  Consider a dynamical system $T:\mathbb{T}\times \mathbb{R}^{d} \rightarrow
\mathbb{T}\times \mathbb{R}^{d} $ given by $ T(x,y) = (E(x), C(y) + f(x))$,
where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear
contracting map of $\mathbb{R}^d$ and $f$ is in $C^2(\mathbb{T},\mathbb{R}^d)$.
We prove that if $T$ is volume expanding and $u\geq d$, then for every $E$
there exists an open set $\mathcal{U}$ of pairs $(C,f)$ for which the
corresponding dynamic $T$ admits an absolutely continuous invariant
probability. A geometrical characteristic of transversality between
self-intersections of images of $\mathbb{T}\times\{ 0 \}$ is present in the
dynamic of the maps in $\mathcal{U}$. In addition, we give a condition between
$E$ and $C$ under which it is possible to perturb $f$ to obtain a pair
$(C,\tilde{f})$ in $\mathcal{U}$.
