03361nam 2200481 450 991080694000332120220327124038.01-4704-4746-0(CKB)4100000006516236(MiAaPQ)EBC5501869(RPAM)20632234(PPN)230400078(EXLCZ)99410000000651623620180924d2018 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierDiophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces /Lior Fishman, David Simmons, Mariusz UrbańskiProvidence, Rhode Island :American Mathematical Society,2018.1 online resource (150 pages)Memoirs of the American Mathematical Society ;Volume 254, number 12151-4704-2886-5 Includes bibliographical references and index.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 Jarník-Besicovitch theorem) -- Proof of theorem 8.1 (generalization of Khinchin's theorem) -- Proof of theorem 9.3 (BA{d} has full dimension in \Lr(G))."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 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"--Provided by publisher.Memoirs of the American Mathematical Society ;Volume 254, number 1215Diophantine analysisGeometry, HyperbolicDiophantine analysis.Geometry, Hyperbolic.512.73Fishman Lior1964-1703797Simmons David1988-Urbański MariuszMiAaPQMiAaPQMiAaPQBOOK9910806940003321Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces4089268UNINA