Limit points badly approximable by horoballs

作者：Mayeda Dustin^*; Merrill Keith

来源：Geometriae Dedicata, 2013, 163(1): 127-140.

DOI：10.1007/s10711-012-9738-9

• Absolute winning
• Winning
• Diophantine approximation
• Kleinian groups
• Badly approximable

For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries I%26quot; 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 I %26gt;(I%26quot;). As an application, we deduce that for a geometrically finite Kleinian group acting on , the limit points badly approximable by parabolics, denoted BA(I%26quot;), is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that BA(I%26quot;) has dimension equal to the critical exponent of the group. Since BA(I%26quot;) can alternatively be described as the limit points representing bounded geodesics in the quotient , we recapture a result originally due to Bishop and Jones.

• 出版日期2013-4