Transversality and Lefschetz Numbers for Foliation Maps

Let F be a smooth foliation on a closed Riemannian manifold M , and let Λ be a transverse invariant measure of F . Suppose that Λ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the Λ-Lefschetz number of any leaf preserving diffeomorphism (M,F) → (M,F) is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological Λ-Lefschetz number is equal to the analytic Λ-Lefschetz number defined by Heitsch and Lazarov which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to F .