermodynamic Formalism of Typical Cocycles: Quasi-multiplicativity and Applications
Abstract: We discuss the subadditive thermodynamic formalism for the class of typical cocycles, as defined in [Bonatti-Viana, 2004]. Following [Park, 2020], we show these cocycles are quasi-multiplicative. In particular, this property ensures the continuous variation of subadditive topological pressure and equilibrium states with respect to a geometric potential associated with the cocycle. We will also discuss some applications.
21/11/2024
Speaker: Mauricio Poletti (Federal University of Ceará, Brasil)
Title: Lyapunov Spectrum of volume preserving partially hyperbolic maps.
Abstract: Given an invariant measure and a diffeomorphism, The Lyapunov spectrum is said to be simple if for almost every point there is the maximal quantity of different exponents possible, (or equivalently the Oseledets decomposition is given by one dimensional spaces). K. Marin proved that conservative partially hyperbolic diffeomorphisms with two dimensional center (with some technical conditions) generically have two different center exponents. In this work we study the simplicity of the full Lyapunov spectrum of these maps.
This is a joint work with K. Marin and D. Obata.
18/11/2024
Speaker: Aline Melo (Federal University of Ceará, Brasil)
Title: Analiticity of Lyapunov exponent of i.i.d. multiplicative process.
Abstract: In this talk, we extend the results and methods of Y. Peres from a finite to an infinite (but compact) space of symbols. In other words, we establish the analiticity of the maximal Lyapunov exponent for i.i.d. random product of matrices as a function of the transition probabilities. Our approach combines the spectral properties of the associated Markov operator with the theory of holomorphic functions in Banach spaces. This is a joint work with Artur Amorim and Marcelo Durães.
31/10/2024 (PT I), 07/11/2024 (PT II) and 14/11/2024 (PT III)
Speaker: Filiphe Veiga (Federal University of Minas Gerais, Brasil)
Title: Continuity Properties of Lyapunov Exponents for Surface Diffeomorphisms
Abstract: In this seminar, we discuss the work of Buzzi, Crovisier, and Sarig, where they prove a relationship between Lyapunov exponents and entropy in smooth surface diffeomorphisms. We outline the main theorem, which asserts that, in a smooth setting, continuity of entropy implies continuity of Lyapunov exponents for ergodic measures.
16/10/2024
Speaker: Moghaddamfar Kambiz (Federal University of Ceará, Brasil)
Title: Introduction to Autonomous Diffeomorphisms and Their Classification on Compact 2-Manifolds
Abstract: The notion of autonomous diffeomorphisms is essentially inspired by a work of A. Zeghib in 1995, where the terminology "autonomous" was introduced. This lecture introduces the concept of autonomous diffeomorphisms on smooth parallelizable manifolds. An autonomous diffeomorphism is defined as a transformation where the derivative cocycle, representing the derivative of the diffeomorphism in matrix form, remains constant with respect to a chosen framing across the manifold. The focus is on the classification of these diffeomorphisms on compact 2-manifolds, their connection to affine geometry, and the classification of the manifolds that admit such a structure. We will also discuss the initial idea of how the method used in two dimensions can be extended to higher dimensions.
08/10/2024
Speaker: Raquel Saraiva (Federal University of Minas Gerais, Brasil)
Title: Example of discontinuity for the Lyapunov exponents for SL(2, R)-valued cocycles in the α-Hölder topology
Abstract: In 2010, Bocker and Viana presented an example of a discontinuity point for the Lyapunov exponents as a function of the cocycle relative to theα-Hölder topology. The linear cocycle taking values in SL(2, R) is defined over a Bernoulli shift and the perturbations are constructed under some conditions related to α and the norm of the cocycle. Butler, in 2018, improved the hypothesis of Bocker-Viana but he restricted the probability measure considered. He constructs locally constant cocycles arbitrarily close to the original one with small Lyapunov exponents. In this talk, with a stronger condition than Butler but without restricting the measure, we show that the original cocycle can be approximated by locally constant cocycles with zero Lyapunov exponents. In particular, this extends Bocker-Viana result. This is a joint work with Edhin Mamani
03/10/2024
Speaker: Moghaddamfar Kambiz (Federal University of Ceará, Brasil)
Title: Exploring Partially Hyperbolic Diffeomorphisms in Three Dimensions via Autonomous Dynamics
Abstract: The classification of partially hyperbolic diffeomorphisms in dimension three has been studied for years and is still an active area of research. Rigidity of these diffeomorphisms, under additional conditions, has been explored in both smooth and topological contexts.
In this talk, we introduce a notion of autonomous dynamical systems proposed by A. Zeghib and use it to prove the rigidity of partially hyperbolic diffeomorphisms on closed compact three-manifolds, assuming certain smoothness conditions on their associated framing.
The geometrical approach used here may allow us to extend the result to higher dimensions.
12/10/2024
Speaker: Ian Melbourne (University of Warwick, Inglaterra)
Title: Wong-Zakai approximation of stochastic integrals
Abstract: A standard question in the theory of stochastic processes is how to interpret stochastic integrals (Ito, Stratonovich, and so on). The idea of Wong-Zakai approximation is to approximate stochastic processes by smooth processes. Studying the limits of the integrals for the smooth processes should give insight into the correct interpretation of the stochastic integrals.
I will review the classical results on success/failure of this program. Then I will discuss recent contributions from the theory of deterministic dynamical systems for the cases when integration is with respect to Brownian motion or a stable Levy process.
10/10/2024
Speaker: Carlos Matheus (CNRS/École Polytechnique, França)
Title: Exponential mixing for frame flows on many pinched negatively curved manifolds
Abstract: Frame flows are isometric extensions of geodesic flows obtained from parallel transport of orthonormal frames along geodesic paths. The dynamics of this kind of flow often provides useful insight in several rigidity questions in Differential Geometry: for a very recent example, one might consult the work of Filip, Fisher and Lowe. In this talk, we shall discuss a joint work with Thibault Lefeuvre and Martin Leguil about the speed of mixing of frame flows associated to pinched negatively curved manifolds.
04/10/2024
Palestrante: Jamerson Bezerra (Nicolaus Copernicus University, Polônia)
Title: A dynamical Thouless formula
Abstract: In the context of dynamically defined Schrodinger operators, the Thouless formula provides a strong relation between the behaviour of the Lyapunov exponent of the associated Schrodinger cocycle and the distribution measure of the elements of the almost sure spectrum - the integrated density of states (IDS). Indeed, this formula is the essential tool to translate the regularity properties of the Lyapunov expoent to the IDS.
In this talk we are going to discuss how to generalize this formula for winding family of affine cocycles and how to use it to obtain information concerning the regularity of the Lyapunov exponents of random cocycles.
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