Geometric Properties of Higher Dimensional Solenoidal Attractors

2026/07/29 by Daniele Galli, Eleonora Passaglia, Andrea Ulliana
Mathematics · #math.DS #math.FA #msc:28A78 #msc:28A80 #msc:37C40 #msc:37C70 #msc:37D20

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22 pages

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

We study the skew product systems T: \mathbbS¹×ℝ^d → \mathbbS¹ ×ℝ^d, T(x,y)=(ℓ x, A y+ϕ(x)), where ℓ≥ 2, A ∈ GL_d(ℝ) with ρ(A)<1, and ϕ∈ C^r(\mathbbS¹, ℝ^d). We allow the fibers to have any dimension and A to be non-conformal. We show that: when |det(A)|ℓ<1, for almost every ϕ, the Hausdorff dimension of the solenoid attractor and the SRB measure equals the affinity dimension; when |det(A)|ℓ>1, for almost every ϕ, the SRB measure is absolutely continuous. We introduce a derivative dispersion condition, replacing the usual transversality condition.

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