We show that, for every compact n-dimensional manifold, n ≥ 1, there is a residual subset of Diff 1 (M ) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either it is contained in the closure of an infinite set of sinks or sources (Newhouse phenomenon), or it presents some weak form of hyperbolicity called dominated splitting (this is a generalization of a bidimensional result of Mañé [Ma3]). In particular, we show that any C 1 -robustly transitive diffeomorphism admits a dominated splitting. RésuméGénéralisant un résultat de Mañé sur les surfaces [Ma3], nous montrons que, en dimension quelconque, il existe un sous-ensemble résiduel de Diff 1 (M ) de difféomorphismes pour lesquels la classe homocline de toute selle périodique hyperbolique possède deux comportements possibles : ou bien elle est incluse dans l'adhérence d'une infinité de puits ou de sources (phénomène de Newhouse), ou bien elle présente une forme affaiblie d'hyperbolicité appelée une décomposition dominée. En particulier nous montrons que tout difféomorphisme C 1 -robustement transitif possède une décomposition dominée.

We prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms, the ones exhibiting a homoclinic tangency are C 1 dense. This represents a step towards the global understanding of dynamics of surface diffeomorphisms.

Inspired by Lorenz' remarkable chaotic flow, we describe in this paper the structure of all C 1 robust transitive sets with singularities for flows on closed 3-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any C 1 robust attractor with singularities for flows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the flow, and has positive Lyapunov exponent at every orbit, showing that any C 1 robust attractor resembles a geometric Lorenz attractor.

Abstract. We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, there exists a physical (or Sinai-RuelleBowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a u-Gibbs state and is an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strongunstable direction.This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are u-Gibbs states.

PACS. 03.65.-w -Quantum mechanics. PACS. 03.65.Ta -Foundations of quantum mechanics; measurement theory. PACS. 05.45.Mt -Quantum chaos; semiclassical methods.Abstract. -Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator.De Broglie-Bohm's (BB) approach to quantum mechanics has experienced an increased popularity in recent years. This is due to the fact that it combines the accuracy of the standard quantum description with the intuitive insight derived from the causal trajectory formalism, thus providing a powerful theoretical tool to understand the physical mechanisms underlying microscopic phenomena [1,2]. Although the behavior of quantum trajectories is very different from classical solutions it can be used to gain intuition in many physical phenomena. Numerous examples can be found in different areas of research. In particular, we can mention studies of barrier tunneling in smooth potentials [3], the quantum back-reaction problem [4] and ballistic transport of electrons in nanowires [5].According to the BB theory of quantum motion, a particle moves in a deterministic orbit under the influence of the external potential and a quantum potential generated by the wave function. This quantum potential can be very intricate because it encodes information on wave interferences. Based on it, Bohm already predicted complex behavior of the quantum trajectories in his seminal work [6]. This was recently confirmed in several studies when presence of chaos in various systems has been shown numerically [7][8][9]. However, the mechanisms that cause such a complex behavior is still lacking. In this letter we show that movement of the zeros of the wave function, commonly known as vortices, implies chaos in the dynamics of quantum trajectories. Such a movement perturbs the velocity field producing transverse homoclinic orbits that generate the well known Smale horseshoes which is the origin of complex behavior. Our assertion is based on an analytical proof in a simplified model which resembled the velocity field near the vortices. In addition, we present a numerical study in a 2-D c EDP Sciences

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Abstract. We construct a class of vector fields on 3-manifolds containing the hyperbolic ones and the geometric Lorenz attractor. Conversely, we shall prove that nonhyperbolic systems in this class resemble the Lorenz attractor: they have Lorenz-like singularities accumulated by periodic orbits and they cannot be approximated by flows with nonhyperbolic critical elements.

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