Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

TY - JOUR

T1 - Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

AU - Fishman, Lior

AU - Simmons, David S.

AU - Urbański, Mariusz

PY - 2018/6/11

Y1 - 2018/6/11

N2 - In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson--Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.

AB - In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson--Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.

KW - math.DS

KW - Hyperbolic geometry

KW - Schmidt's game

KW - Diophantine approximation

KW - Gromov hyperbolic metric spaces

UR - http://www.ams.org/cgi-bin/mstrack/accepted_papers/memo

UR - http://www.scopus.com/inward/record.url?scp=85049911946&partnerID=8YFLogxK

U2 - 10.1090/memo/1215

DO - 10.1090/memo/1215

M3 - Article

SN - 0065-9266

VL - 254

SP - 1

EP - 150

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

IS - 1215

ER -