Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
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| 100 | 1 | |a Fishman |D Lior |d 1964- |0 (DE-588)1165679922 |e auteur |4 aut | |
| 245 | 1 | 0 | |a Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces |c Lior Fishman, David Simmons, Mariusz Urbański |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 2018 | |
| 264 | 4 | |c ©2018 | |
| 300 | |a v, 137 pages |b illustrations |c 25 cm | ||
| 490 | 1 | |a Memoirs of the American Mathematical Society |v number 1215 |i 1215 |w (NEBIS)000023178 |9 574068856 | |
| 504 | |a Includes bibliographical references (pages 133-137) | ||
| 505 | 0 | |a 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)) | |
| 520 | |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. | ||
| 650 | 0 | |a Diophantine approximation | |
| 650 | 0 | |a Hyperbolic spaces | |
| 650 | 0 | |a Metric spaces | |
| 650 | 7 | |a DIOPHANTISCHE APPROXIMATIONEN (ZAHLENTHEORIE) |x ger |0 (ETHUDK)000012560 |2 ethudk | |
| 650 | 7 | |a HYPERBOLISCHE GEOMETRIEN |x ger |0 (ETHUDK)000012888 |2 ethudk | |
| 650 | 7 | |a METRISCHE RÄUME (TOPOLOGIE) |x ger |0 (ETHUDK)000013021 |2 ethudk | |
| 650 | 0 | |a Geometry, Hyperbolic | |
| 650 | 0 | |a Diophantine analysis | |
| 650 | 7 | |a Analyse diophantienne |0 (RERO)A021062911 |2 rero | |
| 650 | 7 | |a Géométrie hyperbolique |0 (RERO)A021029831 |2 rero | |
| 691 | 7 | |B u |a DIOPHANTISCHE APPROXIMATIONEN (ZAHLENTHEORIE) |z ger |u 511.42 |2 nebis E1 | |
| 691 | 7 | |B u |a HYPERBOLISCHE GEOMETRIEN |z ger |u 514.134 |2 nebis E1 | |
| 691 | 7 | |B u |a METRISCHE RÄUME (TOPOLOGIE) |z ger |u 515.124 |2 nebis E1 | |
| 691 | 7 | |B u |a DIOPHANTINE APPROXIMATIONS (NUMBER THEORY) |z eng |u 511.42 |2 nebis E1 | |
| 691 | 7 | |B u |a APPROXIMATIONS DIOPHANTIENNES (THÉORIE DES NOMBRES) |z fre |u 511.42 |2 nebis E1 | |
| 691 | 7 | |B u |a HYPERBOLIC GEOMETRIES |z eng |u 514.134 |2 nebis E1 | |
| 691 | 7 | |B u |a GÉOMÉTRIES HYPERBOLIQUES |z fre |u 514.134 |2 nebis E1 | |
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| 700 | 1 | |a Simmons |D David |d 1988- |0 (DE-588)1137953950 |e auteur |4 aut | |
| 700 | 1 | |a Urbański |D Mariusz |d 1958- |0 (DE-588)120432684 |e auteur |4 aut | |
| 830 | 0 | |a Memoirs of the American Mathematical Society |v 1215 | |
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