Maria José Pacifico

Three-Dimensional Flows

Preliminary Definitions and Results.- Singular Cycles and Robust Singular Attractors.- Robustness on the Whole Ambient Space.- Robust Transitivity and Singular-Hyperbolicity.- Singular-Hyperbolicity and Robustness.- Expansiveness and Physical Measure.- Singular-Hyperbolicity and Volume.- Global Dynamics of Generic 3-Flows.- Related Results and Recent Developments.

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.

Large Deviations for Non-Uniformly Expanding Maps

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states.

Homoclinic classes for generic C^1 vector fields

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We also prove that a homoclinic class of X is isolated if and only if it is Omega-isolated, and that it is the intersection of its stable set and its unstable set. All these properties are well known for structurally stable Axiom A vector fields.

Mixing attractors for 3-flows

We prove that every non-trivial attractor is mixing for a generic 3-flow in 1(M), the interior of the 3-flows for which all periodic orbits and singularities are hyperbolic. This implies an extension of a result by Bowen (1976 Mixing Anosov flows Topology 15 77-9): C1 robust transitive sets with singularities for generic flows in 1(M) are mixing. In particular, generic Lorenz attractors are transitive sets for their corresponding time-t map, t≠0.

Inclination-flip homoclinic orbits arising from orbit-flip

Degenerated homoclinic orbits for C1 flows on closed 3-manifolds are studied. We assume that Γ, the homoclinic orbit, is associated with a singularity σ with real eigenvalues λ2<λ3<0<λ1. Degenerated means that either the centre-unstable and stable manifolds of σ are tangent along Γ (inclination-flip) or Γ belongs to the strong stable manifold of σ (orbit-flip). We prove that if Γ is orbit-flip, and -λ2<λ1, then the corresponding vector field can be C1 approximated by those exhibiting inclination-flip homoclinic orbits.

N-expansive homeomorphisms on surfaces.

In this paper we study N-expansive homeomorphisms on surfaces. We prove that when f is a 2-expansive homeomorphism defined on a compact boundaryless surface M with non-wandering set Ω(f) being the whole of M then f is expansive. This condition on the non-wandering set cannot be relaxed: we present an example of a 2-expansive homeomorphisms on a surface with genus 2 with wandering points that is not expansive.

Infinite-modal maps with global chaotic behavior

We prove that certain parametrized families of one-dimensional maps with infinitely many critical points exhibit global chaotic behavior in a persistent way: For a positive Lebesgue measure set of parameter values the map is transitive and almost every orbit has positive Lyapunov exponent. An application of these methods yields a proof of existence and even persistence of global spiral attractors for smooth flows in three dimensions, to be given in [5].

Robustly expansive codimension-one homoclinic classes are hyperbolic

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