The Heyde theorem on $\bf a$-adic solenoids

Margaryta Myronyuk¹

¹ B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47, Lenin Ave. Kharkov, 61103, Ukraine

Colloquium Mathematicum, Tome 132 (2013) no. 2, pp. 195-210.

Résumé

We prove the following analogue of the Heyde theorem for $\bf a$-adic solenoids. Let $ \xi_1$, $\xi_2$ be independent random variables with values in an ${\bf a}$-adic solenoid $ \varSigma_{\bf a}$ and with distributions $\mu_1$, $\mu_2$. Let $\alpha_j, \beta_j$ be topological automorphisms of $\varSigma_{\bf a}$ such that $\beta_1\alpha^{-1}_1 \pm \beta_2\alpha^{-1}_2$ are topological automorphisms of $\varSigma_{\bf a}$ too. Assuming that the conditional distribution of the linear form $L_2=\beta_1\xi_1 + \beta_2\xi_2$ given $L_1=\alpha_1\xi_1 + \alpha_2\xi_2$ is symmetric, we describe the possible distributions $\mu_1$, $\mu_2$.

Détail

DOI : 10.4064/cm132-2-3

Keywords: prove following analogue heyde theorem a adic solenoids independent random variables values adic solenoid varsigma distributions alpha beta topological automorphisms varsigma beta alpha beta alpha topological automorphisms varsigma too assuming conditional distribution linear form beta beta given alpha alpha symmetric describe possible distributions