}\textrm{
} \lambda_- \leq |dh|\leq \lambda,$$ $$\label{eqn:hypstructure} dh:E^u \rightarrow E^u\textrm{ }\textrm{
}\textrm{ and }\textrm{ }\textrm{ } \mu_-\leq |dh^{-1}|\leq \mu.$$ Note that if $\lambda_-=\mu_-=0$, then we get the standard notion of hyperbolicity.\
We assume that the bundles $E^s$ and $E^u$ are trivialized, i.e. that there exist isomorphisms over $B$: $$\label{eqn:triv} \varphi^s:B \times \rr^{k}
\rightarrow E^s, \textrm{ }\textrm{ }\textrm{ } \varphi^u:B \times
\rr^l \rightarrow E^u$$ for some positive integers $k,l\geq 0$. The above is a technical condition necessary for our proof, but we conjecture that all our results hold without it. Let us note that it holds when $h$ is the Smale-Williams solenoid map or any linear Anosov diffeomorphism of a torus.\

\[def:lmhyp\] An invariant set $\L$ of a map $h$ with the above properties will be called $(\l_-, \l, \mu_-, \mu)$- hyperbolic.

Take another compact manifold $M$, called the *fiber*, and form the Cartesian product $X=B\times M$. A *skew product* over $h$ is defined as any $C^1-$map of the form

$$\label{eqn:skprod} \F:X\rightarrow X,\textrm{ }\textrm{ }\textrm{ }\F(b,m)=(h(b), f_b(m)),$$

where $f_b(m):M \rightarrow M$ is a $C^1$ family of $C^1-$morphisms.

\[def:domsplit\] We say that the skew product satisfies the *modified dominated splitting condition* if

$$\label{eqn:dom} \max\left(\max(\lambda,\mu)+
{\left| \left| \frac {\partial f_b^{\pm 1}}{\partial b}\right |
