We discuss how to realize simple Smale Flows on 3-manifolds. We focus on three questions: (1) What are the topological conjugate classes of Lorenz Smale flows that can be realized on S3? (2) Which 3-manifolds can also admit a Lorenz Smale flow? (3) What are the topological conjugate classes of simple Smale flows whose saddle set can be modeled by &nu(0+,0+,0,0) can be realized on S3? This dissertation extends the work of M. Sullivan and B. Yu.

We study the linking structure of the attractor-repeller pairs in simple Smale flows on the 3-sphere in which the chaotic saddle set is modeled by four-band templates with twisted bands. This is a small step in an attempt to classify simple Smale flows on S³. We obtain three new theorems which illustrate that the dynamics of simple Smale flows are sensitive to half-twists in the bands of the embedded template. Haynes and Sullivan showed that the attractor-repeller pair a∪r in a simple Smale flow with chaotic saddle set modeled by embedded template U⁺ is either a Hopf link or a trefoil and meridian [19]. By placing a single half-twist in a selected band of U⁺, we obtain four new templates that model chaotic saddle sets. For simple Smale flows on S³ with chaotic saddle sets modeled by those templates, we find that such simple Smale flows are realizable and that a∪r must be a Hopf link, a figure-8 knot and meridian, a trefoil and meridian, or a cinquefoil and meridian.

In this dissertation, we study the flows on R3 associated with a nonlinear system differential equation introduced by Clark Robinson in [46]. The periodic orbits are modeled by a semi-flow on the L(−1,−1) template. It is known that these are positive knots, but need not have positive braid presentations. Here we prove that they are fibered. We investigate their linking and we construct a zeta-function that counts periodic orbits according to their twisting. This extends work by M. Sullivan in [55], and [57].

Orientador: Ketty Abaroa de Rezende / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T17:12:31Z (GMT). No. of bitstreams: 1 EspirituLedesma_GuidoGerson_D.pdf: 1229937 bytes, checksum: 00f2d538b5b2a2c4147d828351f4ef16 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho, usamos os grafos de Lyapunov como uma ferramenta combinat{\'o}ria para obter classifica\c{c}{\~o}es completas de fluxos Smale sobre $\ss$ e fluxos Morse-Novikov sobre superf{\'i}cies orient{\'a}veis e n{\~a}o orient{\'a}veis. Esta classifica\c{c}{\~a}o consiste em obter condi\c{c}{\~o}es necess{\'a}rias e suficientes que devem ser satisfeitas por um grafo de Lyapunov abstrato de forma a ser associado a um fluxo Smale sobre $\ss$ ou um fluxo Morse-Novikov sobre uma superf{\'i}cie respectivamente. Assim nesta tese de doutorado obtemos os seguintes resultados: \begin{enumerate} \item As condições locais que devem ser satisfeitas por cada vértice do grafo de Lyapunov, assim como as condições globais que devem ser satisfeitas pelos grafos para estarem associados a um fluxo Smale sobre $\ss$ ou a um fluxo Morse-Novikov sobre uma superfície s{\~a}o determinadas. \item A realização destes grafos abstratos sujeita {\'a}s condições determinadas acima, como fluxos Smale sobre $\ss$ ou fluxos Morse-Novikov sobre superfícies respectivamente, são obtidas. \end{enumerate} / Abstract: In this work Lyapunov graphs are used as a combinatorial tool in order to obtain a complete classification of Smale flows on $\ss$ and Morse-Novikov flows on orientable and non-orientable surfaces. This classification consists in determining necessary and sufficient conditions that must be satisfied by an abstract Lyapunov graph so that it is associated to a Smale flow on $\ss$ or to a Morse-Novikov flow on a surface respectively.\\ In summary in this doctoral thesis we obtain the following results: \begin{enumerate} \item The local conditions that must be satisfied by each vertex on a Lyapunov graph is determinated as well as the global conditions on the graph in order for it to be associated to a Smale flow on $\ss$ or a Morse-Novikov flow on a surface. \item The realization of these graphs subject to the conditions found above as Smale flows on $\ss$ or as Morse-Novikov flows on surfaces respectively is obtained. \end{enumerate} / Doutorado / Matematica / Doutor em Matemática