Preprint A629/2009
ERGODIC MEASURES FAR AWAY FROM TANGENCIES

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.