the operator that maps $\G$ to $\beta_b$ is Lipshitz at $\F$ with constant $C$ (uniformly in $b$). We will consider other (parameter dependent) operators and functionals defined on $\Omega$; the expression $O(\rho)$ has the same meaning for them.\
Small perturbations of skew products are not necessarily skew products anymore. However, in this paper we will show that they are conjugated to skew products, and moreover the conjugation map satisfies a continuity property.\
We will now state our main result.

\[The Main Theorem\] \[thm:mt\] Consider a skew product $\F $ as in over a $(\l_-, \l, \mu_-, \mu
)-$ hyperbolic map in the base, satisfying 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 $\G-$invariant set $Y\subset X$ and a continuous map $p:Y\rightarrow B$ such that the diagram

$$\label{eqn:semi}
\begin{CD} Y @>{\G}>> Y \\
@V{p}VV @V{p}VV \\
\Lambda @>{h}>> \Lambda
\end{CD}$$

commutes. Moreover, the map $$\label{eqn:homeo1}
H: Y \to \Lambda \times M, \textrm{ }\textrm{ }\textrm{ } H(b,m)=(p(b,m),m)$$ is a homeomorphism.\
b) The fibers $W_b = p^{-1}(b)$ are Lipschitz close to vertical (constant) fibers, and continuous in $b$. This means that $W_b$ is the graph of 