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