Odometer actions of the Heisenberg group

Abstract

Let $H_3(\mathbb{R})$ 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 $H_3(\mathbb{R})$-{\it odometer}, i.e. the inverse limit of the $H_3(\mathbb{R})$-actions by rotations on the homogeneous spaces $H_3(\mathbb{R})/{\Gamma }_j$, $j\in \mathbb{N}$. The decomposition of the underlying Koopman unitary representation of $H_3(\mathbb{R})$ 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 $H_3(\mathbb{R})$-odometers are neither isospectral nor spectrally determined.

To appear in J. d'Anal. Math

References in corpus (1)

• $G$-odometers and their almost 1-1 extensions