LEADER 03361nam 2200481 450 001 9910806940003321 005 20220327124038.0 010 $a1-4704-4746-0 035 $a(CKB)4100000006516236 035 $a(MiAaPQ)EBC5501869 035 $a(RPAM)20632234 035 $a(PPN)230400078 035 $a(EXLCZ)994100000006516236 100 $a20180924d2018 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aDiophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces /$fLior Fishman, David Simmons, Mariusz Urban?ski 210 1$aProvidence, Rhode Island :$cAmerican Mathematical Society,$d2018. 215 $a1 online resource (150 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vVolume 254, number 1215 311 $a1-4704-2886-5 320 $aIncludes bibliographical references and index. 327 $aGromov 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 Jarni?k-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)). 330 $a"In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson-Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society ;$vVolume 254, number 1215 606 $aDiophantine analysis 606 $aGeometry, Hyperbolic 615 0$aDiophantine analysis. 615 0$aGeometry, Hyperbolic. 676 $a512.73 700 $aFishman$b Lior$f1964-$01703797 702 $aSimmons$b David$f1988- 702 $aUrban?ski$b Mariusz 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910806940003321 996 $aDiophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces$94089268 997 $aUNINA