Higher-dimensional attractors with absolutely continuous invariant probability

Consider a dynamical system $T:$\mathbb{T}$\times $\mathbb{R}$^{d} \rightarrow $\mathbb{T}$\times $\mathbb{R}$^{d} $givenby$ 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 $u\ge d$, 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 $\mathcal{U}$.