Operators and Complexes

Abstract

Here we consider differential operators and differential complexes on SY andSXwhich are canonically associated to the symplectic Anosov structure of the (twisted) geodesic flows. The main motivation of the constructions discussed here is to find suitable frameworks for characterization of the divisors of the zeta functionsZ _σ in terms of currents onSXwhich are specified byharmonicity conditionswith respect to the foliations P^±. Although we shall prove in Chapter 5 and Chapter 7, that for the Selberg zeta functionZ _s of the geodesic flow of an even-dimensional hyperbolic space and the Ruelle zeta functionZRof the geodesic flow of a 4-dimensional hyperbolic space, the harmonic currents introduced here actually suffice for this purpose, it is not clear whether some of the differential geometrical constructions discussed here already suffice for an analogous characterization of the divisors of the zeta functions of the general twisted geodesic flows.