Preprint A565/2007
Lyapunov stable chain recurrent class

Jiagang Yang

**Keywords: **
dominated splitting | homoclinic class | stable manifold | generic

We show that there exists a generic subset $R\subset
C^1(M)\setminus \overline{HT}$, such that for $f\in R$, the union
of stable manifolds for periodic points of $f$ is dense, it gives
an answer to Bonatti's conjecture in the set $(\overline{HT})^c$.
In fact, we prove that when $f\in R$ and $C$ is any Lyapunov
stable chain recurrent class of $f$, it should be a homoclinic
class, what's more, suppose $i_0$ is the minimal index for the
periodic points in $C$, then $C$ has an index $i_0$ dominated
splitting $E^{cs}_{i_0}\oplus E^{cu}_{i_0+1}$, and either
$E^{cs}_{i_0}|_C$ is contracting or $C$ is an index $i_0-1$
fundamental limit and the bundle $E^{cs}_{i_0}|_C$ has a
codimension-1 sub-dominated splitting
$E^{cs}_{i_0}|_C=E^{s}_{i_0-1}\oplus E^{cs}_1$ where
$E^{s}_{i_0-1}$ is contracting and $dim(E^{cs}_1)$=1