The Ergodic Theory of Lattice Subgroups (AM-172)

By: Alexander Gorodnik, Amos Nevo

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Paperback | 4 January 2010

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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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Non-Fiction Mathematics Groups & Group Theory Calculus & Mathematical Analysis Functional Analysis & Transforms Topology Science Physics Classical Mathematics Dynamics & Statics Engineering & Technology Mechanical Engineering & Materials

Materials Science Mechanics of Solids Dynamics & Vibration Number Theory Probability & Statistics

Preface	p. vii
Main objectives	p. vii
Ergodic theory and amenable groups	p. viii
Ergodic theory and nonamenable groups	p. x
Main results: Semisimple Lie groups case	p. 1
Admissible sets	p. 1
Ergodic theorems on semisimple Lie groups	p. 2
The lattice point-counting problem in admissible domains	p. 4
Ergodic theorems for lattice subgroups	p. 6
Scope of the method	p. 8
Examples and applications	p. 11
Hyperbolic lattice points problem	p. 11
Counting integral unimodular matrices	p. 12
Integral equivalence of general forms	p. 13
Lattice points in S-algebraic groups	p. 15
Examples of ergodic theorems for lattice actions	p. 16
Definitions, preliminaries, and basic tools	p. 19
Maximal and exponential-maximal inequalities	p. 19
S-algebraic groups and upper local dimension	p. 21
Admissible and coarsely admissible sets	p. 21
Absolute continuity and examples of admissible averages	p. 23
Balanced and well-balanced families on product groups	p. 26
Roughly radial and quasi-uniform sets	p. 27
Spectral gap and strong spectral gap	p. 29
Finite-dimensional subrepresentations	p. 30
Main results and an overview of the proofs	p. 33
Statement of ergodic theorems for S-aigebraic groups	p. 33
Ergodic theorems in the absence of a spectral gap: overview	p. 35
Ergodic theorems in the presence of a spectral gap: overview	p. 38
Statement of ergodic theorems for lattice subgroups	p. 40
Ergodic theorems for lattice subgroups; overview	p. 42
Volume regularity and volume asymptotics: overview	p. 44
Proof of ergodic theorems for S-algebraic groups	p. 47
Iwasawa groups and spectral estimates	p. 47
Ergodic theorems in the presence of a spectral gap	p. 50
Ergodic theorems in the absence of a spectral gap, I	p. 56
Ergodic theorems in the absence of a spectral gap, II	p. 57
Ergodic theorems in the absence of a spectral gap, III	p. 60
The invariance principle and stability of admissible averages	p. 67
Proof of ergodic theorems for lattice subgroups	p. 71
Induced action	p. 71
Reduction theorems	p. 74
Strong maximal inequality	p. 75
Mean ergodic theorem	p. 78
Pointwise ergodic theorem	p. 83
Exponential mean ergodic theorem	p. 84
Exponential strong maximal inequality	p. 87
Completion of the proofs	p. 90
Equidistribution in isometric actions	p. 91
Volume estimates and volume regularity	p. 93
Admissibility of standard averages	p. 93
Convolution arguments	p. 98
Admissible, well-balanced, and boundary-regular families	p. 101
Admissible sets on principal homogeneous spaces	p. 105
Tauberian arguments and Holder continuity	p. 107
Comments and complements	p. 113
Lattice point-counting with explicit error term	p. 113
Exponentially fast convergence versus equidistribution	p. 115
Remark about balanced sets	p. 116
Bibliography	p. 117
Index	p. 121
Table of Contents provided by Ingram. All Rights Reserved.

The Ergodic Theory of Lattice Subgroups (AM-172)

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