AXIOM A DIFFEOMORPHISMS WHICH ARE DERIVED FROM ANOSOV FLOWS. CHRISTIAN BONATTI AND NANCY GUELMAN Abstract. Let M be a closed 3-manifold, and X t be a transitive Anosov flow. We build a diffeomorphism of the form f (p)= Y t(p) (p), where Y is an Anosov flow equivalent to X. The diffeomorphism f is structurally stable (i.e. satisfies the Axiom A and the strong transversality condition); the non-wandering set of f is the union of a transitive attractor and a transitive repeller; finally f is also partially hyperbolic (the direction R.Y is the central bundle). 1. Introduction Let X be a transitive Anosov vector field on a 3-manifold M , and X (.,t): M × R → M its flow. In 1975, Palis and Pugh ([14])asked if the time one map p → X (p, 1) of X may be C 1 approximated by Axiom A diffeomorphisms; as they noticed, the answer is positive when X is the suspension of an Anosov diffeomorphism. It was only by the beginning of this century that [11] and [2] give a partial negative answer to this question: a transitive Anosov flow which is not topologically equivalent to a suspension cannot be approximated by Axiom A diffeomorphisms having more than one attractor. A flow is equivalent to a suspension if and only if it admits a closed embedded global cross-section. Fried noticed that transitive Anosov flows on 3-manifolds ”almost admit global cross-sections”. More precisely, a Birkhoff section is an embedded surface with boundary B → M such that: • the interior B \ ∂B of B is transverse to the vector field X ; • the boundary ∂B is the union of finitely many periodic orbits of X ; • there is T> 0 such that for every point x ∈ M there is t ∈ (0,T ] with X (x,t) ∈ B. In [7] Fried built (infinitely many) Birkhoff sections for any transitive Anosov flow on a 3-manifold M . He also proved that the first return map P defined on the interior of B induces a pseudo-Anosov diffeomorphism ˜ P on the closed surface ˜ B obtained from B by replacing each boundary component by a point. In that meaning, Fried noticed that X looks like the suspension of a pseudo-Anosov homeomorphisms and he described a simple surgery on the suspension of ˜ P reconstructing the flow X . Hence, it is tempting to try to build an Axiom A diffeomorphism close to the time one map of the flow of X , in the same way that it has been done in the case of a suspension flow. Indeed, it is what we propose in this paper. However, it cannot be done in a naive way, because a Birkhoff section is not transverse to the flow on its boundary. For this This paper was partially supported by Universit´ e de Bourgogne, the IMB, the grant EGIDE de la R´ egion Bourgogne, PEDECIBA and Universidad de la Rep´ ublica. We thank the warm hospitality of the I.M.B and I.M.E.R.L.. 1