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Equidistribution in the $d$-dimensional $a$-adic solenoids

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

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}.
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