Summary: This paper contains a generalization to the \(d\)-torus \(\mathbb{T}^d\) of the author’s earlier results [ Isr. J. Math. 91, 419–428 (1995; Zbl 0839.11030)]. Given a probability measure \(\mu\) and an endomorphism \(T\) of \(\mathbb{T}^d\), we explore the relations between three properties: the uniform distribution of the sequence \((T^n\mathbf t)\) for \(\mu\)-almost all \(\mathbf t\); the behaviour of \(\mu\) relative to the translations by some rational subgroups of \(\mathbb{T}^d\); and the entropy of \(\mu\) for another endomorphism \(S\) of \(\mathbb{T}^d\).