The ergodic theory of lattice subgroups / Alexander Gorodnik and Amos Nevo.

Format:

Book

Author/Creator:

Gorodnik, Alexander, 1975-

Contributor:

Nevo, Amos, 1966-

Series:

Annals of mathematics studies ; no. 172.

Annals of mathematics studies ; no. 172

Language:

English

Subjects (All):

Ergodic theory.

Lie groups.

Lattice theory.

Harmonic analysis.

Dynamics.

Physical Description:

1 online resource (136 p.)

Edition:

Course Book

Place of Publication:

Princeton, N.J. : Princeton University Press, 2009.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

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

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786612303807

9781282303805

1282303805

9781400831067

1400831067

OCLC:

507428541