Enrique R. Pujals

Homoclinic tangencies and hyperbolicity for surface diffeomorphisms.

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.

Partial hyperbolicity and robust transitivity

Throughout this paper M denotes a three-dimensional boundaryless compact manifold and Diff(M) the space of gl-diffeomorphisms defined on M endowed with the usual Cl-topology. A ~-invariant set A is transitive if A=w(x) for some xEA. Here w(x) is the forward limit set of x (the accumulation points of the positive orbit of x). The maximal invariant set of ~ in an open set U, denoted by A~(U), is the set of points whose whole orbit is contained in U, i.e. A ~ ( U ) = ~ i e z ~i(U). The set A~(U) is robustly transitive if Ar is transitive for every diffeomorphism r CLclose to ~. A diffeomorphism ~EDiff(M) is transitive if M=w(x) for some xEM, i.e. if A ~ ( M ) = M is transitive. Analogously, ~ is robustly transitive if every r gLclose to also is transitive, i.e. if A ~ ( M ) = M is robustly transitive. In this paper we focus our attention on forms of hyperbolicity (uniform, partial and strong partial) of a maximal invariant set A~(U) derived from its robust transitivity. Observe that U can be equal to M, and then ~ is robustly transitive. On one hand, in dimension one there do not exist robustly transitive diffeomorphisms: the diffeomorphisms with finitely many hyperbolic periodic points (Morse~ Smale) are open and dense in Diff(S1). On the other hand, for two-dimensional diffeomorphisms, every robustly transitive set A~(U) is a basic set (i.e. A~(U) is hyperbolic, transitive, and the periodic points of ~ are dense in A~(U)). In particular, every robustly transitive surface diffeomorphism is Anosov and the unique surface which supports such diffeomorphisms is the torus T 2. These assertions follow from [M3] and [M4]. In dimension bigger than or equal to three, besides Anosov (hyperbolic) diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type. As far as we know, three types of such diffeomorphisms have been constructed: skew products,

Singular-hyperbolic attractors are chaotic

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-Ruelle-Bowen) 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 strong-unstable 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.

On C1 robust singular transitive sets for three-dimensional flows

Abstract The main goal of this paper is to study robust invariant transitive sets containing singularities for C 1 flows on three-dimensional compact boundaryless manifolds: they are partially hyperbolic with volume expanding central direction. Moreover, they are either attractors or repellers. Robust here means that this property cannot be destroyed by small C 1 -perturbations of the flow.

Singular strange attractors on the boundary of Morse-Smale systems

Abstract In this paper we introduce bifurcations of Morse-Smale systems that produce strange attractors with singularities in n -manifolds, n ≥3. Some of the attractors are new in the sense that they are not equivalent to any geometric Lorenz attractor. The creation through such bifurcations of hyperbolic dynamics as well as Henon and contracting Lorenz attractors is also investigated

Motion of vortices implies chaos in Bohmian mechanics

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.

Robustly expansive codimension-one homoclinic classes are hyperbolic

We shall prove that C 1 -robustly expansive codimension-one homoclinic classes are hyperbolic.

A remark on conservative diffeomorphisms

We show that a stably ergodic diffeomorphism can be C 1 approximated by a diffeomorphism having stably non-zero Lyapunov

Some simple questions related to the Cr stability conjecture

In this paper we formulate a set of simple questions related to the Cr structural stability conjecture for diffeomorphisms.

Chapter 4 - Homoclinic Bifurcations, Dominated Splitting, and Robust Transitivity

This chapter discusses homoclinic bifurcations, dominated splitting, and robust transitivity. A homoclinic tangency is (locally)easily destroyed by a small perturbation of the invariant manifolds. To get open sets of diffeomorphisms where each system exhibits a homoclinic tangency, newhouse studied systems, where the homoclinic tangency is associated to an invariant hyperbolic set with the property, it has large fractal dimension. In fact, the intersection of the local stable and unstable manifolds of a hyperbolic set (for instance, a classical horseshoes), where this kind of hyperbolic sets, roughly speaking, can be visualized as a product of two Cantor sets with the property that the fractal dimension of these Cantor sets (more specifically, the thickness) is large has been studied. The presence of homoclinic tangencies has many analogies with the presence of critical points for one-dimensional endomorphisms. Homoclinic tangencies correspond in the one-dimensional setting to preperiodic critical points, and it is known that their bifurcation leads to complex dynamics. A robust dynamic phenomenon must be reflected in the tangent map. This turns out to be true, for instance, in the case of structural stability, robust transitivity, and lack of tangencies.