Preprint A221/2003
Dynamics beyond uniform hyperbolicity : A global geometric and probabilistic approach
Marcelo Viana | Bonatti, Christian | Diaz, Lorenzo J.
Keywords: Hyperbolicity | Attractor | physical measure | stochastic stability
About forty years ago, Smale was proposing the notion of \emph{uniformly hyperbolic dynamical system}: the limit set, consisting of all forward or backward accumulation points of orbits, is a \emph{hyperbolic set}: the tangent space at each point splits into two complementary subspaces that are uniformly contracted under, respectively, forward and backward iterations. %See Smale~\cite{Sm67}. The objective was twofold: to characterize structurally stable systems, and to show that most systems (an open dense set) are structurally stable. The notion of \emph{structural stability}, introduced by Andronov, Pontrjagin, %~\cite{AP37} means that the whole orbit structure remains the same when the system is slightly modified: there exists a homeomorphism of the ambient manifold mapping orbits of the initial system into orbits of the modified one, and preserving the time arrow. The first goal was most successful, as uniform hyperbolicity indeed proved to be the key ingredient characterizing structurally stable systems. % Precise statements and a detailed account of this %topic may be found in Palis, Takens.~\cite{PT93}. At the same time, the theory provided the definitive conceptual framework for the phenomenon of transverse \emph{homoclinic orbits}, that is, transverse intersections of stable and unstable manifolds, discovered by Poincaré in the late XIXth century, and immediately recognized by him as a major source of complexity in the dynamics. In the process the importance of the theory of uniformly hyperbolic systems extended much beyond the original objectives. It was part of a revolution in our vision of determinism, which was strongly motivated by observations coming from the experimental sciences, where the classical opposition between deterministic and random evolutions was broken. The hyperbolic theory provided a mathematical foundation for the fact that deterministic systems, even with a small number of degrees of freedom, often present chaotic behavior, due to \emph{robust} mechanisms. Thus, it led to the almost paradoxical conclusion that chaos may be stable. On the other hand, structural stability and uniform hyperbolicity were soon realized to be less universal properties than was initially thought: there exist many classes of systems that are robustly unstable and non-hyperbolic. Dynamics was, once more, left without a general paradigm. As the theory was being extended in several directions -- bifurcation theory, non-uniform hyperbolicity, smooth ergodic theory, -- a good deal of the extraordinary progress attained in these last decades concerned specific classes of systems, such as unimodal maps of the interval, Lorenz flows, %~\cite{Lo63}, and Hénon maps.%~\cite{He76}. \smallskip Building on these advances, a new point of view has emerged and several ideas and results have been put forward recently, in the direction of understanding the behavior of most dynamical systems from a global viewpoint. The present work is an attempt to put such recent developments in a unified perspective, and to point likely directions of further progress. It consists of four main parts: \head{I - Hyperbolicity and beyond} Section~\ref{s.hyperbolic} is a review of fundamental facts about uniformly hyperbolic systems, together with a discussion of robust mechanisms of non-hyperbolicity and other main issues involved in extending the theory beyond the hyperbolic set-up. \head{II - Critical behavior, homoclinic phenomena} The situations discussed in Sections~\ref{s.onedimension} through \ref{s.surfaces} are united by a common mechanism generating non-hyperbolicity: tangencies or criticalities. The global dynamics of multimodal transformations in dimension $1$ is determined, fundamentally, by the presence and recurrence properties of critical points. For dissipative diffeomorphisms modelled by the Hénon family, tangencies between stable and unstable manifolds play a role similar to that of critical points in dimension $1$. A theory developed over the last decade, showed that these diffeomorphisms are amenable to a very complete statistical and topological description. Here and, even more explicitly, in the general context of homoclinic bifurcations of diffeomorphisms, fractal invariants have a determinant role in controlling the geometry and the recurrence of tangencies. For surface diffeomorphisms, critical behavior (homoclinic tangencies) is the only essential obstruction to hyperbolicity. \head{III - Dominated behavior, dynamical decomposition} Sections~\ref{s.cycles} to \ref{s.vector} introduce another robust mechanism of non-hyperbolicity: cycles between periodic points with variable Morse indices (unstable dimensions). One main theme is the connection between robust indecomposability of the dynamics -- transitivity, ergodicity -- and weaker hyperbolicity conditions -- partial hyperbolicity, projective hyperbolicity, -- as well as the behavior of invariant foliations -- accessibility, minimality. Another, are the relations between elementary objects of the dynamics, such as homoclinic classes and maximal transitive sets. This culminates in the extension to the greatest possible generality, \emph{tame systems}, of Smale's classical theorem on decomposition of the non-wandering set into finitely many indecomposable pieces. New mechanisms generating generic diffeomorphisms with infinitely many homoclinic classes have been devised in recent years, but the understanding of such \emph{wild systems} remains a major challenge. \head{IV - Non-uniform hyperbolicity} Sections~\ref{s.ergodic} and \ref{s.lyapunov} focus on ergodic aspects of the theory. The notion of non-uniform hyperbolicity is somewhat vague. Classically, one considers a probability measure invariant under the system, and tries to describe the properties of the corresponding ergodic system, assuming most orbits have only non-zero Lyapunov exponents, that is, they exhibit asymptotic exponential contraction and expansion in complementary directions. We want to view non-uniform hyperbolicity also from another angle, where no invariant probability is assumed a priori. Instead, one tries to construct such probabilities, especially SRB measures, from assuming non-uniformly hyperbolic behavior on most orbits, in the sense of Lebesgue measure. Another fundamental problem is to understand how general is the assumption of non-zero Lyapunov exponents.

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