compact Riemannian manifolds; $F$ and $B$ may be with boundary. We suppose that $h:B \rightarrow B$ is a itself, i.e. there exists a $C^2-$morphism $\zeta:Z\rightarrow Z$ such that the following diagram commutes: $$\label{eqn:CD} \begin{CD} B @>{h}>> B \\ @V{\pi}VV
@V{\pi}VV \\ Z @>{\zeta}>> Z , \end{CD}$$ where $\pi$ is the standard projection. We assume that the map $\zeta$ downstairs is expanding, and that the fibers $\{z\}\times F$ are the stable manifolds of $h$: $$\label{eqn:zetaexp}
\mu_-\leq |d\zeta^{-1}|\leq \mu,$$ $$\lambda_-\leq
|dh|\leq \lambda \textrm{ }\textrm{ }\textrm{ on } \textrm{
}\textrm{ } T(\{z\} \times F) \textrm{ ,}\forall z\in Z.$$ Again, for technical reasons we assume that $E^s=TF$ is trivialized as in . In this setup, Theorem \[thm:mt\] can be partially improved by the following result.

\[thm:st\] Consider a skew product $\F $ as above that also satisfies the modified dominated splitting condition. Then for small enough $\rho>0$, any $\rho-$perturbation $\G $ of $\F$ has the following properties:\
a) There exists a continuous map $q : X \to Z$ such that the diagram $$\label{eqn:semisol}
\begin{CD}
X @>{\G}>> X \\
@V{q}VV @V{q}VV \\
Z @>{\zeta}>> Z
\end{CD}$$ commutes. Moreover, the commutative diagrams and must be compatible, in the sense that $q|_Y=\pi \circ p$.\
b)The fibers $W^s_z=q^{-1}(z)$ 