Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Abstract

Let $\Gamma$ be a geometrically finite discrete subgroup in $\mathrm{SO}(d+1,1{)}^{\circ }$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $T^1(\Gamma \backslash {\mathbb{H}}^{d+1})$ with respect to the Bowen-Margulis-Sullivan measure, which is the unique probability measure on $T^1(\Gamma \backslash {\mathbb{H}}^{d+1})$ with maximal entropy. As an application, we obtain a resonance free region for the resolvent of the Laplacian on $\Gamma \backslash {\mathbb{H}}^{d+1}$. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator.

References in corpus (4)

• Fourier dimension and spectral gaps for hyperbolic surfaces
• Exponential mixing of frame flows for convex cocompact hyperbolic manifolds
• Exponential mixing of frame flows for geometrically finite hyperbolic manifolds
• Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Cited by in corpus (3)

• Exponential mixing of frame flows for geometrically finite hyperbolic manifolds
• Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps
• The fractal uncertainty principle via Dolgopyat's method in higher dimensions