This book presents a new Lefschetz trace formula for a foliated flow on a compact foliated manifold with a foliation of codimension one. The leaves preserved by the flow and its closed orbits are assumed to be transversely simple. The formula equates two distributions on the real line: one is a renormalized trace of the flow's action on two reduced leafwise cohomologies, defined via conormal and dual-conormal currents; the other consists of contributions from the preserved leaves, closed orbits, and a b-trace version of Connes' Euler characteristic, defined using a transverse invariant measure on the complement of the preserved leaves. The usual Euler characteristic is undefined here due to non-compactness. The proofs and definitions combine tools from analysis and foliation theory, like small b-calculus, Witten's deformation of differential complexes, heat invariants, and zeta functions of operators, alongside a description of foliations with this type of flow. The exposition is largely self-contained, with prerequisites and references given for further study. The trace formula solves a conjecture of C. Deninger, motivated by his program which links arithmetic zeta functions to foliated geometry. While further generalization is needed for arithmetic applications, the original and technically deep ideas presented here are valuable in themselves and offer a foundation for future work. The book will be of interest to researchers and graduate students in foliation theory, global analysis on manifolds, and arithmetic geometry.

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Jesús A. Álvarez López is a professor at the University of Santiago de Compostela. His PhD was on the cohomology of Riemannian foliations (those with rigid transverse dynamics). Building on this, he collaborated in the study of the leafwise heat flow of Riemannian foliations, obtaining analytic interpretations of his earlier cohomological results, as well as a trace formula for foliated flows, establishing a connection with Deninger's program on arithmetic zeta functions. His other research interests are in equicontinuous foliated spaces, the generic geometry of leaves, Witten's complex on stratified spaces, hypoelliptic PDEs, singular Kähler foliations on pseudomanifolds, and Kazhdan's property (T) for Riemannian foliations. His work has given rise to numerous publications, including the book "Generic coarse geometry of leaves."

Yuri A. Kordyukov graduated in 1984 from the Moscow State University. His PhD (1988) was on elliptic operators on manifolds of bounded geometry and his DSc (2005) was on the analysis of differential operators on foliated manifolds. From 1988 to 2001 he was a professor at the Ufa State Aviation Technical University. Since 2002 he has been at the Institute of Mathematics of the Ufa Federal Research Centre of the Russian Academy of Sciences, currently as a principal researcher. His main research interests are in global analysis and geometry on foliated manifolds, the spectral theory of differential operators, index theory and noncommutative geometry.

Eric Leichtnam is Director of Research at CNRS, based at l'Institut Mathématique de Jussieu-PRG. He defended his PhD Thesis in 1987. His research is in noncommutative geometry and differential geometry, with a particular interest in Deninger's program linking arithmetic zeta functions to foliated spaces. He has published around 50 papers.