An invariant of basic sets of Smale flows (original) (raw)

Invariants of twist-wise flow equivalence

Discrete & Continuous Dynamical Systems - A, 1998

Twist-wise flow equivalence is a natural generalization of flow equivalence that takes account of twisting in the local stable manifold of the orbits of a flow. Here we announce the discovery of two new invariants in this category.

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.

Unstable manifold, Conley index and fixed points of flows

Journal of Mathematical Analysis and Applications, 2014

We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point. * The authors are supported by MINECO (MTM2012-30719).

Regularity of Invariant Manifolds for Nonuniformly Hyperbolic Dynamics

Journal of Dynamics and Differential Equations, 2008

For any sufficiently small perturbation of a nonuniform exponential dichotomy, we show that there exist invariant stable manifolds as regular as the dynamics. We also consider the general case of a nonautonomous dynamics defined by the composition of a sequence of maps. The proof is based on a geometric argument that avoids any lengthy computations involving the higher order derivatives. In addition, we describe how the invariant manifolds vary with the dynamics.

Flows on Vector Bundles and Hyperbolic Sets

Transactions of the American Mathematical Society, 1988

This note deals with C. Conley's topological approach to hyperbolic invariant sets for continuous flows. It is based on the notions of isolated invariant sets and Morse decompositions and it leads to the concept of weak hyperbolicity.

A Unified View of Topological Invariants of Fluid Flows

Topologica, 2008

The helicity is a topological invariant of an ideal fluid in three dimensions. Two-dimensional ideal flows admit an integral of any function of vorticity as topological invariants. This is extended to axisymmetric flows. We show that these are variants of the cross helicity. Noether's theorem associated with the particle relabeling symmetry underpins this unified view. A comment is given to the bearing of Kelvin's circulation theorem with Noether's second theorem.

The ?-stability theorem for flows

Inventiones Mathematicae, 1970

Let X be a C r tangent vector field, r> 1, on a compact smooth Riemannian manifold M, without boundary, and let ~p = {~Pt} be the flow it generates. We think of q~ as a C r action of R on M. In Smale states and proves the O-stability theorem for diffeomorphisms of Mthat is, C' actions of Z on M. He states the corresponding theorem for flows and says "Presumably there is no difficulty in proving similar theorems for ordinary differential equations by the same method." This paper does just that and, assuming [2, 33 does it without much difficulty.

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 .

A topological conjugacy of invariant flows on some class of Lie groups

Boletim da Sociedade Paranaense de Matemática, 2019

The aim of this paper is to classify invariant flows on Lie group GGG whose Lie algebra mathfrakg\mathfrak{g}mathfrakg is associative or semisimple. Specifically, we present this classification from the hyperbolicity of the lift flows on GtimesmathfrakgG \times \mathfrak{g}Gtimesmathfrakg. Then we apply this construction to some special cases as rmGl(2,BbbR){\rm Gl}(2,{\Bbb R})rmGl(2,BbbR) and affine Lie group.