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+φ(x)), \end{equation*} where $\mathrm{\ell }\ge 2,$ $A\in G{L}_d(\mathbb{R})$ 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 $\mathrm{\mid }\det (A)\mathrm{\mid }\mathrm{\ell }<1$, for almost every $\phi$, the Hausdorff dimension of the solenoid attractor and the SRB measure equals the affinity dimension; when $\mathrm{\mid }\det (A)\mathrm{\mid }\mathrm{\ell }>1$, for almost every $\phi$, the SRB measure is absolutely continuous. We introduce a derivative dispersion condition, replacing the usual transversality condition.