Preprint A629/2009
Yang Jiagang
Keywords: dominated splitting | far from tangencies.
We show that for $C^1$ diffeomorphisms far away from homoclinic tangencies, every ergodic invariant measure has at most one zero Lyapunov exponent, and the Oseledets splitting corresponding to positive, zero, and negative exponents is dominated. When the invariant ergodic measure is hyperbolic (all exponents non-zero), then almost every point has a local stable manifold and a local unstable manifold both of which are differentiable embedded disks. Moreover, a version of the classical shadowing lemma holds, so that the hyperbolic measure is the weak limit of a sequence of atomic measures supported on periodic orbits belonging to the same homoclinic class. Together with a recent result of \cite{1DG1}, this allows us to prove that there exists a residual subset $R$ of $C^1$ diffeomorphisms far away from tangencies such that for any $f\in R$, either it's Axiom A, or it has a non-hyperbolic ergodic measure.