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Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

Part of: Global differential geometry Dynamical systems with hyperbolic behavior Geometric probability and stochastic geometry Classical measure theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  08 March 2021

LAURENT DUFLOUX  and
VILLE SUOMALA
LAURENT DUFLOUX
Affiliation:
Department of Mathematics and Statistics, P.O. Box 35, FI-40014, University of Jyväskylä, Finland e-mail: laurent.dufloux@meritis.fr
VILLE SUOMALA
Affiliation:
Department of Mathematical Sciences, P.O. Box 8000, FI-90014, University of Oulu, Finland e-mail: ville.suomalal@oulu.fi
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Abstract

We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\], endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\]) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\]. 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 \[{\dim _{\text{H}}}(E)\].

We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\], 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 \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 28A80: Fractals 37D35: Thermodynamic formalism, variational principles, equilibrium states 37C45: Dimension theory of dynamical systems 53C17: Sub-Riemannian geometry
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 197 - 230
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported by: Academy of Finland via the Centre of Excellence in Analysis and Dynamics research and the research project Geometry of subRiemannian Groups (#288501), the European Research Council via the ERC Starting Grant #713998 GeoMeG ‘Geometry of Metric Groups’, the Institute Mittag-Leffler via the Fractal Geometry and Dynamics research program.

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