Odometer actions of the Heisenberg group

TL;DR

This paper studies odometer actions of the Heisenberg group, describing their spectral decomposition, self-joinings, and spectral properties, revealing they are generally not uniquely determined by their spectra.

Contribution

It explicitly describes the spectral decomposition and self-joinings of Heisenberg odometers, and shows they are not spectrally unique.

Findings

Decomposition of Koopman representation into irreducibles

Explicit description of 2-fold self-joinings

Heisenberg odometers are not isospectral or spectrally determined

Abstract

Let $H_3(\Bbb R)$ denote the 3-dimensional real Heisenberg group. Given a family of lattices $\Gamma_1\supset\Gamma_2\supset\cdots$ in it, let $T$ stand for the associated uniquely ergodic $H_3(\Bbb R)$-{\it odometer}, i.e. the inverse limit of the $H_3(\Bbb R)$-actions by rotations on the homogeneous spaces $H_3(\Bbb R)/\Gamma_j$, $j\in\Bbb N$. The decomposition of the underlying Koopman unitary representation of $H_3(\Bbb R)$ into a countable direct sum of irreducible components is explicitly described. The ergodic 2-fold self-joinings of $T$ are found. It is shown that in general, the $H_3(\Bbb R)$-odometers are neither isospectral nor spectrally determined.