  Let $F$ be a transversely oriented foliation of codimension 1 on a closed
manifold $M$, and let $\phi=\{\phi^t\}$ be a foliated flow on $(M,F)$. Assume
the closed orbits of $\phi$ are simple and its preserved leaves are
transversely simple. In this case, there are finitely many preserved leaves,
which are compact. Let $M^0$ denote their union, $M^1=M\setminus M^0$ and
$F^1=F|_{M^1}$. We consider two topological vector spaces, $I(F)$ and $I'(F)$,
consisting of the leafwise currents on $M$ that are conormal and dual-conormal
to $M^0$, respectively. They become topological complexes with the differential
operator $d_{F}$ induced by the de~Rham derivative on the leaves, and they have
an $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$ induced by $\phi$. Let $\bar
H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ denote the corresponding leafwise
reduced cohomologies, with the induced $\mathbb{R}$-action
$\phi^*=\{\phi^{t\,*}\}$. We define some kind of Lefschetz distribution
$L_{\text{\rm dis}}(\phi)$ of the actions $\phi^*$ on both $\bar H^\bullet
I(F)$ and $\bar H^\bullet I'(F)$, whose value is a distribution on
$\mathbb{R}$. Its definition involves several renormalization procedures, the
main one being the b-trace of some smoothing b-pseudodifferential operator on
the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also
prove a trace formula describing $L_{\text{\rm dis}}(\phi)$ in terms of
infinitesimal data from the closed orbits and preserved leaves. This solves a
conjecture of C.~Deninger involving two leafwise reduced cohomologies instead
of a single one.
