Escape rate spectra for hyperbolic $\mathbb {Z}$-covers of Schottky surfaces

Abstract Given a Fuchsian group $\Gamma $ , we introduce escape rate spectra associated with $\Gamma $ to investigate the transient behavior of the geodesic flow on the hyperbolic surface $\mathbb {D}/\Gamma $ . Our definition is motivated by Bishop’s linear escape set. In the special case when $\mathbb {D}/\Gamma $ is a hyperbolic $\mathbb {Z}$ -covering of a surface uniformized by a generalized Schottky group, we completely determine the escape rate spectra by the convex conjugate of a generalized Poincaré exponent.