The ergodic theory of lattice subgroups [[electronic resource] /] / Alexander Gorodnik and Amos Nevo

• Autore: Gorodnik Alexander <1975->
• Pubblicazione: Princeton, N.J., : Princeton University Press, 2009
• Edizione: Course Book
• Descrizione fisica: 1 online resource (136 p.)
• Disciplina: 515/.48
• Soggetto topico: Ergodic theory; Lie groups; Lattice theory; Harmonic analysis; Dynamics
• Soggetto non controllato: Absolute continuity; Algebraic group; Amenable group; Asymptote; Asymptotic analysis; Asymptotic expansion; Automorphism; Borel set; Bounded function; Bounded operator; Bounded set (topological vector space); Congruence subgroup; Continuous function; Convergence of random variables; Convolution; Coset; Counting problem (complexity); Counting; Differentiable function; Dimension (vector space); Diophantine approximation; Direct integral; Direct product; Discrete group; Embedding; Equidistribution theorem; Ergodic theory; Ergodicity; Estimation; Explicit formulae (L-function); Family of sets; Haar measure; Hilbert space; Hyperbolic space; Induced representation; Infimum and supremum; Initial condition; Interpolation theorem; Invariance principle (linguistics); Invariant measure; Irreducible representation; Isometry group; Iwasawa group; Lattice (group); Lie algebra; Linear algebraic group; Linear space (geometry); Lipschitz continuity; Mass distribution; Mathematical induction; Maximal compact subgroup; Maximal ergodic theorem; Measure (mathematics); Mellin transform; Metric space; Monotonic function; Neighbourhood (mathematics); Normal subgroup; Number theory; One-parameter group; Operator norm; Orthogonal complement; P-adic number; Parametrization; Parity (mathematics); Pointwise convergence; Pointwise; Principal homogeneous space; Principal series representation; Probability measure; Probability space; Probability; Rate of convergence; Regular representation; Representation theory; Resolution of singularities; Sobolev space; Special case; Spectral gap; Spectral method; Spectral theory; Square (algebra); Subgroup; Subsequence; Subset; Symmetric space; Tensor algebra; Tensor product; Theorem; Transfer principle; Unit sphere; Unit vector; Unitary group; Unitary representation; Upper and lower bounds; Variable (mathematics); Vector group; Vector space; Volume form; Word metric
• Classificazione: SI 830
• Altri autori: NevoAmos <1966->
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Frontmatter -- Contents -- Preface -- Chapter One. Main results: Semisimple Lie groups case -- Chapter Two. Examples and applications -- Chapter Three. Definitions, preliminaries, and basic tools -- Chapter Four. Main results and an overview of the proofs -- Chapter Five. Proof of ergodic theorems for S-algebraic groups -- Chapter Six. Proof of ergodic theorems for lattice subgroups -- Chapter Seven. Volume estimates and volume regularity -- Chapter Eight. Comments and complements -- Bibliography -- Index
• Sommario/riassunto: The 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.
• Titolo autorizzato: The ergodic theory of lattice subgroups
• ISBN: 1-282-30380-5; 9786612303807; 1-4008-3106-7
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910781200803321
• Lo trovi qui: Univ. Federico II
• Serie: Annals of mathematics studies ; ; no. 172.