The Ergodic Theory Of Discrete Groups

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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 : 53,5 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.

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 : 48,6 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.

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 : 41,7 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.

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 : 50,9 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.

Global Aspects of Ergodic Group Actions

A. S. Kechris
Global Aspects of Ergodic Group Actions Book PDF

Author : A. S. Kechris
Publisher : American Mathematical Soc.
Page : 258 pages
File Size : 40,8 Mb
Release : 2010
Category : Automorphisms
ISBN : 9780821848944

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Global Aspects of Ergodic Group Actions by A. S. Kechris Pdf

A study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces.

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 : 44,8 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.

Riemann Surfaces and Related Topics (AM-97), Volume 97

Irwin Kra,Bernard Maskit
Riemann Surfaces and Related Topics AM 97 Volume 97 Book PDF

Author : Irwin Kra,Bernard Maskit
Publisher : Princeton University Press
Page : 533 pages
File Size : 44,6 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881550

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Riemann Surfaces and Related Topics (AM-97), Volume 97 by Irwin Kra,Bernard Maskit Pdf

A classic treatment of Riemann surfaces from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.

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 : 43,5 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 : 43,7 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.

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 : 48,9 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.

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 : 50,6 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.

Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds

Mark Pollicott
Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds Book PDF

Author : Mark Pollicott
Publisher : Cambridge University Press
Page : 176 pages
File Size : 44,6 Mb
Release : 1993-02-04
Category : Mathematics
ISBN : 0521435935

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Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds by Mark Pollicott Pdf

These lecture notes provide a unique introduction to Pesin theory and its applications.

Operator Theoretic Aspects of Ergodic Theory

Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel
Operator Theoretic Aspects of Ergodic Theory Book PDF

Author : Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel
Publisher : Springer
Page : 628 pages
File Size : 40,9 Mb
Release : 2015-11-18
Category : Mathematics
ISBN : 9783319168982

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Operator Theoretic Aspects of Ergodic Theory by Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel Pdf

Stunning recent results by Host–Kra, Green–Tao, and others, highlight the timeliness of this systematic introduction to classical ergodic theory using the tools of operator theory. Assuming no prior exposure to ergodic theory, this book provides a modern foundation for introductory courses on ergodic theory, especially for students or researchers with an interest in functional analysis. While basic analytic notions and results are reviewed in several appendices, more advanced operator theoretic topics are developed in detail, even beyond their immediate connection with ergodic theory. As a consequence, the book is also suitable for advanced or special-topic courses on functional analysis with applications to ergodic theory. Topics include: • an intuitive introduction to ergodic theory • an introduction to the basic notions, constructions, and standard examples of topological dynamical systems • Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the Gelfand–Naimark theorem • measure-preserving dynamical systems • von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem • strongly and weakly mixing systems • an examination of notions of isomorphism for measure-preserving systems • Markov operators, and the related concept of a factor of a measure preserving system • compact groups and semigroups, and a powerful tool in their study, the Jacobs–de Leeuw–Glicksberg decomposition • an introduction to the spectral theory of dynamical systems, the theorems of Furstenberg and Weiss on multiple recurrence, and applications of dynamical systems to combinatorics (theorems of van der Waerden, Gallai,and Hindman, Furstenberg’s Correspondence Principle, theorems of Roth and Furstenberg–Sárközy) Beyond its use in the classroom, Operator Theoretic Aspects of Ergodic Theory can serve as a valuable foundation for doing research at the intersection of ergodic theory and operator theory

Discrete Groups and Geometry

William J. Harvey,C. Maclachlan
Discrete Groups and Geometry Book PDF

Author : William J. Harvey,C. Maclachlan
Publisher : Cambridge University Press
Page : 260 pages
File Size : 46,8 Mb
Release : 1992-07-30
Category : Mathematics
ISBN : 9780521429320

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Discrete Groups and Geometry by William J. Harvey,C. Maclachlan Pdf

This book constitutes the proceedings of a conference held at the University of Birmingham to mark the retirement of Professor A. M. Macbeath. The papers represent up-to-date work on a broad spectrum of topics in the theory of discrete group actions, ranging from presentations of finite groups through the detailed study of Fuchsian and crystallographic groups, to applications of group actions in low dimensional topology, complex analysis, algebraic geometry and number theory. For those wishing to pursue research in these areas, this volume offers a valuable summary of contemporary thought and a source of fresh geometric insights.

An Introduction to Infinite Ergodic Theory

Jon Aaronson
An Introduction to Infinite Ergodic Theory Book PDF

Author : Jon Aaronson
Publisher : American Mathematical Soc.
Page : 298 pages
File Size : 42,7 Mb
Release : 1997
Category : Ergodic theory
ISBN : 9780821804940

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An Introduction to Infinite Ergodic Theory by Jon Aaronson Pdf

Infinite ergodic theory is the study of measure preserving transformations of infinite measure spaces. The book focuses on properties specific to infinite measure preserving transformations. The work begins with an introduction to basic nonsingular ergodic theory, including recurrence behaviour, existence of invariant measures, ergodic theorems, and spectral theory. A wide range of possible "ergodic behaviour" is catalogued in the third chapter mainly according to the yardsticks of intrinsic normalizing constants, laws of large numbers, and return sequences. The rest of the book consists of illustrations of these phenomena, including Markov maps, inner functions, and cocycles and skew products. One chapter presents a start on the classification theory.