Dynamical Systems and Group Actions

Dynamical Systems and Group Actions
Author: Lewis Bowen
Publisher: American Mathematical Soc.
Total Pages: 280
Release: 2012
Genre: Mathematics
ISBN: 0821869221

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This volume contains cutting-edge research from leading experts in ergodic theory, dynamical systems and group actions. A large part of the volume addresses various aspects of ergodic theory of general group actions including local entropy theory, universal minimal spaces, minimal models and rank one transformations. Other papers deal with interval exchange transformations, hyperbolic dynamics, transfer operators, amenable actions and group actions on graphs.

Group Actions in Ergodic Theory, Geometry, and Topology

Group Actions in Ergodic Theory, Geometry, and Topology
Author: Robert J. Zimmer
Publisher: University of Chicago Press
Total Pages: 724
Release: 2019-12-23
Genre: Mathematics
ISBN: 022656827X

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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 Topological Dynamics of Group Actions on Homogeneous Spaces

Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces
Author: M. Bachir Bekka
Publisher: Cambridge University Press
Total Pages: 214
Release: 2000-05-11
Genre: Mathematics
ISBN: 9780521660303

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

Ergodic Theorems for Group Actions

Ergodic Theorems for Group Actions
Author: A.A. Tempelman
Publisher: Springer Science & Business Media
Total Pages: 418
Release: 2013-04-17
Genre: Mathematics
ISBN: 9401714606

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This volume is devoted to generalizations of the classical Birkhoff and von Neuman ergodic theorems to semigroup representations in Banach spaces, semigroup actions in measure spaces, homogeneous random fields and random measures on homogeneous spaces. The ergodicity, mixing and quasimixing of semigroup actions and homogeneous random fields are considered as well. In particular homogeneous spaces, on which all homogeneous random fields are quasimixing are introduced and studied (the n-dimensional Euclidean and Lobachevsky spaces with n>=2, and all simple Lie groups with finite centre are examples of such spaces. Also dealt with are applications of general ergodic theorems for the construction of specific informational and thermodynamical characteristics of homogeneous random fields on amenable groups and for proving general versions of the McMillan, Breiman and Lee-Yang theorems. A variational principle which characterizes the Gibbsian homogeneous random fields in terms of the specific free energy is also proved. The book has eight chapters, a number of appendices and a substantial list of references. For researchers whose works involves probability theory, ergodic theory, harmonic analysis, measure theory and statistical Physics.

Ergodic Theory

Ergodic Theory
Author: Manfred Einsiedler
Publisher: Springer Science & Business Media
Total Pages: 486
Release: 2010-09-11
Genre: Mathematics
ISBN: 0857290215

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This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Partial Dynamical Systems, Fell Bundles and Applications

Partial Dynamical Systems, Fell Bundles and Applications
Author: Ruy Exel
Publisher: American Mathematical Soc.
Total Pages: 330
Release: 2017-09-20
Genre: Mathematics
ISBN: 1470437856

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Partial dynamical systems, originally developed as a tool to study algebras of operators in Hilbert spaces, has recently become an important branch of algebra. Its most powerful results allow for understanding structural properties of algebras, both in the purely algebraic and in the C*-contexts, in terms of the dynamical properties of certain systems which are often hiding behind algebraic structures. The first indication that the study of an algebra using partial dynamical systems may be helpful is the presence of a grading. While the usual theory of graded algebras often requires gradings to be saturated, the theory of partial dynamical systems is especially well suited to treat nonsaturated graded algebras which are in fact the source of the notion of “partiality”. One of the main results of the book states that every graded algebra satisfying suitable conditions may be reconstructed from a partial dynamical system via a process called the partial crossed product. Running in parallel with partial dynamical systems, partial representations of groups are also presented and studied in depth. In addition to presenting main theoretical results, several specific examples are analyzed, including Wiener–Hopf algebras and graph C*-algebras.

Dynamical Systems and Semisimple Groups

Dynamical Systems and Semisimple Groups
Author: Renato Feres
Publisher: Cambridge University Press
Total Pages: 268
Release: 1998-06-13
Genre: Mathematics
ISBN: 9780521591621

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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.

Dynamical Systems of Algebraic Origin

Dynamical Systems of Algebraic Origin
Author: Klaus Schmidt
Publisher: Springer Science & Business Media
Total Pages: 323
Release: 2012-01-05
Genre: Mathematics
ISBN: 3034802765

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Although much of classical ergodic theory is concerned with single transformations and one-parameter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multidimensional symmetry groups. However, the wealth of concrete and natural examples which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. The purpose of this book is to help remedy this scarcity of explicit examples by introducing​ a class of continuous Zd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. One aspect of these actions, not surprising in itself but quite striking in its extent and depth nonetheless, is the connection with commutative algebra and arithmetical algebraic geometry. The algebraic framework resulting from this connection allows the construction of examples with a variety of specified dynamical properties, and by combining algebraic and dynamical tools one obtains a quite detailed understanding of this class of Zd-actions.

Dimension Groups and Dynamical Systems

Dimension Groups and Dynamical Systems
Author: Fabien Durand
Publisher: Cambridge University Press
Total Pages: 593
Release: 2022-02-03
Genre: Mathematics
ISBN: 1108838685

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This is the first self-contained exposition of the connections between symbolic dynamical systems, dimension groups and Bratteli diagrams.

Differential Equations, Dynamical Systems, and an Introduction to Chaos

Differential Equations, Dynamical Systems, and an Introduction to Chaos
Author: Morris W. Hirsch
Publisher: Academic Press
Total Pages: 433
Release: 2004
Genre: Business & Economics
ISBN: 0123497035

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Thirty years in the making, this revised text by three of the world's leading mathematicians covers the dynamical aspects of ordinary differential equations. it explores the relations between dynamical systems and certain fields outside pure mathematics, and has become the standard textbook for graduate courses in this area. The Second Edition now brings students to the brink of contemporary research, starting from a background that includes only calculus and elementary linear algebra. The authors are tops in the field of advanced mathematics, including Steve Smale who is a recipient of.