Applications of Connesʹ Geodesic Flow to Trace Formulas in Noncommutative Geometry

The “trace formula” of Chazarain, Duistermaat, and Guillemin expresses that the singularities of the distribution trace of the wave group on a compact Riemannian manifoldXis included in the set of periods of the geodesic flow restricted toS*X. Most of the objects involved in this trace formula have analogues in Connesʹ Noncommutative Geometry. This paper shows, on several significant examples of Noncommutative Geometry, that Connesʹ definition of geodesic flow leads to statements analogous to the classical trace formula of Chazarain, Duistermaat, and Guillemin.