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Topology and dynamics of a flow that has a non-saddle set or a $W$-set

Héctor Barge, J. J. Sánchez-Gabites, J. M. R. Sanjurjo

Abstract

The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set or a $W$-set in a manifold. These are certain classes of compact invariant sets in whose vicinity the asymptotic behaviour of the flow is somewhat controlled. We are mainly concerned with global properties of the dynamics and establish cohomological relations between the non-saddle set and the manifold. As a consequence we obtain a dynamical classification of surfaces (orientable and non-orientable). We also examine robustness and bifurcation properties of non-saddle-sets and study in detail the behavior of $W$-sets in 2-manifolds.

Topology and dynamics of a flow that has a non-saddle set or a $W$-set

Abstract

The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set or a -set in a manifold. These are certain classes of compact invariant sets in whose vicinity the asymptotic behaviour of the flow is somewhat controlled. We are mainly concerned with global properties of the dynamics and establish cohomological relations between the non-saddle set and the manifold. As a consequence we obtain a dynamical classification of surfaces (orientable and non-orientable). We also examine robustness and bifurcation properties of non-saddle-sets and study in detail the behavior of -sets in 2-manifolds.

Paper Structure

This paper contains 12 sections, 22 theorems, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Attractors
  4. Isolated invariant sets
  5. Isolating blocks
  6. Homotopies induced by the flow
  7. The Conley index
  8. The structure of a flow having a global non-saddle set
  9. Non-saddle decompositions
  10. Continuations and bifurcations of non-saddle sets
  11. $W$-sets
  12. Non-global $W$-sets

Key Result

Theorem 3.1

Let $K$ be a global non-saddle set and $S$ a section of $M \setminus K$. Then, the quotient $M/K$ is homotopy equivalent to $\Sigma S\vee \mathbb{S}^1$, were $\Sigma S$ denotes the suspension of $S$. As a consequence the cup product in $\check{H}^*(M,K)$ is trivial. Moreover, there exists an exact s

Figures (4)

  • Figure 1: Flow on the torus.
  • Figure 2: Continuations of an isolated non-saddle set may be saddle
  • Figure 3: Mendelson flow
  • Figure 4: Modification of the Mendelson flow

Theorems & Definitions (49)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Example 3.3
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • Corollary 3.6: Dynamical characterization of closed surfaces
  • ...and 39 more