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.