On the self-similarity problem for Gaussian-Kronecker flows

Abstract

It is shown that a countable symmetric multiplicative subgroup $G=-H\cup H$ with $H\subset {\mathbb{R}}_{+}^{\ast }$ is the group of self-similarities of a Gaussian-Kronecker flow if and only if $H$ is additively $\mathbb{Q}$-independent. In particular, a real number $s\ne \pm 1$ is a scale of self-similarity of a Gaussian-Kronecker flow if and only if $s$ is transcendental. We also show that each countable symmetric subgroup of ${\mathbb{R}}^{\ast }$ can be realized as the group of self-similarities of a simple spectrum Gaussian flow having the Foias-Stratila property.

15 pages