Siegel-Radon transforms of transverse dynamical systems
Michael Björklund, Tobias Hartnick
TL;DR
This work resolves the Siegel-Radon transform as a flexible, unified apparatus for transverse dynamical systems by embedding it into a double-fibration framework with cross sections and intersection spaces. It develops both untwisted and twisted (via eigencharacters) transforms, establishes Siegel duality, and derives explicit integrability criteria for induced and hull-based systems, including Cut-and-project and aperiodic-order contexts. The authors connect classical transforms (Siegel, Zak, Siegel-Veech, MS transforms) to the general theory, and provide concrete embeddings of Schrödinger representations into $L^2$-spaces of hulls, via aperiodic Zak transforms and unitary intertwiners. These results yield powerful tools for spectral analysis in aperiodic order, translation surfaces, and quasicrystal settings, enabling precise counting, embedding, and transfer of integrability properties across induced and hull systems.
Abstract
We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group $G$, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over $G$ together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or $Θ$-) transforms and Marklof-Strömbergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis. Our main applications concern dynamical systems $(X, μ)$ in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced $G$-representations into $L^p(X, μ)$ for appropriate values of $p$. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case. In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schrödinger representations into the $L^2$-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms.