The ergodic theory of lattice subgroups

✍ Scribed by Alexander Gorodnik, Amos Nevo

• Publisher: PUP
• Year: 2010
• Tongue: English
• Leaves: 136
• Series: Ann.Math.Stud.172
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
0.1 Main objectives......Page 8
0.2 Ergodic theory and amenable groups......Page 9
0.3 Ergodic theory and nonamenable groups......Page 11
1.1 Admissible sets......Page 16
1.2 Ergodic theorems on semisimple Lie groups......Page 17
1.3 The lattice point–counting problem in admissible domains......Page 19
1.4 Ergodic theorems for lattice subgroups......Page 21
1.5 Scope of the method......Page 23
2.1 Hyperbolic lattice points problem......Page 26
2.2 Counting integral unimodular matrices......Page 27
2.3 Integral equivalence of general forms......Page 28
2.4 Lattice points in S-algebraic groups......Page 30
2.5 Examples of ergodic theorems for lattice actions......Page 31
3.1 Maximal and exponential-maximal inequalities......Page 34
3.3 Admissible and coarsely admissible sets......Page 36
3.4 Absolute continuity and examples of admissible averages......Page 38
3.5 Balanced and well-balanced families on product groups......Page 41
3.6 Roughly radial and quasi-uniform sets......Page 42
3.7 Spectral gap and strong spectral gap......Page 44
3.8 Finite-dimensional subrepresentations......Page 45
4.1 Statement of ergodic theorems for S-algebraic groups......Page 48
4.2 Ergodic theorems in the absence of a spectral gap: overview......Page 50
4.3 Ergodic theorems in the presence of a spectral gap: overview......Page 53
4.4 Statement of ergodic theorems for lattice subgroups......Page 55
4.5 Ergodic theorems for lattice subgroups: overview......Page 57
4.6 Volume regularity and volume asymptotics: overview......Page 59
5.1 Iwasawa groups and spectral estimates......Page 62
5.2 Ergodic theorems in the presence of a spectral gap......Page 65
5.3 Ergodic theorems in the absence of a spectral gap, I......Page 71
5.4 Ergodic theorems in the absence of a spectral gap, II......Page 72
5.5 Ergodic theorems in the absence of a spectral gap, III......Page 75
5.6 The invariance principle and stability of admissible averages......Page 82
6.1 Induced action......Page 86
6.2 Reduction theorems......Page 89
6.3 Strong maximal inequality......Page 90
6.4 Mean ergodic theorem......Page 93
6.5 Pointwise ergodic theorem......Page 98
6.6 Exponential mean ergodic theorem......Page 99
6.7 Exponential strong maximal inequality......Page 102
6.8 Completion of the proofs......Page 105
6.9 Equidistribution in isometric actions......Page 106
7.1 Admissibility of standard averages......Page 108
7.2 Convolution arguments......Page 113
7.3 Admissible, well-balanced, and boundary-regular families......Page 116
7.4 Admissible sets on principal homogeneous spaces......Page 120
7.5 Tauberian arguments and Hölder continuity......Page 122
8.1 Lattice point–counting with explicit error term......Page 128
8.2 Exponentially fast convergence versus equidistribution......Page 130
8.3 Remark about balanced sets......Page 131
Bibliography......Page 132
Index......Page 136