LEADER 06042nam 22007815 450 001 9910364957603321 005 20200630152225.0 010 $a3-030-18315-7 024 7 $a10.1007/978-3-030-18315-8 035 $a(CKB)4100000010011831 035 $a(MiAaPQ)EBC5997324 035 $a(DE-He213)978-3-030-18315-8 035 $a(PPN)242818854 035 $a(EXLCZ)994100000010011831 100 $a20191216d2019 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aEquidistribution and Counting Under Equilibrium States in Negative Curvature and Trees $eApplications to Non-Archimedean Diophantine Approximation /$fby Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin 205 $a1st ed. 2019. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2019. 215 $a1 online resource (viii, 413 pages) $cillustrations 225 1 $aProgress in Mathematics,$x0743-1643 ;$v329 311 $a3-030-18314-9 327 $aIntroduction -- Negatively curved geometry -- Potentials, critical exponents and Gibbs cocycles -- Patterson-Sullivan and Bowen-Margulis measures with potential on CAT(-1) spaces -- Symbolic dynamics of geodesic flows on trees -- Random walks on weighted graphs of groups -- Skinning measures with potential on CAT(-1) spaces -- Explicit measure computations for simplicial trees and graphs of groups -- Rate of mixing for the geodesic flow -- Equidistribution of equidistant level sets to Gibbs measures -- Equidistribution of common perpendicular arcs -- Equidistribution and counting of common perpendiculars in quotient spaces -- Geometric applications -- Fields with discrete valuations -- Bruhat-Tits trees and modular groups -- Rational point equidistribution and counting in completed function fields -- Equidistribution and counting of quadratic irrational points in non-Archimedean local fields -- Counting and equidistribution of crossratios -- Counting and equidistribution of integral representations by quadratic norm forms -- A - A weak Gibbs measure is the unique equilibrium, by J. Buzzi -- List of Symbols -- Index -- Bibliography. 330 $aThis book provides a complete exposition of equidistribution and counting problems weighted by a potential function of common perpendicular geodesics in negatively curved manifolds and simplicial trees. Avoiding any compactness assumptions, the authors extend the theory of Patterson-Sullivan, Bowen-Margulis and Oh-Shah (skinning) measures to CAT(-1) spaces with potentials. The work presents a proof for the equidistribution of equidistant hypersurfaces to Gibbs measures, and the equidistribution of common perpendicular arcs between, for instance, closed geodesics. Using tools from ergodic theory (including coding by topological Markov shifts, and an appendix by Buzzi that relates weak Gibbs measures and equilibrium states for them), the authors further prove the variational principle and rate of mixing for the geodesic flow on metric and simplicial trees?again without the need for any compactness or torsionfree assumptions. In a series of applications, using the Bruhat-Tits trees over non-Archimedean local fields, the authors subsequently prove further important results: the Mertens formula and the equidistribution of Farey fractions in function fields, the equidistribution of quadratic irrationals over function fields in their completions, and asymptotic counting results of the representations by quadratic norm forms. One of the book's main benefits is that the authors provide explicit error terms throughout. Given its scope, it will be of interest to graduate students and researchers in a wide range of fields, for instance ergodic theory, dynamical systems, geometric group theory, discrete subgroups of locally compact groups, and the arithmetic of function fields. 410 0$aProgress in Mathematics,$x0743-1643 ;$v329 606 $aDynamics 606 $aErgodic theory 606 $aDifferential geometry 606 $aGroup theory 606 $aNumber theory 606 $aConvex geometry  606 $aDiscrete geometry 606 $aProbabilities 606 $aDynamical Systems and Ergodic Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M1204X 606 $aDifferential Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21022 606 $aGroup Theory and Generalizations$3https://scigraph.springernature.com/ontologies/product-market-codes/M11078 606 $aNumber Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M25001 606 $aConvex and Discrete Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21014 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 615 0$aDynamics. 615 0$aErgodic theory. 615 0$aDifferential geometry. 615 0$aGroup theory. 615 0$aNumber theory. 615 0$aConvex geometry . 615 0$aDiscrete geometry. 615 0$aProbabilities. 615 14$aDynamical Systems and Ergodic Theory. 615 24$aDifferential Geometry. 615 24$aGroup Theory and Generalizations. 615 24$aNumber Theory. 615 24$aConvex and Discrete Geometry. 615 24$aProbability Theory and Stochastic Processes. 676 $a516.362 700 $aBroise-Alamichel$b Anne$4aut$4http://id.loc.gov/vocabulary/relators/aut$0781323 702 $aParkkonen$b Jouni$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aPaulin$b Frédéric$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910364957603321 996 $aEquidistribution and Counting Under Equilibrium States in Negative Curvature and Trees$92514141 997 $aUNINA