Correction to "Connecting invariant manifolds and the solution of the C^1 stability and \Omega-stability conjectures for flows"

Abstract

There is a gap in the proof of Lemma VII.4 in [Ann. of Math. (2) 145 (1997), 81--137]. We present an alternative proof of Theorem B (C^1 Omega-stable vector fields satisfy Axiom A). The novel and essential part in the proof of the stability and Omega-stability conjectures for flows is the connecting lemma introduced previously. A mistake in the proof of the last conjecture was pointed out to me by Toyoshiba, who later also provided an independent proof of it, again based on the connecting lemma and previous arguments by Ma\~n\'e and Palis.