Uniform (projective) hyperbolicity or no hyperbolicity: a dichotomy for generic conservative maps
Marcelo Viana | Bochi, Jairo
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.