Some negatively curved manifolds with cusps, mixing and counting. (English) Zbl 0890.53043

Dal’Bo, Françoise; Peigné, Marc

J. Reine Angew. Math. 497, 141-169 (1998).

Let \(X\) be a Hadamard manifold whose sectional curvature \(K\) satisfies \(-b^2\leq K\leq -1\). We consider a family of free isometry groups \(\Gamma\) acting properly discontinuously on \(X\) and containing parabolic transformations which generate elementary groups of divergence type. We first show that the Patterson measure \(\sigma\) associated with \(\Gamma\) has no atomic part and that \(\Gamma\) is of divergent type. This allows us to describe the dynamic properties of the map \(T\) induced by the action of \(\Gamma\) on the boundary of \(X\). More precisely, since \(\Gamma\) is geometrically finite, the map \(T\) is expanding on \(\Lambda\); furthermore, the fact that \(\sigma\) has no atomic part readily implies that the spectral radius of the transfer operator \(P\) associated with \(T\) is 1, a precise description of the dynamics of the generators of \(\Gamma\) on \(\partial X\) thus allows us to control the spectrum of \(P\) and its Fourier transforms on a Banach space of \(\mathbb C\)-valued Hölder continuous functions on \(\Lambda\). As applications, we establish a mixing property for the geodesic flow on the unit tangent bundle of \(X/ \Gamma\) and we describe the behaviour as \(a\) goes to \(+\infty\) of the number of primitive closed geodesics on \(X/\Gamma\) whose length is not larger than \(a\).

Reviewer: F. Dal’Bo and M. Peigné (Rennes)

MSC:

53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25	Geodesic flows in symplectic geometry and contact geometry
53C22	Geodesics in global differential geometry

Keywords:

number of closed geodesics; Hadamard manifold; free isometry groups; parabolic transformations; Patterson measure; transfer operator; geodesic flow