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Streszczenie

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 $\mu_1$, $\mu_2$.