Dimension Theory
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| Author | : Michael G. Charalambous |
| Publisher | : Springer Nature |
| Total Pages | : 262 |
| Release | : 2019-10-08 |
| Genre | : Mathematics |
| ISBN | : 3030222322 |
This book covers the fundamental results of the dimension theory of metrizable spaces, especially in the separable case. Its distinctive feature is the emphasis on the negative results for more general spaces, presenting a readable account of numerous counterexamples to well-known conjectures that have not been discussed in existing books. Moreover, it includes three new general methods for constructing spaces: Mrowka's psi-spaces, van Douwen's technique of assigning limit points to carefully selected sequences, and Fedorchuk's method of resolutions. Accessible to readers familiar with the standard facts of general topology, the book is written in a reader-friendly style suitable for self-study. It contains enough material for one or more graduate courses in dimension theory and/or general topology. More than half of the contents do not appear in existing books, making it also a good reference for libraries and researchers.
| Author | : Yakov B. Pesin |
| Publisher | : University of Chicago Press |
| Total Pages | : 633 |
| Release | : 2008-04-15 |
| Genre | : Mathematics |
| ISBN | : 0226662233 |
The principles of symmetry and self-similarity structure nature's most beautiful creations. For example, they are expressed in fractals, famous for their beautiful but complicated geometric structure, which is the subject of study in dimension theory. And in dynamics the presence of invariant fractals often results in unstable "turbulent-like" motions and is associated with "chaotic" behavior. In this book, Yakov Pesin introduces a new area of research that has recently appeared in the interface between dimension theory and the theory of dynamical systems. Focusing on invariant fractals and their influence on stochastic properties of systems, Pesin provides a comprehensive and systematic treatment of modern dimension theory in dynamical systems, summarizes the current state of research, and describes the most important accomplishments of this field. Pesin's synthesis of these subjects of broad current research interest will be appreciated both by advanced mathematicians and by a wide range of scientists who depend upon mathematical modeling of dynamical processes.
| Author | : A.V. Arkhangel'skii |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 210 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642612652 |
This is the first of the encyclopaedia volumes devoted to general topology. It has two parts. The first outlines the basic concepts and constructions of general topology, including several topics which have not previously been covered in English language texts. The second part presents a survey of dimension theory, from the very beginnings to the most important recent developments. The principal ideas and methods are treated in detail, and the main results are provided with sketches of proofs. The authors have suceeded admirably in the difficult task of writing a book which will not only be accessible to the general scientist and the undergraduate, but will also appeal to the professional mathematician. The authors' efforts to detail the relationship between more specialized topics and the central themes of topology give the book a broad scholarly appeal which far transcends narrow disciplinary lines.
| Author | : |
| Publisher | : Academic Press |
| Total Pages | : 271 |
| Release | : 1970-05-31 |
| Genre | : Mathematics |
| ISBN | : 0080873502 |
| Author | : Luis Barreira |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 300 |
| Release | : 2011-08-24 |
| Genre | : Mathematics |
| ISBN | : 3034802064 |
This self-contained monograph presents a unified exposition of the thermodynamic formalism and some of its main extensions, with emphasis on the relation to dimension theory and multifractal analysis of dynamical systems. In particular, the book considers three different flavors of the thermodynamic formalism, namely nonadditive, subadditive, and almost additive, and provides a detailed discussion of some of the most significant results in the area, some of them quite recent. It also includes a discussion of the most substantial applications of these flavors of the thermodynamic formalism to dimension theory and multifractal analysis of dynamical systems.
| Author | : Jun-Iti Nagata |
| Publisher | : Elsevier |
| Total Pages | : 268 |
| Release | : 2014-05-12 |
| Genre | : Mathematics |
| ISBN | : 1483275027 |
Bibliotheca Mathematica, Volume 6: Modern Dimension Theory provides a brief account of dimension theory as it has been developed since 1941, including the principal results of the classical theory for separable metric spaces. This book discusses the decomposition theorem, Baire's zero-dimensional spaces, dimension of separable metric spaces, and characterization of dimension by a sequence of coverings. The imbedding of countable-dimensional spaces, sum theorem for strong inductive dimension, and cohomology group of a topological space are also elaborated. This text likewise covers the uniformly zero-dimensional mappings, theorems in euclidean space, transfinite inductive dimension, and dimension of non-metrizable spaces. This volume is recommended to students and specialists researching on dimension theory.
| Author | : John M. Mackay |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 162 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 0821852299 |
Conformal dimension measures the extent to which the Hausdorff dimension of a metric space can be lowered by quasisymmetric deformations. Introduced by Pansu in 1989, this concept has proved extremely fruitful in a diverse range of areas, including geometric function theory, conformal dynamics, and geometric group theory. This survey leads the reader from the definitions and basic theory through to active research applications in geometric function theory, Gromov hyperbolic geometry, and the dynamics of rational maps, amongst other areas. It reviews the theory of dimension in metric spaces and of deformations of metric spaces. It summarizes the basic tools for estimating conformal dimension and illustrates their application to concrete problems of independent interest. Numerous examples and proofs are provided. Working from basic definitions through to current research areas, this book can be used as a guide for graduate students interested in this field, or as a helpful survey for experts. Background needed for a potential reader of the book consists of a working knowledge of real and complex analysis on the level of first- and second-year graduate courses.
| Author | : Luís Barreira |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 295 |
| Release | : 2012-04-28 |
| Genre | : Mathematics |
| ISBN | : 3642280900 |
Over the last two decades, the dimension theory of dynamical systems has progressively developed into an independent and extremely active field of research. The main aim of this volume is to offer a unified, self-contained introduction to the interplay of these three main areas of research: ergodic theory, hyperbolic dynamics, and dimension theory. It starts with the basic notions of the first two topics and ends with a sufficiently high-level introduction to the third. Furthermore, it includes an introduction to the thermodynamic formalism, which is an important tool in dimension theory. The volume is primarily intended for graduate students interested in dynamical systems, as well as researchers in other areas who wish to learn about ergodic theory, thermodynamic formalism, or dimension theory of hyperbolic dynamics at an intermediate level in a sufficiently detailed manner. In particular, it can be used as a basis for graduate courses on any of these three subjects. The text can also be used for self-study: it is self-contained, and with the exception of some well-known basic facts from other areas, all statements include detailed proofs.
| Author | : Roman Vershynin |
| Publisher | : Cambridge University Press |
| Total Pages | : 299 |
| Release | : 2018-09-27 |
| Genre | : Business & Economics |
| ISBN | : 1108415199 |
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
| Author | : Witold Hurewicz |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 654 |
| Release | : 1995 |
| Genre | : Mathematics |
| ISBN | : 0821800116 |
This book contains papers of the outstanding and versatile mathematician, Witold Hurewicz. Preceding the collection are introductory articles describing Hurewicz's contributions to Borel sets, dimension theory, and algebraic topology. Hurewicz first studied set theory and dimension, and his papers on this topic are especially clear and precise, making them accessible to beginning mathematicians. His work in algebraic topology is marked by five fundamental papers which provide an introduction to homotopy groups and the Hurewicz Theorem concerning the relation between homotopy and singular homology. These papers are included here in their original form along with English translations. Each paper in the collection is followed by a review from one of the major reviewing journals. These reviews were written by eminent mathematicians and serve as excellent abstracts for the papers.
