* 장소: Zoom


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 particular, we show that the study of the geodesic flow $(\mathcal{G}_t)_{t\in \mathbb{R}}$ on $T^1(\Gamma\backslash \mathbb{H}^n)$ can be reduced to an expanding map on the boundary. During the talk, I am planning to present the intuitive idea behind this construction and the details. If time permits, I will also discuss how to bridge the geodesic flow on $T^1(\Gamma\backslash \mathbb{H}^n)$ with the expanding map on the boundary.