Geometric Properties of Higher Dimensional Solenoidal Attractors

Abstract

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 $\ell \ge 2,$ $A\in G{L}_d(\mathbb{R})$ with $\rho (A)<1$, and $\varphi \in C^r({\mathbb{S}}^1,{\mathbb{R}}^d)$. 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 $\varphi$, the Hausdorff dimension of the solenoid attractor and the SRB measure equals the affinity dimension; when $|\det (A)|\ell >1$, for almost every $\varphi$, the SRB measure is absolutely continuous. We introduce a derivative dispersion condition, replacing the usual transversality condition.

22 pages