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Three heteroclinic orbits induce a countable family of equivalence classes of regular flows

Elena Gurevich

Abstract

We solve the problem of topological classification for smooth structurally stable flows on closed four-dimensional manifolds, the non-wandering set of which contains exactly two saddle equilibria, and the wandering set contains isolated trajectories connecting these saddle equilibria (heteroclinic curves). In particular, we show that for a flow of the class under consideration on $\mathbb{CP}^2$, the number of heteroclinic curves is a complete topological invariant, while on the sphere $\mathbb S^4$, there exists a countably many equivalence classes with an arbitrary odd number $γ\geq 3$ of heteroclinic curves. These results contrast with a three-dimensional case, where under similar conditions there exists only finite set of equivalence classes for each number of heteroclinic curves.

Three heteroclinic orbits induce a countable family of equivalence classes of regular flows

Abstract

We solve the problem of topological classification for smooth structurally stable flows on closed four-dimensional manifolds, the non-wandering set of which contains exactly two saddle equilibria, and the wandering set contains isolated trajectories connecting these saddle equilibria (heteroclinic curves). In particular, we show that for a flow of the class under consideration on , the number of heteroclinic curves is a complete topological invariant, while on the sphere , there exists a countably many equivalence classes with an arbitrary odd number of heteroclinic curves. These results contrast with a three-dimensional case, where under similar conditions there exists only finite set of equivalence classes for each number of heteroclinic curves.
Paper Structure (14 sections, 19 theorems, 45 equations, 4 figures)

This paper contains 14 sections, 19 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Topological Aspects of the Dynamics of Flows from the Class $G_{\mu, \nu, k}$
  3. Energy Function
  4. Proof of the Theorem \ref{['top']}
  5. Handle decomposition of the ambient manifold
  6. Properties of the scheme of the flow $f^t\in G_{1,2,k}$
  7. Necessary and Sufficient Conditions for Topological Equivalence of Flows from $G_{1,2,k}$
  8. Two auxiliary topological lemmas
  9. Refining the properties of the homeomorphism $h: \Sigma\to \Sigma'$ satisfying the definition of equivalence of schemes
  10. Construction of consistent canonical neighborhoods of saddle equilibria
  11. Local topological conjugacy of flows $f^t, {f'}^t\in G_{1,2,k}$ in consistent canonical neighborhoods of saddle equilibria
  12. End of the proof of the Theorem \ref{['cond']}
  13. Realization of Topological Equivalence Classes
  14. Proof of Theorem \ref{['realiz']}

Key Result

Theorem 1

Let $f^t\in G_{\mu,\nu,k}$. Then exactly one of the following implications holds.

Figures (4)

  • Figure 1: Sphere $\Lambda$, trivial knot $\lambda$, its neighborhood $N_\lambda$, and disk $P_1$
  • Figure 2: a) canonical neighborhood of a saddle; b) consistent canonical neighborhoods of saddles $\sigma_1,\sigma_2$
  • Figure 3: Nontrivial sphere $\Lambda$ and trivial knot $\lambda_{k,i}$ in $\Sigma=\mathbb S^2\times \mathbb S^1$
  • Figure 4: Ball $B=\Sigma\setminus ({\rm int}\, N_{_{\Lambda}}\cup {\rm int}\, C_b)$ and arcs $l_{i,j}, e_{i,j}\subset B$ for $i\in \{0,1\}, j\in \{1,2\}$

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Definition 1
  • Definition 2
  • Definition 3
  • ...and 15 more