Limit points badly approximable by horoballs

Abstract

For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries Γ whose limit set is uniformly perfect, and a disjoint collection of horoballs {H j }, we show that the set of limit points badly approximable by {H j } is absolutely winning in the limit set Λ(Γ). As an application, we deduce that for a geometrically finite Kleinian group acting on $${\mathbb{H}^{n+1}}$$ , the limit points badly approximable by parabolics, denoted BA(Γ), is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that BA(Γ) has dimension equal to the critical exponent of the group. Since BA(Γ) can alternatively be described as the limit points representing bounded geodesics in the quotient $${\mathbb{H}^{n+1}/\Gamma}$$ , we recapture a result originally due to Bishop and Jones.