Year: 2026
Type: preprint
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 $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$ with $ρ(A)<1$, and $ϕ\in C^r(\mathbb{S}^1, \mathbb{R}^d)$, with $r>(d+3)/2. 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 $ϕ$, the Hausdorff dimension of the solenoid attractor and the SRB measure equal 1+ the affinity dimension; when $|\det(A)|\ell>1$, for almost every $ϕ$, the SRB measure is absolutely continuous. In the latter case we investigate the Sobolev regularity of the density of the SRB measure. We introduce a derivative dispersion condition, replacing the usual transversality condition.
Source: arXiv (Cornell University)
Institution: University of Zurich
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171782401