Independent Linear Statistics on the Two-Dimensional Torus

Abstract

Let $X={\bf T}^2$ be the two-dimensional torus, ${\rm Aut}({\bf T}^2)$ be the group of topological automorphisms of ${\bf T}^2$, $\Gamma({\bf T}^2)$ be the set of Gaussian distributions on ${\bf T}^2$, and $\xi_1,\xi_2$ be independent random variables taking on values in ${\bf T}^2$ and having distributions $\mu_j$ with the nonvanishing characteristic functions. Consider $\delta\in{\rm Aut}({\bf T}^2)$ and assume that the linear forms $L_1=\xi_1+\xi_2$ and $L_2=\xi_1+\delta\xi_2$ are independent. We give the description of possible distributions $\mu_j$ depending on $\delta$. In particular we give the complete description of $\delta$ such that the independence of $L_1$ and $L_2$ implies that $\mu_1,\mu_2\in\Gamma({\bf T}^2)$. It turned out that the corresponding Gaussian distributions are either degenerate or concentrated on shifts of the same dense in ${\bf T}^2$ one-parameter subgroup.