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.