A dynamical Lefschetz trace formula for algebraic Anosov diffeomorphisms

Abstract

For algebraic Anosov diffeomorphisms we first express the reduced leafwise cohomology with respect to the unstable foliation in terms of finite dimensional Lie algebra cohomology. We then prove a dynamical Lefschetz trace formula for the induced action on this cohomology using a result of Nomizu. We also consider generalized Anosov maps for which the cohomology in question is no longer finite dimensional in general. Using the representation theory of nilpotent Lie groups we still arrive at a meaningful definition of traces on cohomology. Again we establish a dynamical Lefschetz trace formula.