Local: Auditório do Instituto de Matemática e Estatística -IME. Campus de Ondina. Salvador-BA.
Carlos Siqueira (UFBA)
Cristina Lizana Araneda (UFBA)
Edgar Matias (UFBA)
Elaís Cidely Souza Malheiro (UFBA)
Kleyber Cunha (UFBA)
Thiago Bomfim (UFBA)
Roberto Sant'Anna Sacramento (UFBA)
Cristina Lizana Araneda (UFBA)
Yuri Gomes Lima (UFC)
Fotos
Palestrantes
Aline Melo (UFC)
Title: A convergence rate for Birkhoff means of certain uniquely ergodic toral maps.
Abstract. In this talk, we will present an estimate on the uniform convergence rate of the Birkhoff averages of a higher dimensional torus translation given by a frequency satisfying a generic arithmetic condition and a continuous observable. This convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency. Furthermore, we obtain similar results for affine skew product toral transformations and, in the case of one dimensional torus translation, these estimates are nearly optimal. This is a joint work with Xiao-Chuan Liu and Silvius Klein.
Edhin Franklin Mamani Castillo (PUC-Rio)
Title: On the uniqueness of the measure of maximal entropy for geodesic flows of compact surfaces without conjugate points.
Abstract. We introduce the geodesic flow induced by metrics weaker than negative curvature. It is well-known that the geodesic flow of a compact manifold of negative curvature is Anosov. For this system we have a relatively good understanding of its dynamical and ergodic properties. However, when we weaken the metric, i.e., allowing regions of zero or positive curvature, the system may not be Anosov and little is known about its properties. Recently, a progress was achieved that ensures that for a compact surface without conjugate points, its geodesic flow always admits a unique measure of maximal entropy that sheds some light on its ergodic theory. We sketch one of the methods used to obtain this result.
Hellen Lima de Paula (CEFET-MG)
Title: Entropia generalizada de difeomorfismos Morse-Smale em superfícies
Abstract. Um problema clássico em sistemas dinâmicos consiste em compreender a complexidade de uma aplicação em termos de suas órbitas. Uma das principais ferramentas que nos permite atingir esse objetivo é a entropia. No entanto, muitas famílias interessantes de sistemas dinâmicos têm todos os elementos com entropia zero, dentre elas, a dos difeomorsmos Morse-Smale. J. Correa e E. Pujals, [2], introduziram o conceito de entropia generalizada, que estende a noção clássica de entropia. Usando a entropia generalizada e estendendo as técnicas desenvolvidas por L. Hauseux e F. Le Roux, [3], calculamos a entropia de difeomorfismos Morse-Smale em superfícies,[1]. Trabalho conjunto com Javier Correa.
Referências:
[1] Correa, Javier and de Paula, Hellen, Polynomial entropy of Morse-Smale diffeomorphisms on surfaces, Bull. Sci. Math., 182, (2023), 103225.
[2] Correa, Javier and Pujals, Enrique, Orders of growth and generalized entropy, J. Inst.Math. Jussieu (2021), 1-33.
[3] Hauseux, Louis and Le Roux, Frédéric, Entropie polynomiale des homeomorphismes de Brouwer, Annales Henri Lebesgue 2, (2019), 39-57.
José Santana Campos Costa (UFMA)
Title: Semi rigidity of Lyapunov exponents in higher dimensions
Abstract. We present some results about partially hyperbolic diffeomorphisms f on the torus of dimension d (d > 2) which is homotopic to the linear Anosov A. In this context we show a comparison of their Lyapunov exponents, in fact, we show that the sum of the Lyapunov exponents of f is bounded by the sum of the Lyapunov exponents of the linear Anosov A. We will show this also for non-uniformly Anosov diffeomorphism with dominated decomposition, in particular for examples described by C. Bonatti and M. Viana.
This is a joint work with Ali Tahzibi.
Luis Pedro Piñeyrúa (UFC)
Title: Accessibility for dynamically coherent partially hyperbolic diffeomorphism with 2D center.
Abstract. The Pugh-Shub accessibility conjecture says that for any integer $r \in [2,+\infty]$ stable accessibility is open and dense among the set of $C^r$ partially hyperbolic diffeomorphisms, volume preserving or not.
In a joint work with Martin Leguil, we show that the conjecture is true among the set of stably dynamically coherent partially hyperbolic diffeomorphisms with 2 dimensional center under a strong bunching condition.
Maria José Pacífico (UFRJ)
Title: What's new in Lorenz attractors.
Abstract. Ever since its discovery in 1963 by Lorenz [1], the Lorenz attractor has been playing a central role in the research of singular flows, i.e., flows generated by smooth vector fields with singularities. In this talk I shall survey about old and new results describing the dynamics of this kind of attractors from the topological as well as the ergodic point of view. I will end sketching the proof of my newest result establishing that that in a C^1-open and densely family of vector fields (including the classical Lorenz attractor), if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on Λ. In particular, there exists a unique measure of maximal entropy for the flow X|Λ. This corresponds to a joint work with Fan Yang and Jiagang Yang.
References
[1] E. N. Lorenz. Deterministic nonperiodic flow. J. Atmosph. Sci., 20:130–141, 1963.
[2] M. J. Pacifico, F. Yang and J. Yang. Entropy theory for sectional hyperbolic flows. Ann. Inst. Henri Poincar ́e, Anal. Non Lin ́eaire, 38 (2021), 1001–1030.
[3] M. J. Pacifico, F. Yang and J. Yang. Existence and uniqueness of equilibrium states for systems with specification at a fixed scale: an improved ClimenhagaThompson criterion. Nonlinearity, Volume 35, Number 12.
[4] Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions. https://arxiv.org/abs/2209.10784
Mauricio Poletti (UFC)
Title: Uniqueness of u-Gibbs measures for hyperbolic skew products on T^4.
Abstract. We study the u-Gibbs measures of a certain class of uniformly hyperbolic skew products on the 4 dimensional torus. These systems have a strong unstable and a weak unstable directions. We show that Cr-dense and C2-open in this set every u-Gibbs measure is SRB, in particular, there is only one such measure. As an application of this, we can obtain the minimality of the strong unstable foliation.
This is a Joint work with Sylvain Crovisier and Davi Obata.
Sergey Tikhomirov (PUC-Rio)
Title: Probabilistic aspects of shadowing: connection to gamblers ruin problem.
Abstract. It is well-known that shadowing holds in a neighborhood of a hyperbolic set. It is known that shadowing can hold for non hyperbolic systems, but due to results of Sakai, Abdenur, Diaz, Pilyugin, Tikhomirov shadowing is "almost" equivalent to structural stability.
At the same time numerical experiments by Hammel-Grebogi-Yorke for logistics and Henon maps shows that shadowing holds for relatively long pseudotrajectories. It poses a question which type of shadowing holds for systems, which are not necessarily hyperbolic.
I consider probabilistic approach for the topic. I show that for infinite pseudotrajectories it does not change the notion. At the same time it shows that relatively long pseudotrajectories can be shadowed by exact trajectory with high probability. The main technique is a reduction to special form of gambler's ruin problems and mild form of large deviation principle for random walks. We show that our approach works for several examples -- skew product maps.
The talk is based on joint works with G. Monakov.
Sylvie Oliffson Kamphorst (UFMG)
Title: Hiperbolicidade, estabilidade e convexidade em bilhares.
Abstract. Panorama do aparecimento da hiperbolicidade em bilhares e sua relação com a convexidade da mesa através de exemplos como estádios e bilhares com obstáculos.
Victor Carneiro (UFRB)
Title: Transição de fase em dinâmicas suaves.
Abstract. Sabe-se que dinâmicas uniformemente expansoras ou hiperbólicas não apresentam transição de fase em relação a potenciais Holder. Em trabalhos recentes, T. Bomfim, V. Carneiro e A. Fernandes exploram a existência de transição de fase em dinâmicas suaves na vizinhança da expansividade, a fim de encontrar caracterizações de expansão neste contexto, associando com propriedades de gap espectral, análise multifractal, grandes desvios, dentre outros aspectos. Nesta palestra serão apresentados algumas das principais ideias e resultados do grupo acerca do tema.
Vilton Pinheiro (UFBA)
Title: Statistically recurring sets and absolutely continuous invariant probability.
Abstract. Given a point x and a set U, define the (upper) visit frequency of x to U (with respect to a f) as
$$\tau_x(U)=\limsup_n\frac{1}{n}\#\{0\le j<n\,;\,f^j(x)\in U\}.$$
Consider a finite reference measure m (not necessarily invariant). A set U is m-statically recurrent if $\tau_x(V)>0$ for m almost every point $x\in V$ and all measurable set $V\subset U$. For the Birkhoff theorem, if f admits an invariant probability $\mu$ absolutely continuous with respect to m, i.e., $\mu\circ f^{-1}=\mu\ll m$, then there is a m-statistically recurrent set U with $m(U)>0$. A natural question is whether the reciprocal of this result is true.
That is, the existence of a m-statistically recurrent set is sufficient to ensure the existence of an absolutely continuous invariant probability with respect to m?
In this talk, based on a work in progress with A. Castro (UFBA) and I. Rios (UFF), we will discuss the question above.
Vítor Araújo (UFBA)
Title: Physical measures for non-uniform sectional expanding partially hyperbolic flows.
Abstract. We prove that a partially hyperbolic attracting set for a $C^2$ vector field, having slow recurrence to equilibria, supports an ergodic physical measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical measures.
This extends to continuous time systems, a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set.
Horário / Programa
Participantes
Local
Auditório Maria José Zezé de Oliveira
Instituto de Matemática e Estatística -IME
Av. Milton Santos, s/n, Campus Universitário de Ondina
Universidade Federal da Bahia
Salvador-BA.
Referência: Portão 1 do Campus de Ondina (mais perto da Av. Garibaldi)
As inscrições estarão abertas até o dia 02 de abril de 2023 (ENCERRADO).
O evento será 100% presencial. Há possibilidade de financiamento para hospedagem (em breve os inscritos estarão recebendo as informações sobre o financiamento!).
Apoio
