Preprint A564/2007
Newhouse phenomena and homoclinic class

Jiagang Yang

**Keywords: **
Homoclinic Class | Dominated splitting | chain recurrent class

We show that there exists a generic subset $R$ among the $C^1$
diffeomorphisms set which are $C^1$ far away from tangency, such
that for $f \in R$ and any non-trivial chain recurrent class $C$
of $f$, if $C\bigcap P_0^*\neq \phi$ then $C$ is a homoclinic
class contains index $1$ periodic point and there are a family of
sources converge to $C$ in Hausdorff topology.