Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference

Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts

We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator \CR acting on the space of potentials are such that the pressure function \gamma \mapsto \CP(-\gamma V) exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of \CR, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.Comment: The paper was withdrawn from publication due to an error found in some proof. This is a new version and resubmitted for publication. The occurance of phase transition is proved for a parameter a<1 and it is proved there is no phase transition for a>1. For the value a=1 it is still unkow

RENORMALIZATION, THERMODYNAMIC FORMALISM AND QUASI-CRYSTALS IN SUBSHIFTS.

The paper was withdrawn from publication due to an error found in some proof. This is a new version and resubmitted for publication. The occurance of phase transition is proved for a parameter a1. For the value a=1 it is still unkown.We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator \CR acting on the space of potentials are such that the pressure function \gamma \mapsto \CP(-\gamma V) exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of \CR, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system

A study of the sensitivity of topological dynamical systems and the Fourier spectrum of chaotic interval maps

We study some topological properties of dynamical systems. In particular the rela- tionship between spatio-temporal chaotic and Li-Yorke sensitive dynamical systems establishing that for minimal dynamical systems those properties are equivalent. In the same direction we show that being a Li-Yorke sensitive dynamical system implies that the system is also Li-Yorke chaotic. On the other hand we survey the possibility of lifting some topological properties from a given dynamical system (Y, S) to an- other (X, T). After studying some basic facts about topological dynamical systems, we move to the particular case of interval maps. We know that through the knowl- edge of interval maps, f : I → I, precious information about the chaotic behavior of general nonlinear dynamical systems can be obtained. It is also well known that the analysis of the spectrum of time series encloses important material related to the signal itself. In this work we look for possible connections between chaotic dynamical systems and the behavior of its Fourier coefficients. We have found that a natural bridge between these two concepts is given by the total variation of a function and its connection with the topological entropy associated to the n-th iteration, fn(x), of the map. Working in a natural way using the Sobolev spaces Wp,q(I) we show how the Fourier coefficients are related to the chaoticity of interval maps

Vector fields with heteroclinic networks

Dissertação de Doutoramento em Matemática apresentada à Faculdade de Ciências da Universidade do PortoO trabalho desenvolvido ao longo desta tese tem como ponto de partida uma família de equações diferenciais apresentada e estudada por Field (Ver M.J Field, 1996, Lectures on Bifurcations, Dynamics and Symmetry, Pitman Research Notes in Mathematics Series 356, Longman).Field conjectura, com base no seu estudo analítico e numérico, a existência, para certos valores dos parâmetros, de uma rede heteroclínica envolvendo os equilíbrios e as trajectórias periódicas na dinâmica do sistema. No caso de as variedades invariantes de dimensão 2 dos equilíbrios e das trajectórias periódicas se intersectarem transversalmente, Field conjectura também a existência de dinâmica da ferradura da rede heteroclínica.Nesta tese provamos as conjecturas de Field. O trabalho aqui desenvolvido indica a existência de uma rede heteroclínica de Shilnikov e prova a existência de dinâmica da ferradura na vizinhança de uma tal rede heteroclínica.Usamos a simetria do sistema para definir a rede heteroclínica quociente. Isto sugeriu-nos uma abordagem para estudar a dinâmica na vizinhança da rede heteroclínica de Shilnikov. O estudo da dinâmica é efectuado com recurso a uma codificação da dinâmica ao longo da rede heteroclínica e a uma codificação local na vizinhança dos ciclos heteroclínicos na rede quociente.Construímos exemplos simples contendo ciclos heteroclínicos de Shilnikov que são topologicamente equivalentes a ciclos heteroclínicos quocientes no exemplo de Field. Um facto importante acerca destes exemplos é que, apesar de possuírem dinâmica complexa, pela forma como são construídos, são mais fáceis de manipular analiticamente. Por exemplo, provamos analiticamente a intersecção transversal das variedades invariantes de dimensão 2.Os exemplos que construímos ajudam a compreender o comportamento complexo no exemplo de Field. Provamos a existência de dinâmica da ferradura na vizinhança de ciclos heteroclínicos envolvendo duas selas com autovalores complexos. Isto prova a existência ..