F. Dalbo et M. Peigne, SOME NEGATIVELY CURVED MANIFOLDS WITH CUSPS, MIXING AND COUNTING, Journal fur die Reine und Angewandte Mathematik, 497, 1998, pp. 141-169
Let X be a Hadamard manifold whose sectional curvature K satisfies -b(
2) less than or equal to K less than or equal to -1. We consider a fam
ily of free isometry groups Gamma acting properly discontinuously on X
and containing parabolic transformations of divergence type. We show
that such groups are of divergent type, we describe the dynamic proper
ties of the map T induced by the action of Gamma on the boundary of X
and we explore the spectrum of the transfer operator associated with T
. As applications, we establish a mixing property for the geodesic flo
w on the unit tangent bundle of X/Gamma and we describe the behaviour
as a goes to +infinity of the number of primitive closed geodesics on
X/Gamma whose length is not larger than a.