cam a22 4 4500 528737716 CHVBK 20201023213246.0 180921t20182018riua b||| 000 0 eng|d 978-1-4704-2886-0 1-4704-2886-5 (NEBIS)011266864 (RERO)R008841051 (OCoLC)1048613735 AAA fre SzZuIDS NEBIS EPF-BIB rda QA3 .A57 no.1215 QA242 .F57 2018 s1ma rero 512.7/3 23 Fishman Lior 1964- (DE-588)1165679922 auteur aut Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces Lior Fishman, David Simmons, Mariusz Urbański Providence, Rhode Island American Mathematical Society 2018 ©2018 v, 137 pages illustrations 25 cm Memoirs of the American Mathematical Society number 1215 1215 (NEBIS)000023178 574068856 Includes bibliographical references (pages 133-137) 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. Diophantine approximation Hyperbolic spaces Metric spaces DIOPHANTISCHE APPROXIMATIONEN (ZAHLENTHEORIE) ger (ETHUDK)000012560 ethudk HYPERBOLISCHE GEOMETRIEN ger (ETHUDK)000012888 ethudk METRISCHE RÄUME (TOPOLOGIE) ger (ETHUDK)000013021 ethudk Geometry, Hyperbolic Diophantine analysis Analyse diophantienne (RERO)A021062911 rero Géométrie hyperbolique (RERO)A021029831 rero u DIOPHANTISCHE APPROXIMATIONEN (ZAHLENTHEORIE) ger 511.42 nebis E1 u HYPERBOLISCHE GEOMETRIEN ger 514.134 nebis E1 u METRISCHE RÄUME (TOPOLOGIE) ger 515.124 nebis E1 u DIOPHANTINE APPROXIMATIONS (NUMBER THEORY) eng 511.42 nebis E1 u APPROXIMATIONS DIOPHANTIENNES (THÉORIE DES NOMBRES) fre 511.42 nebis E1 u HYPERBOLIC GEOMETRIES eng 514.134 nebis E1 u GÉOMÉTRIES HYPERBOLIQUES fre 514.134 nebis E1 u METRIC SPACES (TOPOLOGY) eng 515.124 nebis E1 u ESPACES MÉTRIQUES (TOPOLOGIE) fre 515.124 nebis E1 rero ams 10 Simmons David 1988- (DE-588)1137953950 auteur aut Urbański Mariusz 1958- (DE-588)120432684 auteur aut Memoirs of the American Mathematical Society 1215 BK020000 XK020000 XK020000 E64-20181002 nebis EN E02120 E02-20180829 122 E01-20180926 E64S E64-20181002 RERO RE61011 RE61011 RE610110001 00 MEM 1215 NEBIS E64 E64 E64BI S 24.1215 NEBIS E01 E01 MG P 711653: 1215 NEBIS E02 E02 E02RB 28 FIS NEBIS 100 1- Fishman Lior 1964- (DE-588)1165679922 auteur aut NEBIS 490 0- Memoirs of the American Mathematical Society number 1215 1215 (NEBIS)000023178 574068856 NEBIS 700 1- Simmons David 1988- (DE-588)1137953950 auteur aut NEBIS 700 1- Urbański Mariusz 1958- (DE-588)120432684 auteur aut RERO 100 1- Fishman Lior 1964- (IDREF)229879748 cre RERO 490 1- Memoirs of the American Mathematical Society no. 1215 RERO 700 1- Simmons David 1988- (IDREF)203210476 RERO 700 1- Urbański Mariusz (IDREF)07869583X RERO 830 -- Memoirs of the American Mathematical Society 1215 NEBIS EAD50 EBI01 E01 https://opac.nebis.ch/objects/pdf03/e01_978-1-4704-2886-0_01.pdf Titelblatt und Inhaltsverzeichnis VIEW pdf