Ergodic Theory And Semisimple Groups

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Ergodic Theory and Semisimple Groups

R.J. Zimmer
Ergodic Theory and Semisimple Groups Book PDF

Author : R.J. Zimmer
Publisher : Springer Science & Business Media
Page : 219 pages
File Size : 46,9 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781468494884

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Ergodic Theory and Semisimple Groups by R.J. Zimmer Pdf

This book is based on a course given at the University of Chicago in 1980-81. As with the course, the main motivation of this work is to present an accessible treatment, assuming minimal background, of the profound work of G. A. Margulis concerning rigidity, arithmeticity, and structure of lattices in semi simple groups, and related work of the author on the actions of semisimple groups and their lattice subgroups. In doing so, we develop the necessary prerequisites from earlier work of Borel, Furstenberg, Kazhdan, Moore, and others. One of the difficulties involved in an exposition of this material is the continuous interplay between ideas from the theory of algebraic groups on the one hand and ergodic theory on the other. This, of course, is not so much a mathematical difficulty as a cultural one, as the number of persons comfortable in both areas has not traditionally been large. We hope this work will also serve as a contribution towards improving that situation. While there are a number of satisfactory introductory expositions of the ergodic theory of integer or real line actions, there is no such exposition of the type of ergodic theoretic results with which we shall be dealing (concerning actions of more general groups), and hence we have assumed absolutely no knowledge of ergodic theory (not even the definition of "ergodic") on the part of the reader. All results are developed in full detail.

Dynamical Systems and Semisimple Groups

Renato Feres
Dynamical Systems and Semisimple Groups Book PDF

Author : Renato Feres
Publisher : Cambridge University Press
Page : 268 pages
File Size : 40,7 Mb
Release : 1998-06-13
Category : Mathematics
ISBN : 0521591627

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Dynamical Systems and Semisimple Groups by Renato Feres Pdf

The theory of dynamical systems can be described as the study of the global properties of groups of transformations. The historical roots of the subject lie in celestial and statistical mechanics, for which the group is the time parameter. The more general modern theory treats the dynamical properties of the semisimple Lie groups. Some of the most fundamental discoveries in this area are due to the work of G.A. Margulis and R. Zimmer. This book comprises a systematic, self-contained introduction to the Margulis-Zimmer theory, and provides an entry into current research. Assuming only a basic knowledge of manifolds, algebra, and measure theory, this book should appeal to anyone interested in Lie theory, differential geometry and dynamical systems.

Group Actions in Ergodic Theory, Geometry, and Topology

Robert J. Zimmer
Group Actions in Ergodic Theory Geometry and Topology Book PDF

Author : Robert J. Zimmer
Publisher : University of Chicago Press
Page : 724 pages
File Size : 52,6 Mb
Release : 2019-12-23
Category : Mathematics
ISBN : 9780226568133

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Group Actions in Ergodic Theory, Geometry, and Topology by Robert J. Zimmer Pdf

Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.

Ergodic Theory, Groups, and Geometry

Robert J. Zimmer,Dave Witte Morris
Ergodic Theory Groups and Geometry Book PDF

Author : Robert J. Zimmer,Dave Witte Morris
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 46,7 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821809808

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Ergodic Theory, Groups, and Geometry by Robert J. Zimmer,Dave Witte Morris Pdf

This introduction to ergodic theory provides an overview of important methods, major developments and open problems in the subject. The lectures in the book include additional comments at the end of each chapter with references to recent developments. These updates can help lead the graduate student to cutting-edge results in the field.

The Ergodic Theory of Lattice Subgroups (AM-172)

Alexander Gorodnik,Amos Nevo
The Ergodic Theory of Lattice Subgroups AM 172 Book PDF

Author : Alexander Gorodnik,Amos Nevo
Publisher : Princeton University Press
Page : 136 pages
File Size : 45,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9780691141855

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The Ergodic Theory of Lattice Subgroups (AM-172) by Alexander Gorodnik,Amos Nevo Pdf

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.

Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

M. Bachir Bekka,Matthias Mayer
Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces Book PDF

Author : M. Bachir Bekka,Matthias Mayer
Publisher : Cambridge University Press
Page : 214 pages
File Size : 54,9 Mb
Release : 2000-05-11
Category : Mathematics
ISBN : 0521660300

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Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces by M. Bachir Bekka,Matthias Mayer Pdf

This book, first published in 2000, focuses on developments in the study of geodesic flows on homogenous spaces.

Group Actions in Ergodic Theory, Geometry, and Topology

Robert J. Zimmer
Group Actions in Ergodic Theory Geometry and Topology Book PDF

Author : Robert J. Zimmer
Publisher : University of Chicago Press
Page : 724 pages
File Size : 46,7 Mb
Release : 2019-12-23
Category : Mathematics
ISBN : 9780226568270

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Group Actions in Ergodic Theory, Geometry, and Topology by Robert J. Zimmer Pdf

Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.

Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics

Calvin C. Moore
Group Representations Ergodic Theory Operator Algebras and Mathematical Physics Book PDF

Author : Calvin C. Moore
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 50,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461247227

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Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics by Calvin C. Moore Pdf

The Mathematical Sciences Research Institute sponsored a three day conference, May 21-23, 1984 to honor Professor George W. Mackey. The title of the conference, Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics, reflects the interests in science that have characterized Professor wide ranging Mackey's work. The conference provided an opportunity for his students, friends and colleagues to honor him and his contributions. The conference was attended by over one hundred people and the participants included five mathematical generations Professor Mackey's mathematical father, Marshall Stone, many mathematical children, grandchildren, and at least one mathematical great-grandchild. This volume is a compendium of the scientific papers presented at the conference plus some additional papers contributed after the conference. The far ranging scope of the various articles is a further indication of the large number of fields that have been affected by Professor Mackey's work. Calvin C. Moore Berkeley, CA Feb, 1986 Table of Contents Preface vi i Ambiguity Functions and Group L. Auslander and Representations R. Tolimieri Kirillov Orbits and Direct Integral Lawrence Corwin 11 Decompositions on Certain Quotient Spaces Some Homotopy and Shape Calculations Edward G. Effors and 69 for C*-Algebras Jerome Kaminker 121 Small Unitary Representations of Roger Howe Classical Groups Dual Vector Spaces Irving Kaplansky 151 Exponential Decay of Correlation Calvin C. Moore 163 Coefficients for Geodesic Flows Lattices in U(n. I) G. D. Mostow Induced Bundles and Nonlinear Irving E. Segal 199 Wave equations Compact Ahelian Aut.

Discrete Subgroups of Semisimple Lie Groups

Gregori A. Margulis
Discrete Subgroups of Semisimple Lie Groups Book PDF

Author : Gregori A. Margulis
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 48,9 Mb
Release : 1991-02-15
Category : Mathematics
ISBN : 354012179X

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Discrete Subgroups of Semisimple Lie Groups by Gregori A. Margulis Pdf

Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. Since the notion of "Lie group" is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e.g. classification results on abstract homomorphisms of semi-simple algebraic groups over global fields. The treatise of course contains a presentation of the author's fundamental rigidity and arithmeticity theorems. The work in this monograph requires the language and basic results from fields such as algebraic groups, ergodic theory, the theory of unitary representatons, and the theory of amenable groups. The author develops the necessary material from these subjects; so that, while the book is of obvious importance for researchers working in related areas, it is essentially self-contained and therefore is also of great interest for advanced students.

Ergodic Theory, Groups, and Geometry

Robert J. Zimmer,Dave Witte Morris
Ergodic Theory Groups and Geometry Book PDF

Author : Robert J. Zimmer,Dave Witte Morris
Publisher : American Mathematical Soc.
Page : 103 pages
File Size : 41,9 Mb
Release : 2008-01-01
Category : Mathematics
ISBN : 9780821883365

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Ergodic Theory, Groups, and Geometry by Robert J. Zimmer,Dave Witte Morris Pdf

"The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure-theoretic techniques. This book provides an introduction to some of the important methods, major developments, and open problems in the subject. It is slightly expanded from lectures given by Zimmer at the CBMS conference at the University of Minnesota. The main text presents a perspective on the field as it was at that time. Comments at the end of each chapter provide selected suggestions for further reading, including references to recent developments."--BOOK JACKET.

Dynamics, Ergodic Theory and Geometry

Boris Hasselblatt
Dynamics Ergodic Theory and Geometry Book PDF

Author : Boris Hasselblatt
Publisher : Cambridge University Press
Page : 324 pages
File Size : 51,7 Mb
Release : 2007-09-24
Category : Mathematics
ISBN : 9780521875417

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Dynamics, Ergodic Theory and Geometry by Boris Hasselblatt Pdf

Based on the subjects from the Clay Mathematics Institute/Mathematical Sciences Research Institute Workshop titled 'Recent Progress in Dynamics' in September and October 2004, this volume contains surveys and research articles by leading experts in several areas of dynamical systems that have experienced substantial progress. One of the major surveys is on symplectic geometry, which is closely related to classical mechanics and an exciting addition to modern geometry. The survey on local rigidity of group actions gives a broad and up-to-date account of another flourishing subject. Other papers cover hyperbolic, parabolic, and symbolic dynamics as well as ergodic theory. Students and researchers in dynamical systems, geometry, and related areas will find this book fascinating. The book also includes a fifty-page commented problem list that takes the reader beyond the areas covered by the surveys, to inspire and guide further research.

Lectures on Ergodic Theory

Paul R. Halmos
Lectures on Ergodic Theory Book PDF

Author : Paul R. Halmos
Publisher : Courier Dover Publications
Page : 113 pages
File Size : 44,8 Mb
Release : 2017-12-13
Category : Mathematics
ISBN : 9780486814896

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Lectures on Ergodic Theory by Paul R. Halmos Pdf

This concise classic by Paul R. Halmos, a well-known master of mathematical exposition, has served as a basic introduction to aspects of ergodic theory since its first publication in 1956. "The book is written in the pleasant, relaxed, and clear style usually associated with the author," noted the Bulletin of the American Mathematical Society, adding, "The material is organized very well and painlessly presented." Suitable for advanced undergraduates and graduate students in mathematics, the treatment covers recurrence, mean and pointwise convergence, ergodic theorem, measure algebras, and automorphisms of compact groups. Additional topics include weak topology and approximation, uniform topology and approximation, invariant measures, unsolved problems, and other subjects.

Ergodic Theory and Harmonic Analysis

Karl E. Petersen
Ergodic Theory and Harmonic Analysis Book PDF

Author : Karl E. Petersen
Publisher : Cambridge University Press
Page : 452 pages
File Size : 55,5 Mb
Release : 1995-01-27
Category : Mathematics
ISBN : 9780521459990

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Ergodic Theory and Harmonic Analysis by Karl E. Petersen Pdf

Ergodic theory is a field that is stimulating on its own, and also in its interactions with other branches of mathematics and science. In recent years, the interchanges with harmonic analysis have been especially noticeable and productive. This book contains survey papers describing the relationship of ergodic theory with convergence, rigidity theory and the theory of joinings. These papers present the background of each area of interaction, the most outstanding results and promising lines of research. They should form perfect starting points for anyone beginning research in one of these areas. Thirteen related research papers describe the work; several treat questions arising from the Furstenberg multiple recurrence theory, while the remainder deal with convergence and a variety of other topics in dynamics.

Introduction to Arithmetic Groups

Armand Borel
Introduction to Arithmetic Groups Book PDF

Author : Armand Borel
Publisher : American Mathematical Soc.
Page : 118 pages
File Size : 54,5 Mb
Release : 2019-11-07
Category : Education
ISBN : 9781470452315

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Introduction to Arithmetic Groups by Armand Borel Pdf

Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel's Introduction aux groupes arithmétiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces. The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments.

The Ergodic Theory of Discrete Groups

Peter J. Nicholls
The Ergodic Theory of Discrete Groups Book PDF

Author : Peter J. Nicholls
Publisher : Cambridge University Press
Page : 237 pages
File Size : 50,6 Mb
Release : 1989-08-17
Category : Mathematics
ISBN : 9780521376747

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The Ergodic Theory of Discrete Groups by Peter J. Nicholls Pdf

The interaction between ergodic theory and discrete groups has a long history and much work was done in this area by Hedlund, Hopf and Myrberg in the 1930s. There has been a great resurgence of interest in the field, due in large measure to the pioneering work of Dennis Sullivan. Tools have been developed and applied with outstanding success to many deep problems. The ergodic theory of discrete groups has become a substantial field of mathematical research in its own right, and it is the aim of this book to provide a rigorous introduction from first principles to some of the major aspects of the theory. The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson for Fuchsian groups, and later extended and refined by Sullivan.