**26 de Maio, 14h00, sala 6.2.38**

**Paolo Gidoni **(CMAF-CIO - Universidade de Lisboa)

**The notion of twist: from the Poincaré-Birkhoff Theorem towards higher dimension**

**Abstract**: The celebrated Poincaré-Birkhoff Theorem, with its several applications, attests the relevance of twist as boundary condition for planar systems. But what becomes of the twist condition when we try to extend this result to higher dimensions? Whereas on the plane the notion of twist is quite intuitive, the same cannot be said about twist in higher dimensions. Indeed, due to the special nature of the plane, the Poincaré-Birkhoff Theorem can be seen as the superposition of several distinct higher dimensional result. In this seminar we will review the main steps taken in the study of this issue, focusing, in the last part of the talk, on some recent results introducing the notion of “avoiding cones condition”.

**20 de Janeiro, 14h00, sala 6.2.38**

**Alexandre Tavares Baraviera** (Universidade Federal do Rio Grande do Sul)

**Some properties of block transformation maps**

**Abstract: **In this talk I will discuss some dynamical and measure-theoretic properties of a class of maps on the symbolic space that transforms blocks into symbols. As examples we can consider, among others, the decimation map and the map generated by the majority rule.

# 2016

**2 de Dezembro, 14h00, sala 6.2.33 **

** Raquel Ribeiro**

Universidade de São Paulo - IME - USP - Brasil

**Sombreamentos em Sistemas Dinâmicos**

**Abstract:**

Estudos em sistemas dinâmicos muitas vezes necessitam da noção de aproximação de órbitas do sistema. Por exemplo, em simulações computacionais sempre temos um erro numérico ao calcular uma trajetória, mas ao mesmo tempo sempre queremos ter a certeza de que o que vemos na tela do computador é uma boa aproximação da órbita do sistema. A noção clássica de aproximação em sistemas dinâmicos é a seguinte: Dado $\delta>0$, uma sequência de pontos $\{x_j\}_{j\in \mathbb{Z}}$ é uma $\delta$-pseudo-órbita para uma aplicação $f$ se $$ d\left(f(x_j),x_{j+1}\right) < \delta, \;\forall \;\ j \in \mathbb{Z}.$$ Uma $\delta$-pseudo-órbita é $\epsilon$-sombreada por uma órbita de $y \in M$ se $ d\left(f^j(y), x_j\right) < \epsilon,\;\;\forall \,\,\, j\in \mathbb{Z}$. Uma aplicação $f$ tem a propriedade de sombreamento se para todo $\epsilon>0$, existe $\delta>0$ tal que toda $\delta$-pseudo-órbita $\{x_i\}_{i\in \mathbb{Z}}$ é $\epsilon$-sombreada por uma órbita de $f$. Podemos perguntar quando um sistema tem a propriedade de sombreamento. O primeiro resultado foi o Lema do Sombreamento, dado por Anosov, na década de 70, o qual fornece uma condição suficiente para um sistema ter a propriedade de sombreamento. Este lema é uma importante ferramenta no estudo de sistemas dinâmicos hiperbólicos, como por exemplo, para o estudo da estabilidade estrutural, estabilidade topológica, existência de partições de Markov, etc. Determinar quais sistemas possuem a propriedade de sombreamento é um importante problema em dinâmica. Neste seminário nós estudaremos tipos de sombreamento para sistemas em um determinado contexto, a saber, o genérico. Nós mostraremos que qualquer uma das seguintes hipóteses implica que um conjunto isolado é topologicamente transitivo e hiperbólico: (i) o conjunto é transitivo por cadeia e satisfaz a propriedade de sombreamento, (ii) o conjunto satisfaz a propriedade de sombreamento no limite, ou (iii) o conjunto satisfaz a propriedade de sombreamento em média com a hipótese adicional que as variedades instável e estável de qualquer par de órbitas críticas intersectam uma a outra. O resultado se encontra em \cite{Ribeiro}. De acordo com o tempo, nós discutiremos os tipos de sombreamento no atrator geométrico de Lorenz, um importante exemplo na teoria de sistemas dinâmicos. Ribeiro, R. \emph{Hyperbolicity and types of shadowing for $C^1$ generic vector fields}. Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2963-2982.

**7 de Outubro, 14h30, sala 6.2.38**

** Alexandre A. P. Rodrigues**

CMUP, Universidade do Porto

**On Taken's Last Problem: times averages for heteroclinic attractors.**

**Abstract:**

In this talk, after introducing some technical preliminaries about the topic, I will discuss some properties of a robust family of smooth ordinary differential equations exhibiting tangencies for a dense subset of parameters. We use this to find dense subsets of parameter values such that the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. This is a joint work with I. Labouriau (University of Porto).

**16 de Setembro, 15h00, sala 6.2.44**

** Esmeralda Sousa Dias**

CAMGSD and IST Universidade de Lisboa

**The geometric organization of the dynamics of some Poisson cluster maps**

**Abstract:**

Cluster maps are birational maps which preserve a certain presymplectic form. Often, there exist nontrivial Poisson structures for which these maps are Poisson maps. The existence of such Poisson structures lead to foliations of domain of the cluster map and to several "reduced" maps. I will explain how the existence of such Poisson structures enables to understand not only the geometric organization of the dynamics of some cluster maps but also lead to a complete description of their dynamics. This talk is based on the work arXiv:1607.03664 with Inês Cruz and Helena Mena Matos (CMUP, U. Porto)

**1 de julho, 13:30, sala 6.2.33**

**Silvius Klein** (Norwegian University of Science and Technology (NTNU)

**Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles**

**Abstract:**

The purpose of the first part of this talk is to describe a recent result on the continuity of the Lyapunov exponents for analytic quasi-periodic cocycles. The new feature of this work is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible anywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.

The second part of this talk will be concerned with describing some problems and other work in progress that may be investigated using a similar approach.

Joint work with Pedro Duarte.

**22 de Abril, 14h00, sala 6.2.33**

**Pierre Berger**

Laboratoire Analyse, Géométrie & Applications, Université Paris 13

**On the Kolmogorov typicality of dynamics displaying infinitely many coexisting sinks**

**Abstract:**

In this talk we will show that the finiteness of the number of attractor is not typical in the sens of Kolmogorov. Moreover, we will show the existence of an open set of surface map $U$ in which typically in the sens of Kolmogorov-Arnold, the dynamics displays infinitely many attractors. This means that for a $C^r$-generic family $(f_a)_a$ of maps $f_a$ in $U$, for every small parameter $a$, the dynamics $f_a$ displays infinitely many sinks. A part of this work is in collaboration with S. Crovisier et E. Pujals.

**15 de Abril, 14h, sala 6.2.33**

** José Pedro Gaivão**

CEMAPRE and ISEG, Universidade de Lisboa

**Non-differentiable irrational invariant curves and a question of John Mather**

**Abstract:**

In the 20’s Birkhoff proved that any invariant curve of a symplectic twist map of the annulus which is not homotopic to a point (essential curve) is the graph of a Lipschitz function. An open question of Mather asks for an example of a $C^r$ symplectic twist map with an invariant essential curve that is not $C^1$ and that contains no periodic point. In this talk we will discuss the construction of such example when $r=1$. The example is due to Marie-Claude Arnaud. The cases $r>1$ and $r=\omega$ remain open.

**1 de Abril, 11h00, sala 6.2.33**

** Arseniy Akopyan**

IST Austria

**Elementary approach to closed billiard trajectories in asymmetric normed spaces**

**Abstract:**

We apply the technique of Karoly Bezdek and Daniel Bezdek to study the billiards in convex bodies, when the length is measured with a (possibly asymmetric) norm. We give elementary proofs of some known results and prove an estimate for the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. (joint work with A. M. Balitskiy, R. N. Karasev, and A. Sharipova)

**4 de Março, 16h00, sala 6.2.33**

** Jaqueline Siqueira **

CMUP, CNPq Brasil

**On equilibrium states for impulsive semiflows **

**Abstract:**

Impulsive dynamical systems may be interpreted as suitable mathematical models of real world phenomena that display abrupt changes in their behavior, and are described by three objects: a continuous semiflow on a metric space $X$; a set $D$ contained in $X$; where the flow experiments sudden perturbations; and an impulsive function $I : D\to X$ ; which determines the change on a trajectory each time it collides with the impulsive set $D$. We consider impulsive semiflows defined on compact metric spaces and give suficient conditions, both on the semiflows and the potentials, for the existence and uniqueness of equilibrium states. We also generalize the classical notion of topological pressure to our setting of discontinuous semiflows and prove a variational principle. This is a joint work with José Ferreira Alves and Maria de Fátima de Carvalho.

**4 de Março, 14h30, sala 6.2.33**

** Vanessa Ramos**

CMUP/Universidade Federal do Maranhão

**Equilibrium States for Hyperbolic Potentials **

**Abstract:**

Let $f:M\rightarrow M$ be a continuous map defined on a compact metric space $M$ and let $\phi:M\rightarrow\mathbb{R}$ be a real continuous function. In this classical setting, we say that $\mu_{\phi}$ is an equilibrium state associated to $(f,\phi)$, if $\mu_{\phi}$ is an $f$-invariant probability measure characterized by the following variational principle: $$ P_{f}(\phi)=h_{\mu_{\phi}}(f)+\displaystyle\int{\!\phi} \,d\mu_{\phi}=\sup_{\mu\in\mathcal{M}_{f}(M)}\left\{h_{\mu}(f)+\int{\!\phi}\, d\mu\right\}$$ where $P_{f}(\phi)$ denotes the topological pressure, $h_{\mu}(f)$ is the metric entropy and the supremum is taken over all $f$-invariants probabilities measures. Existence is a relatively soft property that can often be established via compactness arguments. Uniqueness is usually more subtle and requires a better understanding of the dynamics. In this talk, we will show uniqueness of equilibrium states associated to local diffeomorphisms $f:M\rightarrow M$ and hyperbolic H\"{o}lder continuous functions $\phi:M\rightarrow\mathbb{R}$.

**2015**

**12 de Outubro, 10h30, sala 6.2.33**

** Nuno Luzia**

Universidade Federal do Rio de Janeiro

**Quantitative recurrence results for random walks**

**Abstract:**

The famous Polya's recurrence theorem says that the simple random walk, in the integer lattice, is recurrent in dimensions 1 and 2 but not in higher dimensions (as a famous mathematician once explained: a drunk man leaving the bar and walking randomly through the streets will eventually come home). In this talk I will give a quantitative version of Polya's recurrence theorem and some related results for weakly dependent random walks.

**5 de Outubro, 10h30, sala 6.2.33**

**Telmo Peixe**

FCUL/ISEG/CMAFCIO

**Polymatrix Replicators and Lotka-Volterra Systems**

**Abstract:**

The Polymatrix replicators form a simple class of o.d.e.’s on prisms defined by simplexes, which describe the evolution of strategical behaviours within a population stratified in $n \geq 1$ social groups. This class of replicator dynamics contains well known classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix) replicator equations, and some replicator equations for n-person games.

In the 1980’s Raymond Redheffer et al. developed a theory on the class of stably dissipative Lotka-Volterra systems. This theory is based on a reduction algorithm that “infers” the localization of the system’s attractor in some affine subspace. Waldyr Oliva et al. in 1998 proven that the dynamics on the attractor of such systems is always embeddable in a Hamiltonian Lotka-Volterra system. We extend these results to Polymatrix replicators.

**17 abril, 11h, 6.2.38**

**Jorge Milhazes de Freitas **

Faculdade de Ciências - Universidade do Porto/ CMUP

**Laws of rare events with convergence rates**

**Abstract:**

In the classical theory of Extreme Values, it is well known that the Extremal Index (EI) measures the intensity of clustering of extreme events in stationary processes. By extreme or rare events, we mean occurrences usually undesired and possibly catastrophic that have a small probability of occurring. We will see that for some certain uniformly expanding systems there exists a dichotomy based on whether the rare events correspond to the entrance in small balls around a periodic point or a non-periodic point. In fact, either there exists EI in (0,1) around (repelling) periodic points or the EI is equal to 1 at every non-periodic point. The main assumption is that the systems have sufficient decay of correlations of observables in some Banach space against all integrable observables. Under the same assumption, we obtain convergence rates for the asymptotic extreme value limit distribution. The dependence of the error terms on the `time' and `length' scales is made very explicit.

**26 de Fevereiro, 14h00, sala 6.2.42**

** Paulo Varandas**

Universidade Federal da Bahia

**Large deviations in dynamical systems**

**Abstract:**

In this talk we shall give a short overview of the application of large deviations techniques for the study of the velocity of convergence of Birkhoff averages and non-additive limit theorems in dynamical systems. Some dynamically relevant quantities that arise in limit theorems include Lyapunov exponents and entropy. We shall report of recent contributions to the theory in the case that the reference probability measure admits some (weak) Gibbs property.

**11 de Fevereiro, 14h00, sala B3-01 Complexo Interdisciplinar UL**

** José Ferreira Alves**

CMUP FCUP

**Propriedades ergódicas de semifluxos impulsivos**

**Abstract:**

Um semifluxo impulsivo é caraterizado por três ingredientes fundamentais: um fluxo contínuo num espaço métrico $X$, um conjunto $D \subset X$ onde o fluxo sofre alguma perturbação abrupta, e uma função impulsiva $I:D\to X$ que determina para onde saltam as trajetórias que atingem $D$. Sistemas dinâmicos com efeitos impulsivos parecem ser um modelo matemático adequado para descrever fenómenos reais que apresentam mudanças bruscas de comportamento. Em trabalhos em colaboração com Maria Carvalho e Carlos Vásquez damos condições suficientes para a existência de medidas de probabilidade invariantes por um semifluxo impulsivo e estabelecemos um Princípio Variacional.

**19 de Janeiro, 16h30, sala A2-25 Complexo Interdisciplinar UL**

** João José Gomes**

FCUL

**Maximum likelihood estimation for linear mixed models**

**Abstract:**

A linear model is $Y=X\beta+\varepsilon$ where $Y$, the response vector, is Gau\ ssian $(X\beta,\sigma I)$. A mixed model incorporates two random variables: $B$ the random effects and $Y\ $ the response vector. In a linear mixed model the unconditional distribution of $B$ and the condition\ al distribution $(Y| B = b)$ are both Gaussian distributions, $(Y| B =b)\sim N(X\beta+Zb,\sigma I)$ and $B \sim N (0,\Sigma\theta)$. Objective: Parameter estimation by maximum likelihood. Problem: Understand the algorithm.

**19 de Janeiro, 15h00, sala A2-25 Complexo Interdisciplinar UL**

** Susana Pinheiro**

**Maximum likelihood estimation for linear mixed models**

**Abstract:**

We consider a logistic growth model with a predation term and a stochastic perturbation given by a diffusive term with power-type coefficient. This SDE has the particularity that the standard conditions for the existence and uniqueness of solutions of SDES (linear growth and Lipschitz) do not hold for a large subset of parameter space. Thus, we start by studying the well posedness of the problem at hand, obtaining a detailed characterization for the existence and uniqueness of solutions. We then provide criteria ensuring extinction and persistence of such population. Additionally, we find subsets of parameter space where (absolutely continuous) stationary measures for the SDE under consideration are guaranteed to exist, providing a description for the corresponding densities. We conclude with an initial application to the optimal management of resources. We consider a real asset such as, for instance, a farm or an aquaculture facility, devoted to the exploration of a unique culture or population whose growth follows a SDE such as described above, and look for the optimal harvesting strategy associated with such culture or population.

**19 de Janeiro, 14h00, sala A2-25 Complexo Interdisciplinar UL**

** Diogo Pinheiro**

CUNY, USA

**Stochastic optimal control under model uncertainty**

**Abstract:**

I will discuss stochastic optimal control problems with model uncertainty either in the form of a discrete sequence of random time horizons or in the form of a parametric dependence on a certain switching process. Such problems are interesting not only for their mathematical novelty, but also for their potential application to subjects such as Finance, Actuarial Science, Economics, Population Dynamics and Engineering. I will focus mainly on the derivation of generalized dynamic programming principles, as well as of the corresponding Hamilton-Jacobi-Bellman equations. Time allowing, I will discuss some applications of these abstract results.