2024/02/07 by Jesús A. Álvarez López, López, Jesús A. Álvarez, Yuri A Kordyukov +3
Mathematics · #57R30 #58A14 #58G11 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2402.06671
openalex publication_date 2024/02/07 · openalex created_date 2024/02/15 · openalex updated_date 2026/08/01
Let F be a transversely oriented foliation of codimension 1 on a closed manifold M, and let ϕ=\ϕt\ be a foliated flow on (M,F). Assume the closed orbits of ϕ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let M0 denote their union, M1=M∖ M0 and F1=F|M1. 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 M0, respectively. They become topological complexes with the differential operator dF induced by the de~Rham derivative on the leaves, and they have an ℝ-action ϕ^*=\ϕt *\ induced by ϕ. Let H^\bullet I(F) and H^\bullet I'(F) denote the corresponding leafwise reduced cohomologies, with the induced ℝ-action ϕ^*=\ϕt *\. We define some kind of Lefschetz distribution L\rm dis(ϕ) of the actions ϕ^* on both H^\bullet I(F) and H^\bullet I'(F), whose value is a distribution on 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 M0. We also prove a trace formula describing L\rm dis(ϕ) 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.