Secondary Invariants and the Singularity of the Ruelle Zeta-function in the Central Critical Point

The Ruelle zeta-function of the geodesic flow on the sphere bundle S(X) of an even-dimensional compact locally symmetric space X of rank 1 is a meromorphic function in the complex plane that satisfies a functional equation relating its values in s and -s . The multiplicity of its singularity in the central critical point 5 = 0 only depends on the hyperbolic structure of the flow and can be calculated by integrating a secondary characteristic class canonically associated to the flow-invariant foliations of S(X) for which a representing differential form is given. Let Y = G/K be a rank one symmetric space of the non-compact type, i.e., Y is a real, complex, or quaternionic hyperbolic space or the ( 16-dimensional) hyperbolic Cayley-plane. Let T be a uniform lattice in the (connected simple) isometry group G of y without torsion. Y acts properly discontinuous on Y = G/K (K a maximal compact subgroup of G), and X = Y\G/K is a compact locally symmetric space. We consider I asa Riemannian manifold with respect to an arbitrary (constant) multiple g of the metric go induced by the Killing form on the Lie algebra of G. Then the Riemannian manifold (X, g) is a space of negative curvature. The negativity of the curvature of the metric g on X implies the existence of an infinite countable set of prime closed geodesies in X with a discrete set of prime periods accumulating at infinity. Let , be the geodesic flow on the unit sphere bundle S(X) of the space (X,g). The prime period of a periodic orbit of O, on S(X) coincides with the length of the closed geodesic in X obtained by projecting the periodic orbit into X. Now we use these periods to define the zeta-function (1) ZÄ(S) = J](l-exp(-S/c))-1 c for 5 g C such that Re(s) > h (h being the topological entropy of the geodesic flow Or on S(X)). The product in (1) runs over all closed oriented geodesies c in X, and lc denotes the length of c as a curve in X. Note that for each (unoriented) closed geodesic c in X there are two lifts of c as periodic orbits of í>( in correspondence with the two possibilities to orient c. Received by the editors August 1, 1992, and, in revised form, March 8, 1993. 1991 Mathematics Subject Classification. Primary 58F17, 58F20, 11F72; Secondary 58F18, 58F06. ©1995 American Mathematical Society 0273-0979/95 $1.00+ $.25 per page