C stability and Ω - stability conjectures for flows

There is a gap in the proof of Lemma VII.4 in [1]. We present an alternative proof of Theorem B (C 1 Ω-stable vector fields satisfy Axiom A) in [1]. The novel and essential part in the proof of the stability and Ω-stability conjectures for flows is the connecting lemma introduced in [1]. A mistake in the proof of the last conjecture was pointed out to me by Toyoshiba [5], who later also provided an independent proof of it, again based on the connecting lemma and previous arguments by Mañé and Palis. The crucial step to the proof of Theorem B is the separation of singu-larities from periodic orbits ([1, Corollary III]) by the C 1 connecting lemma ([1, Theorem A]). After the separation, the proof proceeds based on Mañé's theorems used in [3] and we still rely on Palis's argument in [4], proving first the density of Axiom A diffeomorphisms in the set of C 1 Ω-stable ones to then show that every C 1 Ω-stable diffeomorphism satisfies Axiom A. Let G 1 Ω (M) be the set of C 1 Ω-stable vector fields on a compact smooth boundaryless manifold M with the C 1 topology and X ∈ G 1 Ω (M). As in [1], we prove the hyperbolicity of Per(X) (= Ω(X) − Sing(X)) by induction. In fact, we prove that P j (X) is hyperbolic assuming that j−1 i=0 P i (X) is hyperbolic for some 1 ≤ j ≤ dimM − 1, where P i (X) is the closure of the set of periodic points with index i (dimension of the stable subspace), which is enough to conclude that X satisfies Axiom A. For a dense subset of G 1 Ω (M), we can use the statement of [1, Lemma VII.4] by an already classic argument on set-valued functions of C 1 vector fields. In fact, there is a residual subset of the set of C 1 vector fields (therefore of G 1 Ω (M)) in which the closure of the set of hyperbolic periodic points of saddle type moves continuously with respect to vector fields (see for instance the proof of [1, Corollary II] for this kind of argument). Therefore, as proved in [1], we get the density of Axiom A vector fields in G 1 Ω (M). Then, by Ω-conjugacy, we see that Ω(X) can be decomposed into a finite union of disjoint …