The Ergodic Theory of Lattice Subgroups (AM-172) ：The Ergodic Theory of Lattice Subgroups (AM-172) ( Annals of Mathematics Studies )

Publication subTitle ：The Ergodic Theory of Lattice Subgroups (AM-172)

Publication series ：Annals of Mathematics Studies

Author： Gorodnik Alexander;Nevo Amos;;

Publisher： Princeton University Press‎

Publication year： 2009

E-ISBN: 9781400831067

P-ISBN(Paperback): 9780691141848

Subject： O177.99 Other

Keyword：数理科学和化学,数学

Language： ENG

Access to resources Favorite

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Description

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 approxi

Chapter

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

The users who browse this book also browse