A Lefschetz formula for flows

TL;DR

This paper establishes a Lefschetz type formula for the geodesic flow on odd-dimensional hyperbolic manifolds, linking local indices of closed orbits with the global flow operator on cohomology.

Contribution

It introduces a novel Lefschetz formula connecting local Fuller indices of closed orbits to the global Frobenius operator in hyperbolic geometry.

Findings

Proves a Lefschetz formula for geodesic flows on hyperbolic manifolds.

Relates local indices of closed orbits to the global flow operator.

Provides a new tool for analyzing dynamical systems on hyperbolic spaces.

Abstract

For the geodesic flow of an odd dimensional hyperbolic manifold we prove a Lefschetz type formula. The local terms are Fuller indices of the closed orbits. The global "Frobenius operator" is the generator of the flow and its action on tangential cohomology.