This paper contains a generalization to the $d$-torus $\mathbb{T}^d$ of earlier results (B. Host. Nombres normaux, entropie, translations. Israel J. Math. 91 (1995), 419–428). Given a probability measure $\mu$ on 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 $\mathbb{T}^d$.