Smale Flows on S 2 × S 1 (original) (raw)

Smale flows on mathbbS2timesmathbbS1\mathbb{S}^2\times\mathbb{S}^1mathbbS2timesmathbbS1

arXiv: Dynamical Systems, 2014

In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on S 2 × S 1. This classification gives necessary and sufficient conditions that must be satisfied by an (abstract) Lyapunov graph in order for it to be associated to a Smale flow on S 2 × S 1 .

Smale flows on

Ergodic Theory and Dynamical Systems, 2015

Lyapunov graph in the study of Smale flows and Morse/Novikov flows

2021

In this work Lyapunov graphs are used as a combinatorial tool in order to obtain a complete classification of Smale flows on S 2 × S 1 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 S 2 × S 1 or to a Morse-Novikov flow on a surface respectively. In summary in this doctoral thesis we obtain the following results: 1. 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 S 2 × S 1 or a Morse-Novikov flow on a surface. 2. The realization of these graphs subject to the conditions found above as Smale flows on S 2 × S 1 or as Morse-Novikov flows on surfaces respectively is obtained.

Lyapunov graphs and flows on surfaces

Transactions of the American Mathematical Society, 1993

In this paper, a characterization of Lyapunov graphs associated to smooth flows on surfaces is presented. We first obtain necessary and sufficient conditions for a Lyapunov graph to be associated to Morse-Smale flows and then generalize them to smooth flows. The methods employed in the proofs are of interest in their own right for they introduce the use of the Conley index in this context. Moreover, an algorithmic geometric construction of flows on surfaces is described.

An invariant of basic sets of Smale flows

Ergodic Theory and Dynamical Systems, 1997

We consider one-dimensional flows which arise as hyperbolic invariant sets of a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

The topology and dynamics of flows

Open Problems in Topology II, 2007

After a brief survey of various types of flows (Morse-Smale, Smale, Anosov, & partially hyperbolic) we focus on Smale flows on S 3. However, we do give some consideration to Smale flows on other three-manifolds and to Smale diffeomorphisms.

Gutierrez-Sotomayor Flows on Singular Surfaces

2020

In this work we address the realizability of a Lyapunov graph labeled with GS singularities, namely regular, cone, Whitney, double crossing and triple crossing singularities, as continuous flow on a singular closed 222-manifold mathbfM\mathbf{M}mathbfM. Furthermore, the Euler characteristic is computed with respect to the types of GS singularities of the flow on mathbfM\mathbf{M}mathbfM. Locally, a complete classification theorem for minimal isolating blocks of GS singularities is presented in terms of the branched one manifolds that make up the boundary.

Lyapunov graphs for circle valued functions

Topology and its Applications, 2018

Conley index theory is used to obtain results for flows associated to circular Lyapunov functions defined on general compact smooth n-manifolds. This is done in terms of their underlying circular Lyapunov digraphs, which are generalizations of Morse digraphs, by extensively studying their combinatorics, invariants and realizabilty.

MORSE DECOMPOSITIONS OF SEMIFLOWS ASSOCIATED WITH GRAPHS

EXTENDED ABSTRACT. The mathematical theory of dynamical systems analyzes, from an axiomatic point of view, the common features of many models that describe the behavior of systems in time. In its abstract form, a dynamical system is given by a time set T (with semigroup operation •), a state space M , and a map Φ : T ×M → M that satisfies (i) Φ(0, x) = x for all x ∈ M , describing the initial value, and (ii) Φ(t • s, x) = Φ(t, Φ(s, x)) for all t, s ∈ T and x ∈ M . Common examples for the time set T are the natural numbers N or the nonnegative reals R + as semigroups, and the integers Z or the reals R as groups (under addition). If the state space M carries an additional structure, such a being a measurable space, a topological space or a manifold, the map Φ is required to respect this structure, i.e. it is assumed to be measurable, continuous, or differentiable, respectively. At the heart of the theory of dynamical systems is the study of system behavior as t → ∞ or t → ±∞ (qualitat...

On geometric properties of stochastic flows related to the Lyapunov spectrum

Probability Theory and Related Fields, 2000

We study the geometric properties of two stochastic flows on spheres in Euclidean space. The underlying one-point motion in both cases is Brownian. Both flows arise from the action of a Lie group valued Brownian motion on a quotient. For both flows the curvature of a curve moving under the flow is shown to be a diffusion, null recurrent in one case and transient in the other.

Smale Flows on S 2 × S 1 (original) (raw)

Related papers