Wenyu Pan

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

Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. In the joint work with Jialun LI, we establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Our approach is to construct coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$.