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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Princeton, N.J
Princeton University Press
2009
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| Schriftenreihe: | Annals of Mathematics Studies |
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| ISBN: | 9780691141855, 1400831067, 9781400831067, 0691141843, 9780691141848, 0691141851 |
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| Abstract | 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. |
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| AbstractList | 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 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. No detailed description available for "The Ergodic Theory of Lattice Subgroups (AM-172)". 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. |
| Author | Nevo, Amos Gorodnik, Alexander |
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| Keywords | 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 |
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| Notes | Includes bibliographical references (p. [117]-120) and index |
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| Snippet | The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems... No detailed description available for "The Ergodic Theory of Lattice Subgroups (AM-172)". |
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| SubjectTerms | 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 |
| SubjectTermsDisplay | Ergodic theory |
| TableOfContents | The ergodic theory of lattice subgroups -- 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 Front Matter Table of Contents Preface Chapter One: Main results: 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 Cover -- Title -- Copyright -- Contents -- Preface -- 0.1 Main objectives -- 0.2 Ergodic theory and amenable groups -- 0.3 Ergodic theory and nonamenable groups -- Chapter 1. Main results: Semisimple Lie groups case -- 1.1 Admissible sets -- 1.2 Ergodic theorems on semisimple Lie groups -- 1.3 The lattice point-counting problem in admissible domains -- 1.4 Ergodic theorems for lattice subgroups -- 1.5 Scope of the method -- Chapter 2. Examples and applications -- 2.1 Hyperbolic lattice points problem -- 2.2 Counting integral unimodular matrices -- 2.3 Integral equivalence of general forms -- 2.4 Lattice points in S-algebraic groups -- 2.5 Examples of ergodic theorems for lattice actions -- Chapter 3. Definitions, preliminaries, and basic tools -- 3.1 Maximal and exponential-maximal inequalities -- 3.2 S-algebraic groups and upper local dimension -- 3.3 Admissible and coarsely admissible sets -- 3.4 Absolute continuity and examples of admissible averages -- 3.5 Balanced and well-balanced families on product groups -- 3.6 Roughly radial and quasi-uniform sets -- 3.7 Spectral gap and strong spectral gap -- 3.8 Finite-dimensional subrepresentations -- Chapter 4. Main results and an overview of the proofs -- 4.1 Statement of ergodic theorems for S-algebraic groups -- 4.2 Ergodic theorems in the absence of a spectral gap: overview -- 4.3 Ergodic theorems in the presence of a spectral gap: overview -- 4.4 Statement of ergodic theorems for lattice subgroups -- 4.5 Ergodic theorems for lattice subgroups: overview -- 4.6 Volume regularity and volume asymptotics: overview -- Chapter 5. Proof of ergodic theorems for S-algebraic groups -- 5.1 Iwasawa groups and spectral estimates -- 5.2 Ergodic theorems in the presence of a spectral gap -- 5.3 Ergodic theorems in the absence of a spectral gap, I -- 5.4 Ergodic theorems in the absence of a spectral gap, II 5.5 Ergodic theorems in the absence of a spectral gap, III -- 5.6 The invariance principle and stability of admissible averages -- Chapter 6. Proof of ergodic theorems for lattice subgroups -- 6.1 Induced action -- 6.2 Reduction theorems -- 6.3 Strong maximal inequality -- 6.4 Mean ergodic theorem -- 6.5 Pointwise ergodic theorem -- 6.6 Exponential mean ergodic theorem -- 6.7 Exponential strong maximal inequality -- 6.8 Completion of the proofs -- 6.9 Equidistribution in isometric actions -- Chapter 7. Volume estimates and volume regularity -- 7.1 Admissibility of standard averages -- 7.2 Convolution arguments -- 7.3 Admissible, well-balanced, and boundary-regular families -- 7.4 Admissible sets on principal homogeneous spaces -- 7.5 Tauberian arguments and Hölder continuity -- Chapter 8. Comments and complements -- 8.1 Lattice point-counting with explicit error term -- 8.2 Exponentially fast convergence versus equidistribution -- 8.3 Remark about balanced sets -- Bibliography -- Index Chapter Six. Proof of ergodic theorems for lattice subgroups Chapter Three. Definitions, preliminaries, and basic tools Index Chapter Five. Proof of ergodic theorems for S-algebraic groups - / Chapter Seven. Volume estimates and volume regularity Contents Chapter One. Main results: Semisimple Lie groups case Chapter Two. Examples and applications Chapter Four. Main results and an overview of the proofs Frontmatter -- Preface Chapter Eight. Comments and complements Bibliography |
| Title | The ergodic theory of lattice subgroups |
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| Volume | 172 |
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