Marcelo Viana

Abundance of strange attractors

In 1976 [He] H~non performed a numerical study of the family of diffeomorphisms of the plane ha,b(X, y)=(1-ax2+y, bx) and detected for parameter values a=l.4, b=0.3, what seemed to be a non-trivial attractor with a highly intricate geometric structure. This family has since then been the subject of intense research, both numerical and theoretical, but its dynamics is still far from being completely understood. In particular one could not exclude the possibility that the attractor observed by H6non were just a periodic orbit with a very high period. Recently, in a remarkable paper [BC2], Benedicks and Carleson were able to show that this is not the case, at least for a positive Lebesgue measure set of parameter values near a=2, b=O. More precisely, they showed that if b>0 is small enough then for a positive measure set of a-values near a=2 the corresponding diffeomorphism ha,b exhibits a strange attractor. Their argument is a very creative extension of the techniques they had previously developed in [BUll for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated, nonhyperbolic dynamics. When acquainted in 1985 with the work by Benedicks and Carleson, then in progress, Palls suggested that one should in this context think of the H6non family as a particular, although important, model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies. He proposed that the correct setting for Benedicks-Carlesons results is within this more general framework of homoclinic bifurcations and stated the following

Multidimensional nonhyperbolic attractors

We construct smooth transformations and diffeomorphisms exhibiting nonuniformly hyperbolic attractors with multidimensional sensitiveness on initial conditions: typical orbits in the basin of attraction have several expanding directions. These systems also illustrate a new robust mechanism of sensitive dynamics: despite the nonuniform character of the expansion, the attractor persists in a full neighbourhood of the initial map.

Strange attractors in higher dimensions

We consider generic one-parameter families of diffeomorphisms on a manifold of arbitrary dimension, unfolding a homoclinic tangency associated to a sectionally dissipative saddle point (the product of any pair of eigenvalues has norm less than 1). We prove that such families exhibit strange attractors in a persistent way: for a positive Lebesgue measure set of parameter values. In the two-dimensional case this had been obtained in a joint work with L. Mora, based on and extending the results of Benedicks-Carleson on the quadratic family in the plane.

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.

Statistical stability for robust classes of maps with non-uniform expansion

We consider open sets of transformations in a manifold M, exhibiting nonuniformly expanding behaviour in some forward invariant domain U ‰ M. Assuming that each transformation has a unique SRB measure in U, and some general uniformity conditions, we prove that the SRB measure varies continuously with the dynamics in the L 1 -norm. As an application we show that an open class of maps introduced in [V1] fits this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map.

Lyapunov exponents with multiplicity 1 for deterministic products of matrices

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycles behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Holder continuous potentials), all Oseledets subspaces are one-dimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters. This improves previous results of Bonatti, Gomez–Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset.

Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps

We develop a Ruelle-Perron-Frobenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Holder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, piecewise Holder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a nonlacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here. Dedicated to the memory of William Parry

Attracteurs de Lorenz de variété instable de dimension arbitraire

Abstract We construct the first examples of flows with robust multidimensional Lorenz-like attractors: the singularity contained in the attractor may have any number of expanding eigenvalues, and the attractor remains transitive in a whole neighbourhood of the initial flow. These attractors support a Sinai-Ruelle-Bowen SRB-measure and, contrary to the usual (low-dimensional) Lorenz models, they have infinite modulus of structural stability.

The entropy conjecture for diffeomorphisms away from tangencies

We prove that every C1 diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shubs entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive and have no symbolic extensions.

Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.