A trace formula for foliated flows

Abstract

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.