We consider a large class of one-dimensional maps arising from the contracting Lorenz attractors for three dimensional flows: the eigenvalues λ2 < λ1 < 0 < λ3 of the flow at the singularity satisfy λ1 + λ3 < 0 (instead of λ1 + λ3 > 0 as in the classical geometric Lorenz models). Such flows were studied by A. Rovella who showed that non-uniform expansiveness is a persistent form of behavior (positive Lebesgue measure sets of parameters). Using mainly expansiveness, we prove the existence of absolutely continuous measures invariant under these maps, and from this fact we are able to construct Sinai-Ruelle-Bowen measures for the original flows that generate them. © 2000 Editions scientifiques et médicales Elsevier SAS.