A Remark on Conservative Diffeomorphisms

We show that a stably ergodic diffeomorphism can be C approximated by a diffeomorphism having stably non-zero Lyapunov exponents. 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 the converse direction is investigated. Let M be a compact manifold of dimension d ≥ 2, and let μ be a volume measure in M . Take α > 0 and let Diff μ (M) be the set of μ-preserving C 1+α diffeomorphisms, endowed with the C1 topology. Let SE ⊂ Diff μ (M) be the set of stably ergodic diffeomorphisms (i.e., the set of diffeomorphisms such that every sufficiently C1-close C1+α conservative diffeomorphism is ergodic). Our result answers positively a question of [BuDP]: Theorem 1. There is an open and dense set R ⊂ SE such that if f ∈ R then f is non-uniformly hyperbolic, that is, all Lyapunov exponents of f are non-zero. Moreover, every f ∈ R admits a dominated splitting. Remark. The set SE contains all Anosov diffeomorphisms, and many partially hyperbolic ones – see [GPS]. It is not true that every stably ergodic diffeomorphism can be approximated by a partially hyperbolic system (in the weaker sense), see [T, BnV]. Remark. Let SE ′ be the set of diffeomorphisms f ∈ SE such that every power f, k ≥ 2, is ergodic. Then every f in an open dense subset of SE ′ 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) diffeomorphism f must have a dominated splitting. This is true because if it doesn’t, [BDP] permits us to perturb f and create a periodic point whose derivative is the identity. Then, using the Pasting Lemma from [AM] (for which C1+α regularity is an essential hypothesis), one breaks transitivity. 2. A result of [BB] gives a perturbation of f such that the sum of the Lyapunov exponents “inside” each of the bundles of the (finest) dominated splitting is non-zero. Date: May 4, 2008. J. B. is supported by CNPq-Profix. J.B. wishes to thank the hospitality of the LAGA – Univertité Paris 13. 1 2 J. BOCHI, B. R. FAYAD, E. PUJALS 3. Using a result of [BV], we find another perturbation such that the Lyapunov exponents in each of the bundles become almost equal. (If we attempted to make the exponents exactly equal, we couldn’t guarantee that the perturbation is C1+α.) Since the sum of the exponents in each bundle varies continuously, we conclude there are no zero exponents. Remark. The perturbation techniques of [BB] and [BV] in fact don’t assume ergodicity, but are only able to control the integrated Lyapunov exponents. That’s why we have to assume stable ergodicity (in place of stable transitivity) in theorem 1. Remark. Some ideas of the proof were already present in [DP]. Let us recall briefly the definition and some properties of dominated splittings, see [BDP] for details. Let f ∈ Diffμ(M). A Df -invariant splitting TM = E1⊕· · ·⊕E, with k ≥ 2, is called a dominated splitting (over M) if there are constants c, τ > 0 such that (1) ‖Df(x) · vj‖ ‖Dfn(x) · vi‖ < ce for all x ∈ M , all n ≥ 1, and all unit vectors vi ∈ E (x) and vj ∈ E (x), provided i < j. (One can also define in the same way a dominated splitting over an f invariant set.) A dominated splitting is always continuous, that is, the spaces Ei(x) depend continuously on x. Also, a dominated splitting persists under C1-perturbations of the map. More precisely, if g is sufficiently close to f , then g has a dominated splitting E1 g ⊕ · · · ⊕ E k g , called the continuation, which coincides with the given one when g = f . Moreover, E g(x) depends continuously on g (and x). A dominated splitting E1 ⊕ · · · ⊕ E is called the finest dominated splitting if there is no dominated splitting defined over all M with more than k bundles. If some dominated splitting exists, then the finest dominated splitting exists, is unique, and refines every dominated splitting. The continuation of the finest dominated splitting is not necessarily the finest dominated splitting of the perturbed diffeomorphism. Nevertheless, we have: Lemma 2. Assume f ∈ Diff μ (M) has a dominated splitting and U ⊂ Diff 1+α μ (M) is a neighborhood of f . Then there is an open set V ⊂ U and k ∈ N such that every g ∈ V, the finest dominated splitting exists and has k bundles. Moreover, this splitting E1 g ⊕ · · · ⊕ E k g varies continuously with g over V. The proof of the lemma is obvious. Let us call a dominated splitting for f ∈ Diff μ (M) stably finest if it has a continuation which is the finest dominated splitting of every sufficiently C1-close diffeomorphism of class C1+α. Thus, lemma 2 says that diffeomorphisms with stably finest dominated splittings are (open and) dense among C1+α diffeomorphisms with a dominated splitting. A REMARK ON CONSERVATIVE DIFFEOMORPHISMS 3 Let λ1(f, x) ≥ · · · ≥ λd(f, x) be the Lyapunov exponents of f (counted with multiplicity), defined for almost all x. We write also