Preprint A101/2001
Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps

Marcelo Viana | Bochi, Jairo

**Keywords: **
Lyapunov Exponents | projective hyperbolicity | symplectic map | volume-preserving map | linear cocycle

We announce and outline the proofs of recent results
exploiting a connection between the following
a priori
loosely related problems:
How do Lyapunov exponents of conservative (symplectic
or
volume preserving) diffeomorphisms depend on the
underlying dynamics?
How typical is it for Lyapunov exponents to vanish?
We prove that Lyapunov exponents can be simultaneously
continuous at a given diffeomorphism only if the
corresponding Oseledets splitting is dominated or else
trivial,
almost everywhere.
Trivial splitting means that all Lyapunov exponents are
equal to zero. Domination is a property of uniform
hyperbolicity on the projective bundle.
As a consequence one gets a surprising dichotomy for a
residual
(dense $G_\delta$) subset of volume preserving
$C^1$ diffeomorphisms
on any compact manifold:
the Oseledets splitting is either dominated or trivial,
almost everywhere.
Analogous results hold for symplectic $C^1$ diffeomorphisms,
where the conclusion is even stronger: domination is
replaced
by partial hyperbolicity.