a) of the above theorem implies that $G:\Lambda \times M \rightarrow
\Lambda \times M$ is indeed a skew product: $$G(b,m)=(h(b),
g_b(m)).$$ One can then study the dynamical properties of the more mysterious map $\G|_Y$ by studying the dynamical properties of its conjugate skew product $G$.\
The fiber maps $g_b$ of the skew product $G$ are $C^1$-close to those of the skew product $\F|_Y$, in the following sense: $$\label{eqn:close} d(g_b^{\pm 1},f_b^{\pm 1})_{C^1}
\leq O(\rho).$$

But what can be said about the fiber maps $g_b$ for different $b$’s? Since $\F$ is a $C^1-$morphism, the fiber maps $f_b$ depend in a $C^1$ fashion on the point $b\in \Lambda$. Such a result fails for the fiber maps $g_b$, but statement b) of Theorem \[thm:mt\] implies that the fiber maps $g_b$ depend continuously on the point $b\in \Lambda$: $$\label{eqn:holfibermaps}
d(g_b^{\pm 1},g_{b'}^{\pm 1})_{C^0} \leq O(d(b,b')^\a),$$ where $\a$ is given by . A skew product $G$ whose fiber maps satisfy will be called a *skew product*. Thus Theorem \[thm:mt\] can be summarized as follows:

**Let $\G$ be any small perturbation of a $C^2$ skew product $\F$ over a $(\l_-, \l, \mu_-, \mu)-$ hyperbolic map $h$, satisfying the modified dominated splitting condition . Then $\G$ has an invariant 