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We prove that expanding endomorphisms on arbitrary tori are 1-sided Bernoulli with respect to their corresponding measure of maximal entropy and are thus, measurably, as far from invertible as possible; this applies in particular to expanding linear toral endomorphisms and their smooth perturbations. Then we study toral extensions of expanding toral endomorphisms, in particular probabilistic systems on skew products, and prove that under certain, not too restrictive conditions on the extensiondoi:10.1090/s0002-9939-2013-11597-2 fatcat:46or7m6adnerllhdw3jbjse3my