Limit Points Badly Approximable by Horoballs

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