Geometric Properties of Higher Dimensional Solenoidal Attractors

We study the skew product systems $T: \mathbb{S}^1\times\mathbb{R}^d \to \mathbb{S}^1 \times\mathbb{R}^d$, \begin{equation*} T(x,y)=(\ell x, A y+\phi(x)), \end{equation*} where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$ with $\rho(A)<1$, and $\phi \in C^r(\mathbb{S}^1, \mathbb{R}^d)$. We allow the fibers to have any dimension and $A$ to be non-conformal. We show that: when $|\det(A)|\ell<1$, for almost every $\phi$, the Hausdorff dimension of the solenoid attractor and the SRB measure equals the affinity dimension; when $|\det(A)|\ell>1$, for almost every $\phi$, the SRB measure is absolutely continuous. We introduce a derivative dispersion condition, replacing the usual transversality condition.