The new contributions of this memoir begin in this chapter. We consider the leafwise complex of currents conormal at the leaves preserved by the simple foliated flow. In this case, the conormal sequence induces a long exact sequence in cohomology, and a sequence in reduced cohomology (taking the maximal Hausdorff quotient of the terms). The first main result states that the sequence in reduced cohomology splits into short exact sequences. A description of the left and right terms of this sequence is also given, as well as the action induced by the flow on them. This allows us to compute the contribution of the conormal distributions supported on the preserved leaves to the trace formula.