A remark on conservative diffeomorphisms

Abstract. We show that a stably ergodic dieomorphism can be C1 approx-imated by a dieomorphism having stably non-zero Lyapunov exponents. The proof is a simple application of several recent results, by Bonatti-Daz-Pujals, Arbieto-Matheus, Bonatti-Baraviera and Bochi-Viana. Two central notions in Dynamical Systems are ergodicity and hyperbolicity. In many works showing that certain systems are ergodic, some kind of hyperbolicity (e.g. uniform, non-uniform or partial) is a main ingredient in the proof. In this note we do something in the converse direction. Let M be a compact manifold of dimension d 2, and let be a volume measure in M. Take > 0 and let Di1+ (M) be the set of -preserving C 1+ dieomorphisms, endowed with the C1 topology. Let SE Di1+ (M) be the set of stably ergodic dieomorphisms (i.e., the set of dieomorphisms such that every suciently C1-close C1+ conservative dieomorphism is ergodic). Our result answers positively a question of Burns, Dolgopyat, and Pesin [BuDP]: Theorem 1. There is open and dense set R SE such that if f 2 R then f is non-uniformly hyperbolic, that is, all Lyapunov exponents of f are non-zero. Moreover, every f 2 R admits a dominated splitting. Remark. It is not true that every stably ergodic dieomorphism can be approxi-mated by a partially hyperbolic system (in the weaker sense), by the examples of Tahzibi [T]. Remark. Let SE 0 be the set of dieomorphisms f 2 SE such that every power f k, k 2, is ergodic. Then every f in an open dense subset of SE 0 is Bernoulli. This follows from theorem 1 and Pesin theory. The proof of theorem 1 has three steps. 1. A stably ergodic (or stably transitive) dieomorphism f must have a dom-inated splitting. This is true because if it doesn't, Bonatti, Daz, and Pujals [BDP] permits us to perturb f and create a periodic point whose derivative is the identity. Then, using the Pasting Lemma of Arbieto and Matheus [AM] (for which C1+ regularity is an essential hypothesis), one breaks transitivity. 2. A result of Bonatti and Baraviera [BB] { which is a renement of a tech-nique developed by Shub and Wilkinson [SW] { gives a perturbation of