LEADER 07878nam 2201969Ia 450 001 9910781200803321 005 20200520144314.0 010 $a1-282-30380-5 010 $a9786612303807 010 $a1-4008-3106-7 024 7 $a10.1515/9781400831067 035 $a(CKB)2550000000002880 035 $a(EBL)475845 035 $a(OCoLC)507428541 035 $a(SSID)ssj0000337335 035 $a(PQKBManifestationID)11297311 035 $a(PQKBTitleCode)TC0000337335 035 $a(PQKBWorkID)10287892 035 $a(PQKB)10258020 035 $a(DE-B1597)446614 035 $a(OCoLC)979685624 035 $a(DE-B1597)9781400831067 035 $a(Au-PeEL)EBL475845 035 $a(CaPaEBR)ebr10333494 035 $a(CaONFJC)MIL230380 035 $a(MiAaPQ)EBC475845 035 $a(PPN)15099625X 035 $a(EXLCZ)992550000000002880 100 $a20090202d2009 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 14$aThe ergodic theory of lattice subgroups$b[electronic resource] /$fAlexander Gorodnik and Amos Nevo 205 $aCourse Book 210 $aPrinceton, N.J. $cPrinceton University Press$d2009 215 $a1 online resource (136 p.) 225 1 $aAnnals of mathematics studies ;$vno. 172 300 $aDescription based upon print version of record. 311 $a0-691-14184-3 311 $a0-691-14185-1 320 $aIncludes bibliographical references and index. 327 $t Frontmatter -- $tContents -- $tPreface -- $tChapter One. Main results: Semisimple Lie groups case -- $tChapter Two. Examples and applications -- $tChapter Three. Definitions, preliminaries, and basic tools -- $tChapter Four. Main results and an overview of the proofs -- $tChapter Five. Proof of ergodic theorems for S-algebraic groups -- $tChapter Six. Proof of ergodic theorems for lattice subgroups -- $tChapter Seven. Volume estimates and volume regularity -- $tChapter Eight. Comments and complements -- $tBibliography -- $tIndex 330 $aThe results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established. 410 0$aAnnals of mathematics studies ;$vno. 172. 606 $aErgodic theory 606 $aLie groups 606 $aLattice theory 606 $aHarmonic analysis 606 $aDynamics 610 $aAbsolute continuity. 610 $aAlgebraic group. 610 $aAmenable group. 610 $aAsymptote. 610 $aAsymptotic analysis. 610 $aAsymptotic expansion. 610 $aAutomorphism. 610 $aBorel set. 610 $aBounded function. 610 $aBounded operator. 610 $aBounded set (topological vector space). 610 $aCongruence subgroup. 610 $aContinuous function. 610 $aConvergence of random variables. 610 $aConvolution. 610 $aCoset. 610 $aCounting problem (complexity). 610 $aCounting. 610 $aDifferentiable function. 610 $aDimension (vector space). 610 $aDiophantine approximation. 610 $aDirect integral. 610 $aDirect product. 610 $aDiscrete group. 610 $aEmbedding. 610 $aEquidistribution theorem. 610 $aErgodic theory. 610 $aErgodicity. 610 $aEstimation. 610 $aExplicit formulae (L-function). 610 $aFamily of sets. 610 $aHaar measure. 610 $aHilbert space. 610 $aHyperbolic space. 610 $aInduced representation. 610 $aInfimum and supremum. 610 $aInitial condition. 610 $aInterpolation theorem. 610 $aInvariance principle (linguistics). 610 $aInvariant measure. 610 $aIrreducible representation. 610 $aIsometry group. 610 $aIwasawa group. 610 $aLattice (group). 610 $aLie algebra. 610 $aLinear algebraic group. 610 $aLinear space (geometry). 610 $aLipschitz continuity. 610 $aMass distribution. 610 $aMathematical induction. 610 $aMaximal compact subgroup. 610 $aMaximal ergodic theorem. 610 $aMeasure (mathematics). 610 $aMellin transform. 610 $aMetric space. 610 $aMonotonic function. 610 $aNeighbourhood (mathematics). 610 $aNormal subgroup. 610 $aNumber theory. 610 $aOne-parameter group. 610 $aOperator norm. 610 $aOrthogonal complement. 610 $aP-adic number. 610 $aParametrization. 610 $aParity (mathematics). 610 $aPointwise convergence. 610 $aPointwise. 610 $aPrincipal homogeneous space. 610 $aPrincipal series representation. 610 $aProbability measure. 610 $aProbability space. 610 $aProbability. 610 $aRate of convergence. 610 $aRegular representation. 610 $aRepresentation theory. 610 $aResolution of singularities. 610 $aSobolev space. 610 $aSpecial case. 610 $aSpectral gap. 610 $aSpectral method. 610 $aSpectral theory. 610 $aSquare (algebra). 610 $aSubgroup. 610 $aSubsequence. 610 $aSubset. 610 $aSymmetric space. 610 $aTensor algebra. 610 $aTensor product. 610 $aTheorem. 610 $aTransfer principle. 610 $aUnit sphere. 610 $aUnit vector. 610 $aUnitary group. 610 $aUnitary representation. 610 $aUpper and lower bounds. 610 $aVariable (mathematics). 610 $aVector group. 610 $aVector space. 610 $aVolume form. 610 $aWord metric. 615 0$aErgodic theory. 615 0$aLie groups. 615 0$aLattice theory. 615 0$aHarmonic analysis. 615 0$aDynamics. 676 $a515/.48 686 $aSI 830$2rvk 700 $aGorodnik$b Alexander$f1975-$01476747 701 $aNevo$b Amos$f1966-$01476748 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910781200803321 996 $aThe ergodic theory of lattice subgroups$93691524 997 $aUNINA