03928cam  22004218i 450000100090000000300050000900500170001400600190003100700150005000800410006502000270010604000370013305000210017008200160019110000350020724501710024226300090041326400610042230000340048333600210051733700250053833800230056349000740058650508440066050600500150452013270155450400510288153300950293253800360302758800470306365000260311065000260313670000360316270000330319877601830323185600440341485600480345820632234RPAM20180821135731.0m      b    001 0 cr/|||||||||||180821s2018    riu     ob    001 0 eng c  a9781470447465 (online)  aLBSOR/DLCbengerdacLBSORdRPAM00aQA242b.F57 201800a512.7/32231 aFishman, Lior,d1964-eauthor.10aDiophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces /h[electronic resource] cLior Fishman, David Simmons, Mariusz Urbaânski.  a1807 1aProvidence, RI :bAmerican Mathematical Society,c[2018]  a1 online resource (pages cm.)  atext2rdacontent  aunmediated2rdamedia  avolume2rdacarrier0 aMemoirs of the American Mathematical Society, x1947-6221 ; vv. 121500tChapter 1. IntroductiontChapter 2. Gromov hyperbolic metric spacestChapter 3. Basic facts about Diophantine approximationtChapter 4. Schmidt's game and McMullen's absolute gametChapter 5. Partition structurestChapter 6. Proof of Theorem 6.1 (Absolute winning of $\BA _\xi $)tChapter 7. Proof of Theorem 7.1 (Generalization of the Jarnâik-Besicovitch Theorem)tChapter 8. Proof of Theorem 8.1 (Generalization of Khinchin's Theorem)tChapter 9. Proof of Theorem 9.3 ($\BA _d$ has full dimension in $\Lr (G)$)tAppendix A. Any function is an orbital counting function for some parabolic grouptAppendix B. Real, complex, and quaternionic hyperbolic spacestAppendix C. The potential function gametAppendix D. Proof of Theorem 6.1 using the $\HH $-potential game, where $\HH $ = pointstAppendix E. Winning sets and partition structures1 aAccess is restricted to licensed institutions  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.  aIncludes bibliographical references and index.  aElectronic reproduction.bProvidence, Rhode Island :cAmerican Mathematical Society.d2018  aMode of access : World Wide Web  aDescription based on print version record. 0aDiophantine analysis. 0aGeometry, Hyperbolic.1 aSimmons, David,d1988-eauthor.1 aUrbaânski, Mariusz,eauthor.0 iPrint version: aFishman, Lior, 1964-tDiophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces /w(DLC)   2018030061x0065-9266z97814704288604 3Contentsuhttp://www.ams.org/memo/1215/4 3Contentsuhttps://doi.org/10.1090/memo/1215