Secondary invariants and the singularity of the Ruelle zeta-function in the central critical point

Abstract

Abstract.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 <I>, 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

The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps

“…Theorem 1.2 is a generalization of Corollary 1 and Theorem 3 in [19] and Theorem 3 in [9] to the case of a noncompact hyperbolic manifold with cusps, where the second term (the scattering contribution) and the third term (the cuspidal contribution from the unipotent term) appear. Theorem 1.2 is an important ingredient in the study of the analytic torsion for hyperbolic manifolds with cusps [30,31].…”

Periodic orbits and dynamical spectra (Survey)

“…We shall not present Patterson's (1990) dynamical approach to understand the connection of the divisor (zeros and poles) of the Selberg zeta function associated to certain Kleinian groups with the cohomology of the group (see Deitmar (1996), Juhl (1995), and Patterson and Perry (1996) for recent results). A significant breakthrough in Patterson's program has recently been accomplished by Bunke and Olbrich (1996).…”