Limit Points Badly Approximable by Horoballs

TL;DR

This paper proves that the set of limit points badly approximable by horoballs in hyperbolic spaces is absolutely winning, implying it has full dimension and generalizes previous results on Kleinian groups and bounded geodesics.

Contribution

It establishes the absolute winning property for badly approximable limit points in hyperbolic spaces, extending earlier results to more general settings.

Findings

The set of badly approximable limit points has full Hausdorff dimension.

The property of being badly approximable is shown to be absolutely winning.

This generalizes previous results for Kleinian groups and bounded geodesics.

Abstract

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.