Connecting invariant manifolds and the solution of the \(C^ 1\) stability and \(\Omega\)-stability conjectures for flows.

*(English)*Zbl 0871.58067
Ann. Math. (2) 145, No. 1, 81-137 (1997); correction ibid. 150, No. 1, 353-356 (1999).

This spectacular paper contains a complete proof of the famous \(C^1\)–Stability Conjecture and \(\Omega\)–Stability Conjecture for flows. The general idea of the proof is similar to the proof of the same conjectures for diffeomorphisms presented by Mañé and Palis in 1988. Namely, one proves that the stability assumption yields the hyperbolicity of the nonwandering set. The biggest difficulty in the flow case was the existence of singularities, because the methods used for diffeomorphisms could not be applied when the set of periodic points would accumulate on singularities. The author has overcome this problem by proving that if a \(C^1\)-vector field \(X\) is \(C^1\) \(\Omega\)-stable then Sing\((X) \cap \overline{\text{Per}(X)} = \emptyset\). This result is formulated as a corollary of the following powerful theorem.

\(C^1\)-Connecting Lemma. Let \(f\in\text{Diff}^1(M)\), resp \(X \in \chi^1(M)\), a neighborhood \({\mathcal U}\) of \(f\), resp. of \(X\), and an isolated hyperbolic set \(\Lambda\) be given. If \(f\), resp. \(X\), has an almost homoclinic sequence associated to \(\Lambda\) then there exists \(g \in {\mathcal U}\), resp. \(Y \in {\mathcal U}\), coinciding with \(f\), resp. \(X\), in a neighborhood of \(\Lambda\) and having a homoclinic point associated to \(\Lambda\).

The author announced the above results and their proofs during conferences in 1993 and 1994.

In the paper by L. Wen [J. Differ. Equations 129, No. 2, 334-357 (1996; Zbl 0866.58050)] the stability conjecture was also derived from the connecting lemma.

\(C^1\)-Connecting Lemma. Let \(f\in\text{Diff}^1(M)\), resp \(X \in \chi^1(M)\), a neighborhood \({\mathcal U}\) of \(f\), resp. of \(X\), and an isolated hyperbolic set \(\Lambda\) be given. If \(f\), resp. \(X\), has an almost homoclinic sequence associated to \(\Lambda\) then there exists \(g \in {\mathcal U}\), resp. \(Y \in {\mathcal U}\), coinciding with \(f\), resp. \(X\), in a neighborhood of \(\Lambda\) and having a homoclinic point associated to \(\Lambda\).

The author announced the above results and their proofs during conferences in 1993 and 1994.

In the paper by L. Wen [J. Differ. Equations 129, No. 2, 334-357 (1996; Zbl 0866.58050)] the stability conjecture was also derived from the connecting lemma.

Reviewer: J.Ombach (Kraków)

##### MSC:

37D99 | Dynamical systems with hyperbolic behavior |

37C75 | Stability theory for smooth dynamical systems |

37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |

34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |