A trace formula for foliated flows via adiabatic limits

Let F be a C∞ foliation of codimension one on a closed manifold M , and let φ be a foliated flow on (M,F). The subbundle of vectors tangent to the leaves will be denoted by TF ⊂ TM . Suppose that each fixed point x of φ is non-degenerate in the sense that the map 1 − Txφ : TxM → TxM is an isomorphism. Thus the fixed point set of φ is finite. For each fixed point x, there is some real number κx 6= 0 such that the map TxM/TxF → TxM/TxF induced by Txφ is multiplication by ex, and the sign of the determinant of 1−Txφ : TxF → TxF will be denoted by x. For simplicity, suppose that φ has no periodic orbits different from fixed points. Now, pick any riemannian metric g on M , which may not be bundle-like (F may not be given by riemannian submersions). Consider the decomposition g = g⊥ ⊕ gF according to the decomposition TM = TF⊥ ⊕ TF . We can also consider the family of metrics gh = h g⊥ ⊕ gF for h > 0, whose limit as h → 0 is called the adiabatic limit. Observe that the local distance between the leaves blows up at the adiabatic limit.