The ergodic theory of lattice subgroups

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 ba...

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Hlavní autoři:	Gorodnik, Alexander, Nevo, Amos
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Princeton, N.J Princeton University Press 2009
Vydání:	1
Edice:	Annals of Mathematics Studies
Témata:	Amenable group Asymptotic analysis Asymptotic expansion Automorphism Bounded operator Bounded set (topological vector space) Congruence subgroup Coset Counting problem (complexity) Dimension (vector space) Dynamics Equidistribution theorem Ergodic theory Estimation Explicit formulae (L-function) Family of sets Haar measure Harmonic analysis Hilbert space Induced representation Infimum and supremum Interpolation theorem Invariant measure Isometry group Iwasawa group Lattice (group) Lattice theory Lie algebra Lie groups Linear algebraic group Linear space (geometry) Lipschitz continuity Mathematical induction MATHEMATICS MATHEMATICS / Group Theory MATHEMATICS / Number Theory MATHEMATICS / Probability & Statistics / General Maximal compact subgroup Maximal ergodic theorem Measure (mathematics) Metric space Monotonic function Number theory One-parameter group Operator norm Orthogonal complement P-adic number Parity (mathematics) Pointwise Pointwise convergence Principal homogeneous space Principal series representation Probability measure Probability space Rate of convergence Representation theory Resolution of singularities Sobolev space Special case Spectral gap Spectral method Spectral theory Square (algebra) Statistics 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) Volume form Subset P-adic number Counting problem (complexity) Bounded operator Subgroup Maximal ergodic theorem One-parameter group Spectral gap Mathematical induction Invariant measure Hilbert space Explicit formulae (L-function) Monotonic function Spectral method Volume form Amenable group Estimation Transfer principle Principal homogeneous space Unitary representation Isometry group Linear algebraic group Probability space Dimension (vector space) Infimum and supremum Coset Lattice (group) Asymptotic expansion Haar measure Congruence subgroup Subsequence Representation theory Special case Probability measure Unit sphere Ergodic theory Equidistribution theorem Metric space Maximal compact subgroup Tensor product Sobolev space Unit vector Operator norm Induced representation Upper and lower bounds Lipschitz continuity Unitary group Family of sets Principal series representation Symmetric space Pointwise Square (algebra) Theorem Interpolation theorem Rate of convergence Asymptotic analysis Automorphism Orthogonal complement Iwasawa group Tensor algebra Bounded set (topological vector space) Linear space (geometry) Pointwise convergence Variable (mathematics) Resolution of singularities Spectral theory Parity (mathematics) Lie algebra Measure (mathematics) Number theory
ISBN:	9780691141855, 1400831067, 9781400831067, 0691141843, 9780691141848, 0691141851
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Popis
Shrnutí:	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.
Bibliografie:	Includes bibliographical references (p. [117]-120) and index
ISBN:	9780691141855 1400831067 9781400831067 0691141843 9780691141848 0691141851
DOI:	10.1515/9781400831067