Some results of uniform distribution in this multidimensional torus

This paper contains a generalization to the d-torus 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 T-d , we explore the relations between three properties: the uniform distributi on of the sequence (T(n)t) for mu-almost all t; the behaviour of mu relativ e to the translations by some rational subgroups of T-d; and the entropy of mu for another endomorphism S of T-d.