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512.73 FISH

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 Description
 v, 137 pages ; 26 cm.
 Additional Authors
 Simmons, David,Urbański, Mariusz,
 Notes
 Contents: Gromov hyperbolic metric spaces  Basic facts about Diophantine approximation  Schmidt's game and Mcmullen's absolute game  Partition structures  Proof of theorem 6.1 (absolute winning of \BA [xi])  Proof of theorem 7.1 (generalization of the JarníkBesicovitch theorem)  Proof of theorem 8.1 (generalization of Khinchin's theorem)  Proof of theorem 9.3 (BA{d} has full dimension in \Lr(G)).Includes bibliographical references and index.Summary: "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 any element of the group, and groups acting on infinitedimensional 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 PattersonSullivan measure unless the group is quasiconvexcocompact. The latter is an application of a Diophantine theorem"