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 $s\neq \pm 1$ 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 Foiaş-Stratila property.