On linear escape and the dimension of limit sets in variable negative curvature

Abstract

In 2004, Bishop proved that for Kleinian groups acting on hyperbolic space, the Hausdorff dimension of the limit set is completely determined by two extremal dynamical behaviors: recurrent geodesics and geodesics escaping linearly to infinity. In this paper, we extend this phenomenon to arbitrary discrete groups of isometries of complete simply connected Riemannian manifolds with pinched negative sectional curvatures $-b^2\leq k\leq -1$. More precisely, we show that the Hausdorff dimension of the limit set coincides with the maximum of the Hausdorff dimensions of the radial limit set and the linear escape limit set.