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Geometric Properties of Higher Dimensional Solenoidal Attractors

  • Daniele Galli
  • Eleonora Passaglia
  • Andrea Ulliana
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Abstract

We study the skew product systems T: π•Š1×ℝd β†’ π•Š1 ×ℝd, T(x,y)=(β„“ x, A y+Ο•(x)), where β„“β‰₯ 2, A ∈ GLd(ℝ) with ρ(A)<1, and Ο•βˆˆ Cr(π•Š1, ℝd), 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)|β„“<1, for almost every Ο•, the Hausdorff dimension of the solenoid attractor and the SRB measure equal 1+ the affinity dimension; when |det(A)|β„“>1, for almost every Ο•, 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.

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Cite

BibTeXRIS
BibTeX
@misc{galli2026geometric,
  title = {{Geometric Properties of Higher Dimensional Solenoidal Attractors}},
  author = {Daniele Galli and Eleonora Passaglia and Andrea Ulliana},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.27089},
  eprint = {2607.27089},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.27089},
}
RIS
TY  - UNPB
TI  - Geometric Properties of Higher Dimensional Solenoidal Attractors
AU  - Daniele Galli
AU  - Eleonora Passaglia
AU  - Andrea Ulliana
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2607.27089
UR  - https://arxiv.org/abs/2607.27089
AB  - We study the skew product systems T: π•Š^1×ℝ^d β†’ π•Š^1 ×ℝ^d, T(x,y)=(β„“ x, A y+Ο•(x)), where β„“β‰₯ 2, A ∈ GL_d(ℝ) with ρ(A)<1, and Ο•βˆˆ C^r(π•Š^1, 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)|β„“<1, for almost every Ο•, the Hausdorff dimension of the solenoid attractor and the SRB measure equal 1+ the affinity dimension; when |det(A)|β„“>1, for almost every Ο•, 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.
ER  -