**Author(s):**Roman Urban

**Journal:**Uniform Distribution Theory

ISSN 1336-913X

**Volume:**6;

**Issue:**1;

**Start page:**21;

**Date:**2011;

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**Keywords:**Uniform distribution modulo 1 | equidistribution in probability | a-adic solenoid | topological dynamics

**ABSTRACT**

Given a probability measure $mu$ on the $d$-dimensional $a$-adic solenoid $\Omega_a^d$ and an endomorphism $T$ of $\Omega_a^d,$ we consider the relation between uniform distribution of the sequence $T^n\mathbf{x}$ for $\mu$-almost all $\mathbf{x}\in\Omega_a^d$ and the behavior of $\mu$ relative to the translations by some rational subgroups of $\Omega_a^d.$ The main result of this note is an extension of the corresponding result for the $d$-dimensional torus $\mathbb T^d$ due to B. Host \cite{H}.