Quarta-feira, dia 9 de março
10h - Salvador Addas-Zanata, Universidade de São Paulo
Título: Regiões de instabilidade de Mather para difeomorfismos do anel.
Resumo: Seja $f$ um difeomorfismo $C^{1+\varepsilon}$ do anel fechado $A$ que preserva orientação e as componentes do bordo, e seja $\widetilde{f}$ um levantamento de $f$ para seu recobrimento universal. Suponha que $A$ seja uma região de instabilidade de Birkhoff para $f$, e o conjunto de rotação de $\widetilde{f}$ seja um intervalo não degenerado. Então existe um anel $f$-invariante e aberto $A^*$ cujo bordo intercepta ambas as componentes do bordo de de $A$, e pontos $z^+$ e $z^-$ em $A^*$, de modo que a órbita positiva (resp. negativa) de $z^+$ converge para um conjunto contido na componente superior ( resp. inferior) do bordo de $A^*$ e a órbita positiva (resp. negativa) de $z^-$ converge para um conjunto contido na componente inferior (resp. superior) do bordo de $A^*$. Isso estende um importante resultado originalmente provado por Mather para difeomorfismos twist que preservam área no anel. Trabalho em conjunto com Fabio Tal (IME-USP).
10h - 14h Intervalo
14h - Aline Melo, PUC-Rio
Título: Taxa de convergência uniforme para as médias de Birkhoff de certas aplicações unicamente ergódicas no toro.
Resumo: Nesta palestra, vamos apresentar uma estimativa de taxa de convergência uniforme das médias de Birkhoff de um observável contínuo sob a translação no toro. Essa taxa de convergência depende explicitamente do módulo de continuidade do observável e das propriedades aritméticas da frequência que define a aplicação. Além disso, obtemos resultados similares para aplicações skew product afim no toro e, no caso da translação no toro unidimensional, essa estimativa é quase otimal. Esse é um trabalho em conjunto com Silvius Klein e Xiao-Chuan Liu.
15h - Luis Hernández-Corbato, Universidad Complutense de Madrid
Título: An integral to interpret duality in Cech type theories.
Resumo: Cech-type homology and cohomology theories are more suitable than singular theories to study spaces whose local topology is not ``nice''. Thus, the algebraic topological properties of (strange) attractors or invariant sets, in general, are better expressed in terms of their Cech homology and cohomology groups.
In the talk, we will introduce Cech theory from its definition. We use the approach of Alexander-Spanier to define the cohomology group and a version of Vietoris homology to obtain what we call homology at scale $\mathcal U$, for an arbitrary open cover $\mathcal U$ of the space. In these terms, an evaluation of cochains over chains appears naturally and leads to an integral of cohomology classes over ``$\mathcal U$--small'' cycles that reminds to De Rham's integration of differential forms. The integral defines a pairing that proves the duality between Cech homology and cohomology groups. Time permitting, we will explain techniques and results that relate the topologies of an attractor and its basin of attraction.
The talk is mainly based in a work joint with D. Nieves-Rivera, F. Ruiz del Portal and J. Sanchez-Gabites (arXiv:2112.14181).
Quinta-feira, 10 de março
10h - Verónica De Martino, Universidad de la República
Título: Spectral theory on trees.
Resumo: One can think of a tree and its group of automorphisms as a discrete version of the hyperbolic plane and its group of isometries, $SL(2,\R)$. In the continuous context, spectral theory has several applications, from geometry to number theory. In the discrete case, some computations are simplified and allow us to study carefully some examples and obtain very specific results.
In this talk we are going to focus on a specific self-adjoint operator on the tree, the discretized Laplacian. This operator will be intimately related to the geometry of the tree and the representation theory of its group of isometries.
11h - Carlos Fabián Álvarez Escorcia
Universidad del Sinú, Seccional Cartagena
Título: Hyperbolic measures for partially hyperbolic systems.
Resumo: In this talk, we study ergodic maximal entropy measures for certain partially hyperbolic systems of $T^d$, which have a two-dimensional center foliation. We propose a scenario where there exists at most a finite number (non-zero) of ergodic maximal entropy measures and all are hyperbolic.
12h - 14h Intervalo
14h - Marisa Cantarino, Universidade Estadual de Campinas
Título: Endomorfismos de Anosov em superfícies: regularidade de folheações e rigidez.
Resumo: Introduzimos com exemplos a dinâmica uniformemente hiperbólica para o caso não-inversível e suas principais propriedades. Apresentamos um resultado que caracteriza em superfícies a conjugação suave entre um endomorfismo de Anosov especial e sua linearização em termos da regularidade das folheações estável e instável. Esta regularidade é a continuidade absoluta em uma formulação uniformemente limitada, que caracterizamos usando as holonomias.
15h - Santiago Martinchich, Universidad de República
Título: Discretized Anosov flows.
Resumo: The idea of the talk is to present a class of partially hyperbolic diffeomorphisms called \emph{discretized Anosov flows} that naturally generalizes the time 1 map of Anosov flows and all its sufficiently small $C^1$ perturbations. We will see that this class is $C^1$ open and closed inside the set of partially hyperbolic diffeomorphisms in any ambient dimension. Moreover, many classical properties are satisfied for whole connected components of these systems: dynamically coherence, plaque expansivity, uniqueness of invariant foliations. Similar results happen for one-dimensional center partially hyperbolic skew-products.
16h - 16h20 Intervalo
16h20 - Sergio Fenley, Florida State University
Título: A new model for partial hyperbolic dynamics in dimension 3.
Resumo: We discuss a new model for partial hyperbolicity called collapsed Anosov flows. Roughly there is a topological Anosov flow and a collapsing map homotopic to the identity which sends orbits of the flow tangent to curves tangent to the center bundle. There is also a self orbit equivalence of the flow which semi conjugates to the action of the partial hyperbolic diffeomorphism. We discuss properties of collapsed Anosov flows, including several definitions and their equivalence. Some definitions involve only geometric concepts. This class is very large and possibly includes most PH in dimension 3 under certain conditions.
Sexta-feira, 11 de março
Homenagem ao Professor Saponga
14h - Christian Bonatti, Université de Bourgogne
Título: Folheações genéricas das variedades de dimensão 3.
Resumo: Pouco é conhecido sobre a topologia do espaço das folheações, alem dos resultados de conexidade de H. Eynard-Bontemps. Mas as folheações formam um conjunto fechado do espaço dos campos de planos, então formam um espaço completo pelas topologia Cr: isso permite considerar folheações genéricas. Isso foi o ponto de vista do meu único artigo com o Saponga e que vou apresentar aqui.
Provamos que, genericamente, uma folheação :
- seja é uma fibração em esferas S2
- seja tem as suas folhas compactas que são todas toros T2 e são organizadas em um número finito de classes de folhas paralelas.
Além dos resultados apresentados, esse artigo apresenta perguntas para entender melhor essas folheações genéricas: em particular conjecturamos que todas as folhas não compactas das folheações genéricas (orientadas e transversalmente orientadas) tem gênero <2.
Todas as dificuldades vivem nas ações de grupos na reta, que é uma área que conheceu recentemente um desenvolvimento espectacular: talvez já seja tempo de considerar as consequências para folheações?
15h - Javier Ribón Herguedas, Universidade Federal Fluminense
Título: About Saponga's work on fixed points of groups of diffeomorphisms on surfaces
Resumo: Let us consider some celebrated results in dynamics on surfaces:
- An orientation-preserving homeomorphism of the sphere has a fixed point.
- A homeomorphism of the torus with non-vanishing Lefschetz number has a fixed point.
- An orientation-preserving homeomorphism of the plane, that preserves an invariant measure, has a fixed point.
The two first properties are a consequence of Lefschetz fixed point theorem whereas the third one is a consequence of Brouwer's translation theorem and Poincaré's recurrence theorem. These results apply to the dynamics of a single homeomorphism, or in other words to a cyclic group. Saponga, together with his collaborators, obtained generalizations of the above results in the form of theorems of existence of fixed points of nilpotent groups of diffeomorphisms on surfaces of non-negative Euler characteristic. We will provide an overview of Saponga's work and the ideas involved.
16h - 16h20 Intervalo
16h20 - Abramo Hefez, Universidade Federal Fluminense
Título: Contribuição de Sebastião Firmo ao desenvolvimento da matemática na UFF
Resumo: Nesta palestra traçaremos um histórico da pós-graduação em Matemática na UFF mostrando o ambiente que Saponga encontrou ao chegar e a sua contribuição à criação do ambiente de pesquisa hoje vigente.
Segunda-feira, 14 de março
10h - Xuan Zhang, Universidade de São Paulo
Título: Quenched and annealed equilibrium states for random Ruelle expanding maps.
Resumo: (with Manuel Stadlbauer and Paulo Varandas) We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi and Carvalho-Rodrigues-Varandas, where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact.
11h - Graccyela Salcedo, UFRJ
Título: Contração em média de sistemas iterados de funções.
Resumo: Nesta palestra, vou apresentar alguns resultados obtidos na elaboração de minha tese de doutorado, sob a supervisão da Professora Katrin Gelfert.
Considerando um sistema iterado de funções contínuas em um espaço métrico que são escolhidos aleatoriamente, de forma idêntica e independente. Descrevemos condições que garantem contração em média, com uma possível mudança de métrica que preserve a estrutura topológica do espaço. Para o caso particular de um sistema de C^1-difeomorfismos do círculo que é proximal e não tem uma medida de probabilidade simultaneamente invariante por cada função, derivamos uma métrica fortemente equivalente que contrai em média.
12h - 14h Intervalo
14h - Jamerson Bezerra, Nicolaus Copernicus University
Título: Positivity of the Hausdorff dimension of generic dynamically defined Markov and Lagrange.
Resumo: Dynamically defined Markov and Lagrange spectra are subsets of the real line which describes the asymptotic behaviour of orbits of the system with respect to a reference potential. Those spectra are motivated by classical problems in number theory but can be used to understanding the long range behaviour of orbits of generic dynamics. The purpose of this talk is to present some ideas in how to guarantee richness (positivity of the Hausdorff dimension) of dynamically defined Markov and Lagrange spectra for generic dynamics admitting transverse homoclinic intersection in a compact manifold of any dimension large or equal than 2.
15h - Yaya Tall, Universidade de São Paulo
Título: Analyticity of the Lyapunov exponents for random products of quasi-periodic cocycles.
Resumo: In this talk after, some general preliminaries on Linear cocycles and Lyapunov exponents we will discuss the following result: The Lyapunov exponent , with for each $\lambda_{+}(p)$, $p=(p_1,...,p_N)$, with $p_i>0$ for each $i$ associated with a random product of quasi-periodic cocycles depends real analytically on the transition probabilities whenever $\lambda_{+}(p)$ is simple. Moreover if the spectrum at $p$ is simple then all Lyapunov exponents depend real analyticically on $p$. This is a joint work with Jamerson Bezzera (Universidade de Lisboa, Portugal), and Adriana Sánchez (Centro de investigación de Matemática Pura y Aplicada UCR, Costa Rica).
Terça-feira, 15 de março
10h - Emília Alves, Universidade Federal Fluminense
Título: Intersections of Bruhat cells: stratification and applications.
Resumo: We discuss arbitrary intersections of big Bruhat cells. Those objects arise naturally in several problems, across a number of disciplines, such as in singularity theory and in the study of the homotopy type of spaces of locally convex curves. We put forward a stratification of an arbitrary pairwise intersection of big Bruhat cells and show that the dual CW-complex of such stratification is homotopically equivalent to the pairwise intersection under analysis. Furthermore, we resort to our techniques to produce classical and new topological results about such intersections. We include a number of examples where explicit computations can be easily performed to illustrate the methods. This is joint work with N. Saldanha (PUC-Rio).
11h - Rafał Siejakowski, Universidade de São Paulo
Título: Meromorphic 3D-index and its asymptotics.
Resumo: The meromorphic 3D-index is a new and somewhat mysterious topological invariant of orientable 3-manifolds with toroidal boundary, defined as a state integral of Turaev–Viro type on ideal triangulations. The "states" in this integral are formal assignments of circle-valued dihedral angles to the edges of the tetrahedra. In this talk, we explain the predicted asymptotic behaviour of this invariant when the quantisation parameter q tends to 1. Conjecturally, our asymptotic limit contains "classical" information about certain flat PSL(2,C)-bundles on the manifold, including the hyperbolic volume and adjoint Reidemeister torsion whenever the manifold admits a complete hyperbolic structure of finite volume.
12h - 14h Intervalo
14h - Ana Anusic, Universidade de São Paulo
Título: Strongly commuting interval maps.
Resumo: Two maps $f,g\colon X\to X$ on a compact, connected, metric space $X$ are called strongly commuting if $f^{-1}\circ g=g\circ f^{-1}$. Note that if $f$ and $g$ commute (i.e. $f\circ g=g\circ f$), then they strongly commute, but not conversely. The notion of strong commutativity arose in the study of maps on inverse limit spaces which are given by certain commutative "diagonal" diagrams. For such maps, the topological entropy is completely determined by "straight-down" components which are often set-valued maps. In this talk we will explain this connection and show its applications. Furthermore, we will characterize piecewise monotone strongly commuting maps on the unit interval and show that they always have a common fixed point. In general, there exist commuting maps on the interval without common fixed points (Boyce and Huneke 1967). The question of characterizing such maps is still open.
This is a joint work with Chris Mouron.
15h - Jaime Jorge Sánchez-Gabites,
Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid
Título: Revisiting Manning's theorem on entropy in non-manifold spaces.
Resumo: Let $f$ be a continuous map of a compact manifold $M$. A well known theorem of Manning provides the following lower bound on the topological entropy $h(f)$ of $f$ in homological terms: $h(f) \geq \log |\lambda|$ for every complex eigenvalue $\lambda$ of the homomorphism $f_*$ induced by $f$ in $H_1(X;\mathbb{C})$. In particular if any of the eigenvalues has a modulus bigger than one, then $h(f) > 0$ and so the dynamics of $f$ is complicated.
Sometimes one would like to have a bound similar to Manning's when $M$ is not a compact manifold but an arbitrary compact metric space (for instance, a compact invariant set of some bigger dynamical system). The proof of Manning's bound breaks down in that context because it requires a condition that ensures that the $1$--dimensional homology of $M$ is locally trivial. However, it is possible to obtain a bound of the form $h(f) \geq C \log |\lambda|$ where $C$ is a positive constant which can be made explicit. This is accomplished by using \v{C}ech cohomology instead of homology (the preferred theory for badly behaved spaces) and a notion of integration of cochains over chains as an indirect measure of the ``length of a path''. These have to be used together with a result in number theory regarding the approximation of irrational numbers by algebraic ones.
In our talk we will introduce the necessary definitions, show how Manning's bound fails in non-manifold spaces, justify the formulation in terms of \v{C}ech cohomology and, without entering into technical details (and time permitting), give a general idea on how the bound $h(f) \geq C \log |\lambda|$ can be obtained.
This is joint work with Luis Hernández Corbato, David Jesús Nieves Rivera, and Francisco Romero Ruiz del Portal (Universidad Complutense de Madrid, Spain).
Quarta-feira, 16 de março
10h - Dahisy V. S. Lima, Universidade Federal do ABC
Título: Covering Action on Pullback Flows for Attractor-Repeller Decompositions.
Resumo: We are concerned about the topological structure of the set of flows lines connecting invariant sets within a Morse decomposition of an invariant set $S$ with respect to continuous flow on a topological space $X$. By considering the pullback flow defined on a regular covering space $(\widetilde{X},p)$ of $X$, one obtains an algebraic setting that arises from the ambient space in order to distinguish these connections. More specifically, we use the covering action to build a topological separation of $S$ in order to distinguish all connections up to the action of the covering translation group. All this algebraic-topological information is encoded in a matrix, called $p$-connection matrix. The algebraic framework developed herein generalizes the notion of the Novikov differential considered in the case of an infinite cyclic covering induced by a circle-valued Morse function $f$.
11h - 14h Intervalo
14h - Ulisses Lakatos, Universidade de São Paulo
Título: Sobre extensões do grupo de difeomorfismos conformes da esfera.
Resumo: Um programa de classificação completo dos grupos fechados e transitivos agindo no círculo unitário foi proposto por E. Ghys, no início dos anos 2000, e, mais tarde, concluído por J. Giblin e V. Markovic. Uma iniciativa similar para a 2-esfera foi iniciada por F. Kwakkel e F.A. Tal em um preprint de 2014. Inspirados por esses resultados prévios, apresentaremos um trabalho recente -- em conjunto com F.A. Tal -- no qual mostramos que qualquer grupo de difeomorfismos estendendo propriamente o grupo de Möbius (das transformações conformes) é 4-transitivo ou, mais precisamente, 4-transitivo por arcos. Em outras palavras, quaisquer duas listas de quatro pontos da esfera podem ser ordenadamente conectadas por trajetórias de uma isotopia começando na identidade. Mostraremos, ainda, que tal grupo precisa conter uma transformação de entropia topológica positiva.
15h - Fabio Tal, Universidade de São Paulo
Título: Zero Entropy area preserving homeomorphisms on surfaces.
Resumo: We review some recent results describing the behaviour of homeomorphisms of surfaces with zero topological entropy. Using mostly techniques from Brouwer theory, we show that the dynamics of such maps in the sphere is very restricted and in many ways similar to that of an integrable flow. We also show that many of these restrictions are still valid for $2$-torus homeomorphisms.