On Lefschetz formulas for flows on foliated manifolds

Abstract

The paper is devoted to the Lefschetz formulas for flows on compact manifolds, preserving a codimension one foliation and having fixed points. We develop an approach to the Lefschetz formulae based on the notion of the regularized trace on some algebra of singular integral operators introduced in a previous paper. The Lefschetz formula is proved in the case when the flow preserves a foliation given by the fibers of a fiber bundle over a circle. For a particular example of a flow on a two-dimensional torus, preserving a Reeb type foliation, we prove an analogue of the McKean–Singer formula for smoothed regularized Lefschetz functions.