Dynamical Systems And Group Actions
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| Author | : Lewis Bowen |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 280 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 0821869221 |
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.
| Author | : M. Bachir Bekka |
| Publisher | : Cambridge University Press |
| Total Pages | : 214 |
| Release | : 2000-05-11 |
| Genre | : Mathematics |
| ISBN | : 9780521660303 |
This book, first published in 2000, focuses on developments in the study of geodesic flows on homogenous spaces.
| Author | : Ruy Exel |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 330 |
| Release | : 2017-09-20 |
| Genre | : Mathematics |
| ISBN | : 1470437856 |
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.
| Author | : A.A. Tempelman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 418 |
| Release | : 2013-04-17 |
| Genre | : Mathematics |
| ISBN | : 9401714606 |
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.
| Author | : Robert J. Zimmer |
| Publisher | : University of Chicago Press |
| Total Pages | : 724 |
| Release | : 2019-12-23 |
| Genre | : Mathematics |
| ISBN | : 022656827X |
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.
| Author | : Manfred Einsiedler |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 486 |
| Release | : 2010-09-11 |
| Genre | : Mathematics |
| ISBN | : 0857290215 |
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.
| Author | : Renato Feres |
| Publisher | : Cambridge University Press |
| Total Pages | : 268 |
| Release | : 1998-06-13 |
| Genre | : Mathematics |
| ISBN | : 9780521591621 |
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.
| Author | : Fabien Durand |
| Publisher | : Cambridge University Press |
| Total Pages | : 593 |
| Release | : 2022-02-03 |
| Genre | : Mathematics |
| ISBN | : 1108838685 |
This is the first self-contained exposition of the connections between symbolic dynamical systems, dimension groups and Bratteli diagrams.
| Author | : Ethan Akin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 292 |
| Release | : 1997-07-31 |
| Genre | : Mathematics |
| ISBN | : 9780306455506 |
This groundbreaking volume is the first to elaborate the theory of set families as a tool for studying the phenomenon of recurrence. The theory is implicit in such seminal works as Hillel Furstenberg's Recurrence in Ergodic Theory and Combinational Number Theory, but Ethan Akin's study elaborates it in detail, defining such elements of theory as: open families of special subsets the unification of several ideas associated with transitivity, ergodicity, and mixing the Ellis theory of enveloping semigroups for compact dynamical systems and new notions of equicontinuity, distality, and rigidity.
| Author | : Raymond Reiter |
| Publisher | : MIT Press |
| Total Pages | : 462 |
| Release | : 2001-07-27 |
| Genre | : Computers |
| ISBN | : 9780262264310 |
Specifying and implementing dynamical systems with the situation calculus. Modeling and implementing dynamical systems is a central problem in artificial intelligence, robotics, software agents, simulation, decision and control theory, and many other disciplines. In recent years, a new approach to representing such systems, grounded in mathematical logic, has been developed within the AI knowledge-representation community. This book presents a comprehensive treatment of these ideas, basing its theoretical and implementation foundations on the situation calculus, a dialect of first-order logic. Within this framework, it develops many features of dynamical systems modeling, including time, processes, concurrency, exogenous events, reactivity, sensing and knowledge, probabilistic uncertainty, and decision theory. It also describes and implements a new family of high-level programming languages suitable for writing control programs for dynamical systems. Finally, it includes situation calculus specifications for a wide range of examples drawn from cognitive robotics, planning, simulation, databases, and decision theory, together with all the implementation code for these examples. This code is available on the book's Web site.
