Preprint A628/2009
APERIODIC CLASSES

Yang JIagang

**Keywords: **
aperiodic class | Dominated splitting | index complement

We show that there exists a $C^1$ residual subset $R\subset
C^1(M)\setminus \overline{HT}$, such that for $f\in R$ and $C$ an
aperiodic class of $f$, $C$ has a non-trivial partial hyperbolic
splitting with 1-dimensional central bundle: $T_CM=E^s_{i_0}\oplus
E^c_1\oplus E^u_{i_0+2}$ where $E^s_{i_0}(C),E^u_{i_0+2}(C)\neq
\phi$ and $C$ is an index $i_0$ and $i_0+1$ fundamental limit.
With \cite{1BGW}'s argument, we show $C$ is Hausdorff limit of a
family of non-trivial homoclinic classes. As a corollary, we give
a new proof for the following two results which have been proved
in \cite{1Y2}, \cite{1Y3} respectively: suppose $C$ is a
non-trivial chain recurrent class of $f$, if $C\bigcap P^*_0\neq
\phi$ or $C$ is Lyapunov stable, $C$ is a homoclinic class.