A1 Journal article (refereed)
Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere (2022)


Dufloux, L., & Suomala, V. (2022). Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere. Mathematical Proceedings of the Cambridge Philosophical Society, 172(1), 197-230. https://doi.org/10.1017/S0305004121000177


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Publication details

All authors or editors: Dufloux, Laurent; Suomala, Ville

Journal or series: Mathematical Proceedings of the Cambridge Philosophical Society

ISSN: 0305-0041

eISSN: 1469-8064

Publication year: 2022

Volume: 172

Issue number: 1

Pages range: 197-230

Publisher: Cambridge University Press

Publication country: United Kingdom

Publication language: English

DOI: https://doi.org/10.1017/S0305004121000177

Publication open access: Not open

Publication channel open access:

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1812.00731


Abstract

We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group H = C × R, endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection π(E) (projection along the center of H) almost surely equals min{2, dimH(E)} and that π(E) has non-empty interior if dimH(E) > 2. As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension dimH(E).

We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere S3 endowed with the visual metric d obtained by identifying S3 with the boundary of the complex hyperbolic plane. In S3, we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in S3 satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions


Keywords: probability calculation; measure theory; differential geometry; fractals; dynamical systems


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Last updated on 2021-22-12 at 09:27