Search

Neste trabalho, objetivo é mostrar que mapas topologicamente transitivos na reta real tem Entropia Topológica positiva. Para tal são necessárias algumas noções básicas sobre Topologia, Análise na Reta e Sistemas Dinâmicos que serão essenciais para o desenrolar da dissertação. Para a continuidade do trabalho será necessário apresentar algumas das defi- nições de Entropia Topológica que serão via coberturas abertas e por conjuntos geradores e separados em conjuntos compactos. No que se refere a espaços métricos compactos, as definições citadas serão equivalentes. Com o foco de se trabalhar na reta real será necessário uma nova definição de Entropia Topológica para que seja possível trabalhar no contexto desejado, esta definição será aplicada em mapas em espaços métricos. Nos dois últimos capítulos será trabalhado Entropia topológica em contextos diferentes, no penúl- timo utilizando as definições de Entropia Topológica relacionadas a conjuntos compactos. Enquanto no último capítulo será utilizada a definição em Espaços Métricos. Em ambos os capítulos será feito o estudo para dar uma nova definição de Entropia Topológica en- volvendo mapas monótonos. No capítulo sobre Entropia Topológica em intervalos, haverá um seção que trata o conceito de ferradura, em especial o Teorema de Misiurewicz, que é de fundamental importância para quando o mesmo tema for tratado na reta real. Nos capítulos finais, nos dois últimos, são estudados mapas topologicamente transitivos, com a finalidade de mostrar que suas Entropias Topológicas são positivas. Palavras-chave: Entropia Topológica. Mapas Transitivos. Ferraduras In this work, the objective is to show that topologically transitive maps on the real line have positive Topological Entropy. For that, some basic notions about Topology, Analysis in the Line and Dynamic Systems are necessary, which will be essential for the develop- ment of the dissertation. For the continuity of the work it will be necessary to present some of the definitions of Topological Entropy that will be via open covers and by gene- rating sets and separated into compact sets. With regard to compact metric spaces, the definitions mentioned will be equivalent. With the focus on working on the real line, a new definition of Topological Entropy will be necessary so that it is possible to work in the desired context, this definition will be applied to maps in metric spaces. In the last two chapters, topological entropy will be worked on in different contexts, in the penultimate one using the definitions of topological entropy related to compact sets. While in the last chapter the definition in Metric Spaces will be used. In both chapters the study will be done to give a new definition of Topological Entropy involving monotonous maps. In the chapter on Topological Entropy in intervals, there will be a section that deals with the concept of horseshoe, especially the Misiurewicz Theorem, which is of fundamental im- portance when the same theme is dealt with in the real line. In the final chapters, in the last two, topologically transitive maps are studied, in order to show that their Topological Entropies are positive. Keywords: Topological Entropy. Transitive Maps. Horseshoes.

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2021. Propomos um novo tipo de espaços shift, chamados blur shift spaces. Estes espaços são construídos a partir de espaços shift clássicos, aumentando-se o alfabeto com símbolos que representam subconjuntos infinitos do alfabeto original e então definindo-se uma topologia conveniente. Os espaços obtidos são inspirados e generalizam os espaços shift apresentados por Ott, Tomforde e Willis e por Gonçalves e Royer, os quais foram usados para encontrar isomorfismos entre C*-álgebras associadas a algumas classes de espaços shift. Estudamos algumas propriedades topológicas dos \blur shift spaces, como axiomas de separação, axiomas de enumerabilidade, metrizabilidade e caracterizamos quando os espaços são compactos ou localmente compactos, dependendo da escolha dos conjuntos utilizados como novos símbolos do alfabeto. Em particular, a construção apresentada dos blur shif spaces pode ser utilizada como um esquema multi-pontos para compactificar espaços shift clássicos. Finalizamos caracterizando as funções contínuas que comutam com a função shift, e os sliding block codes generalizados, definidos sobre blur shift spaces. Abstract: We propose a new type of shift spaces, called blur shift spaces. These spaces are constructed from classical shift spaces, enlarging the alphabet with symbols that represent infinite subsets of the original alphabet, then defining a convenient topology. The resulting spaces are inspired and generalize those shift spaces presented by Ott, Tomforde and Willis, and by Gonçalves and Royer, which were used to find isomorphisms between C*-algebras associated to some classes of shift spaces. We study some topological properties of the blur shift spaces, such as separation axioms, countability axioms, metrizability and we characterize when the spaces are compact, or locally compact, depending on the choice of the sets used as new symbols of the extended alphabet. In particular, the presented construction of blur shift spaces may be used as a multi-point compactification scheme for classical shift spaces. We close this thesis characterizing continuous shift commuting maps and generalized sliding block codes defined over blur shift spaces.

Let $E_{\la}(z)=\la {\rm exp}(z), \ \lambda\in \mathbb C$ be the complex exponential family. For all functions in the family there is a unique asymptotic value at 0 (and no critical values). For a fixed $\la$, the set of points in $\mathbb C$ with orbit tending to infinity is called the escaping set. We prove that the escaping set of $E_{\la}$ with $\la$ Misiurewicz (that is, a parameter for which the orbit of the singular value is strictly preperiodic) is a connected set., Comment: 10 pages, 1 figure

We establish a one-to-one correspondence between the renormalizations and proper totally invariant closed sets (i.e., α-limit sets) of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. We describe the minimal renormalization by constructing the minimal totally invariant closed set, so that we can define the renormalization operator. Using consecutive renormalizations, we obtain complete topological characteriza- tion of α-limit sets and nonwandering set decomposition. For piecewise linear Lorenz map with slopes ≥ 1, we show that each renormalization is periodic and every proper α-limit set is countable.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Ernest Fontich, [en] The aim of this project is to prove the Smale-Birkhoff Theorem, which provides sufficient conditions for a dynamical system to show chaotic behavior. In order to achieve this purpose, it will be apparent that symbolic dynamics play an essential role. This technique consists in characterizing the orbits of a dynamical system with bi-infinite sequences of symbols, so its structure is displayed in a simpler way, while preserving the information concerning to the dynamics. It gives the chance to perform qualitative analysis without the need of manipulating directly the complicated structure that these systems have.

In this paper we present a-posteriori KAM results for existence of $d$-dimensional isotropic invariant tori for $n$-DOF Hamiltonian systems with additional $n-d$ independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs., 59 pages, 5 tables

This paper is concerned with the Timoshenko system, a recognized model for vibrations of thin prismatic beams. The corresponding autonomous system has been widely studied. However, there are only a few works dedicated to its non-autonomous counterpart. Here, we investigate the long-time dynamics of Timoshenko systems involving a nonlinear foundation and subjected to perturbations of time-dependent external forces. The main result establishes the existence of a pullback exponential attractor, which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. The upper-semicontinuity of attractors, as the non-autonomous forces tend to zero, is also studied.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Núria Fagella Rabionet, [en] Chaos theory is a branch of mathematics focusing on dynamic systems with irregular behavior. Despite being deterministic dynamical systems, their behavior cannot be predicted since small differences in initial conditions can cause the system to evolve very differently. In this paper we will see some examples of very simple dynamical systems but with chaotic dynamics. We will focus mainly on the study of the family $Q_{c}(x)=x^{2}+c$ on the real and complex case. We will talk about symbolic dynamics and topological conjugacy as a useful tool to compare dynamical systems and transfer information from one to another. These two concepts will be used to prove that a dynamical system is chaotic.

In this article, we use Conley Index Theory to set up a framework to associate topological-dynamical invariants to a Morse–Bott flow in a neighborhood of a critical manifold. Isolating blocks N for a critical manifold S are characterized in terms of conditions on the ranks of its homology Conley index. The necessity of these conditions follows from the generalized Morse–Bott inequalities for isolating blocks. Morse–Bott semi-graphs turn out to be a useful combinatorial device to record these topological conditions. One goal is to verify that these conditions when imposed on an n-abstract Morse–Bott semi-graph are sufficient for its realization. This is attained by introducing Morse–Bott handle surgeries in order to construct isolating blocks for critical manifolds in dimensions 2 and 3, and for a large class in higher dimensions. Stronger results are obtained in dimensions 2 and 3 where necessary and sufficient conditions for a Morse–Bott graph to be associated to a Morse–Bott flow on some manifold M are determined.

Tichler proved (Tischler D. 1970 Topology 9, 153-154. (doi:10.1016/0040-9383(70)90037-6)) that a manifold admitting a smooth non-vanishing and closed oneform fibres over a circle. More generally, a manifold admitting k-independent closed one-form fibres over a torus Tk. In this article, we explain a version of this construction for manifolds with boundary using the techniques of b-calculus (Melrose R. 1993 The Atiyah Patodi Singer index theorem. Research Notes in Mathematics. Wellesley, MA: A. K. Peters; Guillemin V, Miranda E, Pires AR. 2014 Adv. Math. (N. Y.) 264, 864-896. (doi:10.1016/j.aim.2014.07.032)). We explore new applications of this idea to fluid dynamics and more concretely in the study of stationary solutions of the Euler equations. In the study of Euler flows on manifolds, two dichotomic situations appear. For the first one, in which the Bernoulli function is not constant, we provide a new proof of Arnold's structure theorem and describe b-symplectic structures on some of the singular sets of the Bernoulli function. When the Bernoulli function is constant, a correspondence between contact structures with singularities (Miranda E, Oms C. 2018 Contact structures with singularities. https://arxiv.org/abs/1806.05638) and what we call b-Beltrami fields is established, thus mimicking the classical correspondence between Beltrami fields and contact structures (see for instance Etnyre J, Ghrist R. 2000 Trans. Am. Math. Soc. 352, 5781-5794. (doi:10.1090/S0002-9947-00-02651-9)). These results provide a new technique to analyse the geometry of steady fluid flows on non-compact manifolds with cylindrical ends., Robert Cardona is supported by FPI-BGSMath doctoral grant. Eva Miranda is supported by the CatalanInstitution for Research and Advanced Studies via an ICREA Academia Prize 2016 and partially supported bythe grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR).Daniel Peralta-Salas is supported by the ERC Starting Grant 335079, the MTM grant 2016-76702-P, and partiallysupported by the ICMAT–Severo Ochoa grant SEV-2015-0554. This material is based upon work supported by theNational Science Foundation under Grant No. DMS-1440140 while Eva Miranda was in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Fall 2018 semester.

We study chiral models in one spatial dimension, both static and periodically driven. We demonstrate that their topological properties may be read out through the long time limit of a bulk observable, the mean chiral displacement. The derivation of this result is done in terms of spectral projectors, allowing for a detailed understanding of the physics. We show that the proposed detection converges rapidly and it can be implemented in a wide class of chiral systems. Furthermore, it can measure arbitrary winding numbers and topological boundaries, it applies to all non-interacting systems, independently of their quantum statistics, and it requires no additional elements, such as external fields, nor filled bands.

Let X = ( X , B , μ ) be a σ-finite measure space and f : X → X be a measurable transformation such that the composition operator T f : φ ↦ φ ∘ f is a bounded linear operator acting on L p ( X , B , μ ) , 1 ≤ p ∞ . We provide a necessary and sufficient condition on f for T f to be topologically transitive or topologically mixing. We also characterize the topological dynamics of composition operators induced by weighted shifts, non-singular odometers and inner functions. The results provided in this article hold for composition operators acting on more general Banach spaces of functions.

Orientador: Ketty Abaroa de Rezende Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica Resumo: Apresentamos as variedades GS (variedades 2-dimensionais localmente difeomorfas a uma ou mais das seguintes vizinhanças: regular, cone, guarda-chuva de Whitney, ponto duplo e ponto triplo), introduzidas por Gutierrez e Sotomayor. Em seguida, consideramos campos de vetores contínuos no espaço Euclidiano tridimensional, que restritos às vizinhanças das singularidades GS são tangentes. E com o auxílio das cartas locais, obtemos uma representação matricial destes campos para cada um dos tipos de singularidade GS, chamadas de formas algébricas locais. Exploramos a transição entre essas formas algébricas, variando os parâmetros de suas entradas através de famílias de campos a 1-parâmetro tendo uma das vizinhanças das singularidades GS como subvariedade invariante, processo que denominamos de continuação algébrica. Posteriormente, analisamos os efeitos das continuações no aspecto dinâmico-topológico dos campos numa vizinhança de cada singularidade GS, descrevendo as mudanças observadas na natureza das singularidades, bem como a relação entre sua hiperbolicidade e a escolha das famílias de campos a 1-parâmetro. Por fim, introduzimos os blocos GS para definir o seu Índice de Conley, explicitando-o de acordo com os parâmetros. E finalmente, apresentamos uma aplicação nos espaços de configuração de braços de robô Abstract: In this dissertation, GS manifolds (2-dimensional manifolds locally diffeomorphic to one or more of the following neighbourhoods: regular, cone, Whitney's umbrella, double crossing and triple crossing) introduced by Gutierrez and Sotomayor are presented. We consider vector fields in the tridimensional Euclidian space that are tangent vector fields when restricted to the neighbourhoods of GS singularities. Using local charts we obtain a matrix representation of these vector fields for each GS singularity type, which we refer to as the local algebraic form. We explore the transition between algebraic forms that maintain the neighbourhood of a GS singularity invariant. We refer to this process as algebraic continuation. Subsequently, we analyze the dynamical topological effect on the vector fields in a neighbourhood of each GS singularity type, describing the changes in the nature of the singularity in a 1-parameter family of vector fields as it goes through parameter values. We also study the relation between algebraic continuations of a family of vector fields and the hiperbolicity of the singularity. The GS basic blocks are introduced in order to define the Conley Index with respect to the parameters. Lastly, we present an application in the configuration space of a robot arm Mestrado Matemática Mestre em Matemática CAPES

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: Núria Fagella Rabionet, When a holomorphic function is iterated, it generates a dynamic system on the complex plane. In this project, we describe both the local and global behavior of the different orbits of a rational map on the complex plane (or the Riemann sphere). We mainly concentrate in the study of the dynamical plane (where initial conditions and orbits live) although we briefly discuss one parameter families of polynomials and their bifurcation loci, like the well known Mandelbrot set. Towards the end, we experiment with a singular perturbation of a family of cubic polynomials and explore the drastic changes that occur in the topology of their Julia sets.

Agraïments: The three actors were supported by MTM2011-26995-C02-02 and EU Project MRTN-CT-2006-035651 and FPU AP-2009-4564 The goal of this paper is to investigate the parameter plane of a rational family of perturbations of the doubling map given by the Blaschke products. First we study the basic properties of these maps such as the connectivity of the Julia set as a function of the parameter a. We use techniques of quasiconformal surgery to explore the relation between certain members of the family and the degree 4 polynomials. In parameter space, we classify the different hyperbolic components according to the critical orbits and we show how to parametrize those of disjoint type.

[cat] Aquesta tesi doctoral pertany a l’àmbit dels sistemes dinàmics discrets al pla complex, és a dir, la iteració de funcions analítiques en una variable complexa. Donada una funció racional f de l'esfera de Riemann en ella mateixa, considerem el sistema dinàmic donat pels seus iterats. L'esfera de Riemann es divideix en dos conjunts completament invariants per f el conjunt de Fatou, definit com el conjunt de punts z on la família {f^n} és normal en algun entorn de z, i el seu complement, el conjunt de Julià. La dinàmica de les òrbites del conjunt de Fatou és estable en el sentit de normalitat o equicontinuitat mentre que la dinàmica al conjunt de Julià presenta un caràcter caòtic. Aquesta tesi se centra en l'estudi de la família de productes de Blaschke Ba(z)=z^3(z-a)/(1-\bar{a}z), on a i z són nombres complexos. Estudiem el seu pla de paràmetres i el seu pla dinàmic fent us intensiu de les eines de cirurgia quasiconforme, que ens permeten construir funcions racionals amb una dinàmica prescrita fent servir funcions quasiregulars com a models. Al capítol 1 fem un repàs dels resultats preliminars usats al llarg del text. Primer expliquem els conceptes bàsics de la dinàmica de les funcions racionals. Després fem un repàs de les aplicacions del cercle, tot introduint els conceptes de producte de Blaschke i llengües. Finalment, presentem la fórmula de Riemann-Hurwitz i com s’aplica a la dinàmica de funcions racionals. Al capítol 2 donem una introducció a la cirurgia quasiconforme. Primer de tot definim els conceptes d’aplicació quasiconforme, estructures quasiconformes i “pullback” sota funcions que preserven l’orientació i introduïm el Teorema Mesurable de Riemann. Tot seguit mostrem com els conceptes previs són generalitzats per a funcions que giren l’orientació i veiem com això s’aplica a aplicacions que són simètriques respecte del cercle unitat. Finalment introduïm els conceptes d’aplicació polynomial-like i antipolynomial-like. Al capítol 3 donem una visió general del pla dinàmic dels productes de Blaschke Ba. Comencem estudiant les seves propietats bàsiques. Tot seguit mostrem que les funcions Ba. no poden tenir dominis de rotació doblement connexos (anells de Herman) (Proposició 3.2.3) i provem un criteri de connectivitat del conjunt de Julià dels Ba (Teorema 3.2.1). Al capítol 4 introduïm la família Mb de polinomis cúbics amb un punt fix superatractor. A continuació veiem com construir polinomis Mb a partir de productes de Blaschke Ba, tot obtenint una aplicació Γ que envia un subconjunt de l’espai de paràmetres de Ba a l’espai de paràmetres dels polinomis Mb. També provem que l’aplicació Γ és continua i és un homeomorfisme restringida a cada component hiperbòlica disjunta. Al capítol 5 estudiem l’espai de paràmetres dels productes de Blaschke Ba. Primer de tot en descrivim les simetries. A continuació classifiquem els diferents tipus de comportaments hiperbòlics que es poden donar i veiem a quines regions de l’espai de paràmetres poden aparèixer. Tot seguit construïm una aplicació polynomial-like al voltant de tot paràmetre de no escapament contingut en una regió d’intercanvi que, sota certes condicions, pot relacionar la dinàmica de Ba amb la dels antipolinomis pc(z)=\bar{z}^2+c (Teorema 5.3.4). Finalment parametritzem tota component hiperbòlica disjunta els cicles atractors de la qual són acotats i no rauen al cercle unitat (Teorema 5.4.2). Al capítol 6 estudiem les llengües dels productes de Blaschke Ba. Inicialment provem algunes de les seves propietats topològiques bàsiques com ara la seva connectivitat mòdul simetria, la seva connectivitat simple i l’existència d’una única punta per a cada llengua (Teorema 6.2.1). Tot seguit mostrem com es produeixen les bifurcacions en un entorn de la punta de cada llengua (Teorema 6.3.2). Finalment estudiem com les llengües s’estenen per a paràmetres a tals que 12., [eng] This PhD thesis belongs to the area of discrete dynamical systems in the complex plane, i.e. the iteration of analytic functions in one complex variable. Given a rational map f from the Riemann sphere onto itself, we consider the dynamical system given by its iterates. The Riemann sphere splits into two totally f-invariant subsets: the Fatou set, which is defined to be the set of points z where the family {f^n} is normal in some neighborhood of z, and its complement, the Julia set. The dynamics of the points in the Fatou set are stable in the sense of normality or equicontinuity whereas the dynamics in the Julia set present chaotic behavior. This thesis focuses on the study of the family of Blaschke products Ba(z)=z^3(z-a)/(1-\bar{a}z), where a and z are complex numbers. We study its parameter and its dynamical planes using intensive use of quasiconformal surgery techinques, which allow us to build rational maps with prescribed dynamics using quasiregular maps as models. The thesis is structured as follows. In Chapter 1 we give an overview on the preliminary results used throughout the thesis. In Chapter 2 we give an introduction to quasiconformal surgery. In Chapter 3 we give an overview of the dynamical plane of the Blaschke products Ba. We begin by studying their basic properties. Afterwards we show that the maps Ba cannot have doubly connected rotation domains (Herman rings) (Proposition 3.2.3) and prove a criterion of connectivity of the Julia set of Ba (Theorem 3.2.1). In Chapter 4 we introduce the family Mb of cubic polynomials with a superattracting fixed point. Then we show how to build polynomials Mb from Blaschke products Ba , obtaining a map Γ from a subset of the parameter plane of the Ba to the parameter plane of the polynomials Mb. We also prove that the map Γ is continuous and restricts to a homeomorphism on every disjoint hyperbolic component. In Chapter 5 we study the parameter plane of the Blaschke products Ba. We first describe the symmetries in the parameter plane. Then we classify the different hyperbolic dynamics which may take place and the sets of parameters for which they may happen. Afterwards we build a polynomial-like map for all non-escaping parameters contained in swapping regions which, under certain conditions, may relate the dynamics of Ba with the one of the antipolynomials pc(z) =\bar{z}^2+c (Theorem 5.3.4). Finally we parametrize all disjoint hyperbolic components whose disjoint cycles are bounded and do not lie on the unit circle (Theorem 5.4.2). In Chapter 6 we study the tongues of the Blaschke products Ba. We first prove some of their topological properties such as their connectivity modulo symmetry, their simple connectivity and the existence of a unique tip for every tongue (Theorem 6.2.1). Then we show how bifurcations take place along curves in a neighborhood of every tongue (Theorem 6.3.2). Finally we study how tongues extend in the annulus of parameters a such that 12.

In this paper we consider the non local evolution problem $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u =- u + K(f\circ u ) \ \ in \ \ \Omega ,\\ u = 0 \ \ in \ \ \mathbb {R}^N\backslash \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^N\), \(f: \mathbb {R}\rightarrow \mathbb {R}\) and K is an integral operator with a symmetric kernel. We prove existence and some regularity properties of the global attractor. We also characterize the global attractor, using the properties of a Lyapunov functional for this model.

En este trabajo desarrollaremos dos aspectos de Dinámica: El primero que trata sobre la dinámica de funciones que van de un intervalo en si mismo, introduciremos las cadenas de Markov y algunos resultados previos para alcanzar al final el teorema de Sharkovsky demostrado con grafos, el cual lo haremos en la primera parte de este trabajo. La segunda parte de este trabajo tratará sobre la teoría ergódica, nos enfocaremos en dos de los teoremas fundamentales que son el teorema de recurrencia de Poincaré y el teorema de Birkhoff. Tesis

Agraïments: The first and fourth authors were partially supported by P11B2011-30 In this paper we study the topology of the hyperbolic component of the parameter plane for the Newton's method applied to n-th degree Bring-Jerrard polynomials given by P_n(z) = z^n-cz 1, \ c. For n=5, using the Tschirnhaus-Bring-Jerrard nonlinear transformations, this family controls, at least theoretically, the roots of all quintic polynomials. We also study a bifurcation cascade of the bifurcation locus by considering c .

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2013 Neste trabalho serão descritas propriedades de ladrilhamentos nas mais diversas áreas da matemática como topologia, sistemas dinâmicos e topologia algébrica. Veremos um método para construir ladrilhamentos que não admitem simetrias de translação, isto é, não são periódicos. Tais ladrilhamentos são chamados de ladrihamentos de substituição e iremos construir um complexo celular associado e determinar sua cohomologia. O estudo será aplicado a alguns exemplos. Abstract : In this essay we show properties of tilings in many areas of mathematics like topology, dynamic systems and algebraic topology. We describe a method to build a tiling that doesn't admit a symmetry of translation, i.e., it is not periodic. Such tilings are called substitution tilings and we will construct an associated cell complex in order to determine its cohomology. The study will be applied to some examples.

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-11-21Bitstream added on 2014-06-13T18:47:58Z : No. of bitstreams: 1 malvezzi_mf_me_rcla.pdf: 740200 bytes, checksum: 1e2dd81c00969cce13ef462cc3d0888b (MD5) O objetivo deste trabalho é apresentar a teoria de sistemas dinâmicos em espaços topológicos uniformes. Como uma aplicação, mostraremos a construção de um fluxo para equações diferenciais ordinárias não autônomas e, utilizando a teoria de dinâmica topológica abordada, daremos enfoque ao estudo de estabilidade de soluções e à investigação de condições que garantem a existência de soluções periódicas ou quase periódicas The purpose of this work is to present the theory of dynamical systems in uniform topological spaces. As an application, we will show the construction of a flow for nonautonomous ordinary differential equations and, by using the theory of topological dynamics discussed, we will focus on the study of stability of solutions and on the investigation for conditions that guarantee the existence of periodic or almost periodic solutions

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-11-22Bitstream added on 2014-06-13T20:08:00Z : No. of bitstreams: 1 sacramento_aj_me_sjrp.pdf: 3231856 bytes, checksum: 0136158c9dd1d9766f0bd327e206e676 (MD5) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) O objetivo deste trabalho é estudar a equivalência de contato topológica dos germes de aplicações diferenciáveis tendo como plano de fundo o estudo da equivalência de contato clássica (ou C∞-K-equivalência). Neste sentido, apresentamos inicialmente uma análise detalhada sobre alguns invariantes e propriedades clássicas da equivalência de contato e, em seguida, introduzimos o estudo da versão topológica desta relação de equivalência. A equivalência de contato topológica (ou C0-K-equivalência) é um tema que recentemente ganhou o interesse de vários pesquisadores por se tratar de uma relação de equivalência cujos invariantes, propriedades e classi cações são pouco conhecidos ou inexistentes. Sob esta ótica, investigamos se alguns invariantes encontrados no caso clássico poderiam ser reproduzidos ou adaptados para o caso topológico. Como parte principal do trabalho, apresentaremos um invariante completo para a equivalência de contato topológica introduzido por T. Nishimura [22]. Este invariante é dado para germes de aplicações nitamente determinadas cujas dimensões da fonte e da meta coincidem The goal of this work is to study the topological contact equivalence of smooth map germs having as background the study of the classical contact equivalence (or C∞-Kequivalence). In this sense, we rstly present a detailed analysis of some invariants and classical properties of the contact equivalence, and then we introduce the study of the topological version of this equivalence relation. Recently several researchers have been interested in this subject because it is an equivalence relation whose invariants, properties and classi cations are unknown or nonexistent. In this work we investigate if some invariants of contact equivalence could be reproduced or adapted for the topological case. In chapter 3 we present a complete invariant for the topological contact equivalence introduced by T. Nishimura [22]. This invariant is given to nitely determined map germs whose dimensions of the source and target are equal

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-08-10Bitstream added on 2014-06-13T18:07:02Z : No. of bitstreams: 1 marchioli_ab_me_sjrp.pdf: 317494 bytes, checksum: 518683b62d488d3433a0bee79ecd4f53 (MD5) Neste trabalho, estudaremos as propriedades dinâmicas de endomorfismos do plano complexo C. Provaremos e o teorema de Montel e mostraremos algumas propriedades topológicas do conjunto de Julia J(f), onde f : C seta C é uma aplicação racional de grau > ou = 2 In this work, we will study the dynamical properties of endomorfisms of complex plane C. We will also prove Montel's theorem and show some topological properties of Julia set J(f), where f : C 'seta' C is a rational map of degree > ou = 2.

We consider semidynamical systems with impulse effects at variable times and we discuss some properties of the limit sets of orbits of these systems such as invariancy, compactness and connectedness. As a consequence we obtain a version of the Poincaré–Bendixson Theorem for impulsive semidynamical systems.

It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.

We establish a one-to-one correspondence between the renormalizations and proper totally invariant closed sets (i.e., α-limit sets) of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. We describe the minimal renormalization by constructing the minimal totally invariant closed set, so that we can define the renormalization operator. Using consecutive renormalizations, we obtain complete topological characteriza- tion of α-limit sets and nonwandering set decomposition. For piecewise linear Lorenz map with slopes ≥ 1, we show that each renormalization is periodic and every proper α-limit set is countable.

Neste trabalho, mostraremos que o algoritmo que determina digito a digito a raiz quadrada de um número real positivo, corresponde a um sistema dinâmico no plano com um comportamento dinâmico complexo. Uma relação de equivalência pode ser obtida e através dela determinamos um novo sistema dinâmico definido no espaço quociente. Tal sistema dinâmico será estudado a partir de dois pontos de vista: Dinâmica Topológica e Teoria Ergódiga. Mostraremos que tal sistema dinâmico é topologicamente conjugado ao shift map no espaço de Bernoulli sobre 10 símbolos. Além disso, mostraremos que existe uma medida invariante natural a qual ergódiga para este sistema dinâmico. In this work, we will show that the algorithm, which determines digit by digit the square root of a positive real number, corresponds to a dynamical system in the plane with complex dynamical behaviour. A relation of equivalence can be obtained and through it we determine a new dynamical system in the quotient space. Such dynamical system will be study from two points of view: Topological Dynamics and Ergodic Theory. We will show that such dynamical system is topologically conjugated to a shift map in the Bernoulli’s space on 10 symbols. Furthermore we will show that there exists a natural invariant measure which is ergodic for this dynamical system.

The problem of searchability in decentralized complex networks is of great importance in computer science, economy and sociology. We present a formalism that is able to cope simultaneously with the problem of search and the congestion effects that arise when parallel searches are performed, and obtain expressions for the average search cost--written in terms of the search algorithm and the topological properties of the network--both in presence and abscence of congestion. This formalism is used to obtain optimal network structures for a system using a local search algorithm. It is found that only two classes of networks can be optimal: star-like configurations, when the number of parallel searches is small, and homogeneous-isotropic configurations, when the number of parallel searches is large., Comment: 4 pages. Final version accepted in PRL

The purpose of this paper is to introduce and discuss the concept of gα-compactness for topological spaces. An example is considered to show that it is strictly stronger than that of compactness. El artículo no presenta resumen

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

We consider the time evolution of the cellular automaton CA 184 with random initial conditions. We derive a partial differential equation that describes the macroscopic time evolution of the coarse-grained local density and we study some solutions of this equation, in particular shock solutions and rarefaction-type solutions for initial step profiles. In order to elucidate the emergence of the large-scale hydrodynamic behaviour we define a microscopic position of a shock and find its mean velocity, its diffusion coefficient and the microscopic structure of the shock as seen from this position.

Orientador: Euclides de Mesquisa Neto Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica Resumo: Neste trabalho é feita uma análise comparativa entre o Método dos Elementos Finitos, o Método dos Elementos de Contorno e o acoplamento entre os dois métodos aplicados a problemas dinâmicos estacionário entre solo-estrutura (DSSI Dynamic soil-Structure Interaction). Apresentamos a formulação do MEF e MEC para obter o sistema final aplicado ao DSSI, bem como para o processo de acoplamento entre os métodos. Finalmente, algumas aplicações práticas entre as três formas .de modelar o problema do DSSI são mostradas com análise crítica dos resultados Abstract: This Work presents a comparison among the MEF, the MEC and the coupling between them applied to dynamic soil-structure interaction (DSS1). Formulations for the MEC and for the MEF to obtain the final system applied to DSS1 and for the process of coupling between them are also presented. Finaly,some practical aplications of the three ways of modelling the DSS1 are shown with a critical results Mestrado Mestre em Engenharia Mecânica

Preprint enviat per a la seva publicació en una revista científica: Astérisque, 1982, num. 98-99, p. 184-194, It js well known that hyperbolic points of an analytic area preserving mapping (APM) T are unstable. As a Corollary of Moser's twist thcorem the elliptic ones are stable provided the eigenvalues l. of DT at the fixed point are nota k-th root of t.he unity, k~ lf2p+2 p ~l. and any of the first p coefficients of the Birkhoff normal form is non-zero. To end the study of the stability of fixed μoints we study the parabolic ar degenerated case. Elliptic points far which stability can not be decided using directly Moser' s theorem (specially if " is a third or fourth root of the uni ty) can be reduced to the parabolic case taking a suitable power of T. The main result is that a degenerated fixed point of an analytic APM is stable if and only if the generating function of T, with the part which generates the identity suppressed, has a strict extremum at the fixed point. Sorne examples and comment are included.

Birkhoff conjectured that the elliptic billiard was the only integrable convex billiard. Here we prove a local version of this conjecture: any non-trivial symmetric entire perturbation of an elliptic billiard is non-integrable.

Agraïments: grant 346300 for IMPAN from the Simons Foundation The goal of this paper is to investigate the family of Blasche products B_a(z)=z^3-a1- which is a rational family of perturbations of the doubling map. We focus on the tongue-like sets which appear in its parameter plane. We first study their basic topological properties and afterwords we investigate how bifurcations take place in a neighborhood of their tips. Finally we see how the period one tongue extends beyond its natural domain of definition.