# Seminários de Sistemas Dinâmicos

As informações dos seminários do Grupo de Sistemas Dinâmicos estão disponíveis em

### Seminários 2019

#### Março

• 09/03/2019 (15h45 - Sala 13)
Palestrante: Vilton Pinheiro (DMAT/UFBA)
Titulo: Caracterização das medidas levantáveis
Resumo: Aplicações induzidas fazem parte do dia a dia de muitos das pessoas que trabalham em Sistemas  Dinâmicos e Teoria Ergódica. No caso de Teoria Ergódica um problema imediato na presença de uma aplicação induzida é caracterização da medidas invariantes levantáveis. Aqui abordaremos, o que acreditamos ser, as três  questões básicas do assunto:
(1) Dar uma condição necessária, suficiente e relativamente testável para que uma medida seja levantável.
(2) Caracterizar os levantamentos (a decomposição ergódica do lift).
(3) Dar condições para o levantamento seja uma medida ergódica.

### Seminários 2018

#### Novembro

• 06/11/2018 (13h30)
Palestrante: Vinícius Coelho (PhD Student - UFBA)
Titulo: Adapted metrics for singular hyperbolic flows
Resumo: We show the existence of singular adapted metrics for any singular hyperbolic set with respect to a $C^{1}$ vector field on finite dimensional compact manifolds. This is a joint work with V. Araújo and L. Salgado.

#### Outubro

• 16/10/2018 (15h00)
Palestrante: Leandro Cioletti (UnB)
Titulo: Thermodynamic Formalism for Topological Markov Chains on Standard Borel Spaces
Resumo: In this talk we present a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space $X\equiv E^{\mathbb{N}}$, where $E$ is any Borel standard space. We introduce a concept of entropy and pressure for shifts acting on $X$ and obtain the existence of equilibrium states as additive probability measures for any bounded continuous potential. Afterwards, we establish convexity and other structural properties of the set of equilibrium states and obtain a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential. We discuss about the Yosida-Hewitt decomposition of such equilibrium states, associated to H\"older potentials. At the end, we briefly mention two applications: the construction of invariant measures for time-homogeneous Markov chains and obtain asymptotic stability for a class of Markov operators. This is a joint work with E. Silva and M. Stadlbauer.

#### Julho

• 03/07/2018,
Palestrante: Vilton Pinheiro (UFBA)
Titulo: Hyperbolic Blocks with smooth metric
Resumo: Let f : M → M be a C1 diffeomorphism defined on a Riemannian manifold M and let 〈., .〉 be its Riemannian metric. Given an f invariant probability μ without zero Lyapunov exponents, we obtain a Riemannian metric 〈.,.〉l, defined on the whole M, equivalent Riemannian metric 〈., .〉 and a Hyperbolic Block H with respect to 〈.,.〉l such that μ(H) > 0. We emphasize that we do not use any induced Finsler norm/metric. We want to open the discussion about how far one can go in Pesin Theory using (H,〈.,.〉l). We recall that the Pugh’s example of a regular point p for a C1 diffeomorphism on the Torus without an injectively immersed sta- ble submanifold tangent to the stable direction does not say that p is “typical” with respect to an invariant measure.
• 11/07/2018,
Palestrante: Vinícius Coelho (PhD Student - UFBA)
Titulo: Kingman-Like Theorem for finite measures.
Resumo: We provided a Kingman-Like Theorem for an arbitray finite measure assuming certain conditions. As an application we proved a version of Birkhoff's Theorem for bounded subinvariant observables and arbitrary finite measure.
• 13/07/2018 (10h00)
Palestrante: Marcus Morro (PhD Student - UFBA)
Titulo: Banca para defesa de tese de doutorado
• 16/07/2018 (15h30)
Palestrante: Wescley Bonomo (UFES)
Minicurso: Difeomorfismos no círculo: De Poincaré a Denjoy - Parte I
Resumo: Neste minicurso serão abordados aspectos elementares da teoria clássica dos difeomorfismos no círculo. Será introduzido o conceito de número de rotação de um difeomorfismo do círculo, o qual é um invariante por conjugações topológicas. Neste caso, o Teorema de Poincaré estabelece que um difeomorfismo f, de classe C1 no círculo, com número de rotação irracional, é semi-conjugado a uma rotação irracional. Adicionalmente é sabido que se todo o círculo for um conjunto minimal para a dinâmica de f, então na verdade f é topologicamente conjugada a uma rotação irracional. Se um difeomorfismos de classe C1 não admite todo o círculo como um conjunto minimal, então o mesmo é necessariamente um conjunto de Cantor. Como ilustrativo deste fenômeno, será apresentado um exemplo de Denjoy e também um teorema de sua autoria, o qual estabelece que difeomorfismos de classe C2 com número de rotação irracional são de fato topologicamente conjugados a rotações irracionais no círculo.
• 17/07/2018,
Palestrante: Diego Daltro (PhD Student - UFBA)
Titulo: Exponential decay of correlations for Gibbs measures for semiflows over C1+\alpha piecewise expanding maps Resumo: Exponential decay of correlations for flows have been considered by Dolgopyat, Baladi-Vallée and Avila-Gouëzel-Yoccoz. Building over the latter, Araujo and Melbourne proved exponential decay of correlations for nonuniformly hyperbolic flows with $C^{1+α}$ stable foliation. A key point in the proof of this results lies on the so called Federer Property, a kind of ‘geometric’ property for the SRB measure. Unfortunately, the Federer property is not true for arbitrary Gibbs measures (see Baladi-Vallée) and seems unlikely to make use this ideas to obtain exponential decay of correlations in Gibbs context. We show that the Gibbs property is enough to exponential decay of correlations for Gibbs measures suspension semiflows over $C^{1+α}$ piecewise expanding maps and to obtain other applications (joint work with P .Varandas).
• 18/07/2018 (13h30)
Palestrante: Shintaro Suzuki (Postdoc UFBA)
Titulo: Birkhoff cone methods for random non-uniformly expanding maps
Resumo: We consider a random dynamical system generated by non-uniformly expanding maps on a compact connected Riemannian manifold, which have contractive and expanding behavior on the manifold. For such a random dynamical system, under suitable integrability conditions for some quantities of a map on each fiber, we prove a version of the Ruelle-Peron-Frobenius theorem for corresponding random transfer operators in the case where their potential function is a C^1 function on each fiber and satisfies a kind of "mean expanding" condition. Our proof here is based on Birkhoff cone methods and we have fiberwise exponential decay of random correlation functions as an application (joint work with P .Varandas and M. Stadlbauer).
• 18/07/2018 (15h30)
Palestrante: Wescley Bonomo (UFES)
Minicurso: Difeomorfismos no círculo: De Poincaré a Denjoy - Parte II
• 19/07/2018 (15h30)
Palestrante: Wescley Bonomo (UFES)
Minicurso: Difeomorfismos no círculo: De Poincaré a Denjoy - Parte III

#### Junho

• 12/06/2018, 13h30, Audiótrio do IME;
Palestrante: Vitor Araújo (UFBA)
Título: Lyapunov spectrum with constant sign
Resumo: Let $f:M\to M$ be a $C^1$ map of a compact manifold $M$ admitting some point whose orbit has only negative Lyapunov exponents. Then the orbit of this point is in the basin of a periodic sink. We need only assume that $Df$ is never the null map at any point (in particular, we need no extra smoothness assumption on $Df$), encompassing a wide class of possible critical behavior. Similarly, a trajectory having only positive Lyapunov exponents for a $C^1$ diffeomorphism is itself a periodic repeller (source).   Analogously for a $C^1$ open and dense subset of vector field on finite dimensional manifolds: for a flow $\phi_t$ generated by such a vector field, if a trajectory admits weak asymptotic sectional contraction (the extreme rates of expansion of the Linear Poincar\'e Flow are all negative), then this trajectory belongs either to the basin of attraction of a periodic hyperbolic attracting orbit (a periodic sink or an attracting equilibrium); or the trajectory accumulates a codimension one saddle singularity. Similar results hold for weak sectional expanding trajectories. Both results extend part of the non-uniform hyperbolic theory (Pesin's Theory) from the $C^{1+}$ diffeomorphism setting to $C^1$ endomorphisms and $C^1$ flows. Some ergodic theoretical consequences are discussed. The  proofs use version of Pliss Lemma for maps and flows.

• 20/06/2018
Palestrante: Marcus Morro (PhD Student - UFBA)
Titulo: Ergodic optimization for flows
Resumo: Contreras proved that for an expanding transformation the max- imizing measures of a generic Lipschitz function is supported on a single pe- riodic orbit. We use this result to prove that for a hyperbolic flow there is a open and dense set of H ̈older functions with a single maximizing measure, which is supported on a periodic orbit. Young proved that The Poincar ́e map of the Lorenz attractor flow is a limit of hyperbolic subshifts of finite type. We use that to show that there is an open set W of Lorenz-like flows, such that for each (Xt)t ⊂ W there is a open and dense set of Ho ̈lder functions with a single maximizing measure, which is supported on a periodic orbit.

• 26/06/2018,
Palestrante:Nicolai Haydn (University of Southern California)
Titulo: Local escape rates for φ-mixing systems
Resumo: If one places a hole of positive measure in an ergodic dynamical system, then almost every point will eventually hit the hole and disappear. The exponential decay rate of the left-over set is the escape rate to the hole. Naturally a smaller hole will have a smaller escape rate. However, if one divides the escape rate by the size (measure) of the hole and takes a limit as the size goes to zero, then one obtains the local escape rate. In this talk we show that if the invariant measure is φ-mixing (with respect to a generating partition) then the local escape rate is equal to one at every point except at periodic points, where it is given by one minus the extremal index. We apply this result to Young towers, equilibrium states for Axiom A systems, interval maps and conformal maps.
Data do Evento:
ter, 02/01/2018 - 08:00
Imagem Evento: