Dimension Theory Of Hyperbolic Flows
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| Author | : Luís Barreira |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 155 |
| Release | : 2013-06-12 |
| Genre | : Mathematics |
| ISBN | : 3319005480 |
The dimension theory of dynamical systems has progressively developed, especially over the last two decades, into an independent and extremely active field of research. Its main aim is to study the complexity of sets and measures that are invariant under the dynamics. In particular, it is essential to characterizing chaotic strange attractors. To date, some parts of the theory have either only been outlined, because they can be reduced to the case of maps, or are too technical for a wider audience. In this respect, the present monograph is intended to provide a comprehensive guide. Moreover, the text is self-contained and with the exception of some basic results in Chapters 3 and 4, all the results in the book include detailed proofs. The book is intended for researchers and graduate students specializing in dynamical systems who wish to have a sufficiently comprehensive view of the theory together with a working knowledge of its main techniques. The discussion of some open problems is also included in the hope that it may lead to further developments. Ideally, readers should have some familiarity with the basic notions and results of ergodic theory and hyperbolic dynamics at the level of an introductory course in the area, though the initial chapters also review all the necessary material.
| 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 | : 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 | : Welington de Melo |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 616 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642780431 |
One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarize the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearizability of circle diffeomorphisms due to M. Herman, J.-C. Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurstonj also included are an exposition on Hofbauer's tower construction and a result on fuB multimodal families (this last result solves a question posed by J. Milnor).
| Author | : Pierre N.V. Tu |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 257 |
| Release | : 2013-11-11 |
| Genre | : Business & Economics |
| ISBN | : 3662027798 |
Dynamic tools of analysis and modelling are increasingly used in Economics and Biology and have become more and more sophisticated in recent years, to the point where the general students without training in Dynamic Systems (DS) would be at a loss. No doubt they are referred to the original sources of mathematical theorems used in the various proofs, but the level of mathematics is generally beyond them. Students are thus left with the burden of somehow understanding advanced mathematics by themselves, with· very little help. It is to these general students, equipped only with a modest background of Calculus and Matrix Algebra that this book is dedicated. It aims at providing them with a fairly comprehensive box of dynamical tools they are expected to have at their disposal. The first three Chapters start with the most elementary notions of first and second order Differential and Difference Equations. For these, no matrix theory and hardly any calculus are needed. Then, before embarking on linear and nonlinear DS, a review of some Linear Algebra in Chapter 4 provides the bulk of matrix theory required for the study of later Chapters. Systems of Linear Differ ential Equations (Ch. 5) and Difference Equations (Ch. 6) then follow to provide students with a good background in linear DS, necessary for the subsequent study of nonlinear systems. Linear Algebra, reviewed in Ch. 4, is used freely in these and subsequent chapters to save space and time.
| Author | : Robert A. Meyers |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 1885 |
| Release | : 2011-10-05 |
| Genre | : Mathematics |
| ISBN | : 1461418054 |
Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
| Author | : Vítor Araújo |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 369 |
| Release | : 2010-06-10 |
| Genre | : Mathematics |
| ISBN | : 3642114148 |
In this book, the authors present the elements of a general theory for flows on three-dimensional compact boundaryless manifolds, encompassing flows with equilibria accumulated by regular orbits. The book aims to provide a global perspective of this theory and make it easier for the reader to digest the growing literature on this subject. This is not the first book on the subject of dynamical systems, but there are distinct aspects which together make this book unique. Firstly, this book treats mostly continuous time dynamical systems, instead of its discrete counterpart, exhaustively treated in some other texts. Secondly, this book treats all the subjects from a mathematical perspective with proofs of most of the results included. Thirdly, this book is meant to be an advanced graduate textbook and not just a reference book or monograph on the subject. This aspect is reflected in the way the cover material is presented, with careful and complete proofs, and precise references to topics in the book.
| Author | : Tomasz Downarowicz |
| Publisher | : Cambridge University Press |
| Total Pages | : 405 |
| Release | : 2011-05-12 |
| Genre | : Mathematics |
| ISBN | : 1139500872 |
This comprehensive text on entropy covers three major types of dynamics: measure preserving transformations; continuous maps on compact spaces; and operators on function spaces. Part I contains proofs of the Shannon–McMillan–Breiman Theorem, the Ornstein–Weiss Return Time Theorem, the Krieger Generator Theorem and, among the newest developments, the ergodic law of series. In Part II, after an expanded exposition of classical topological entropy, the book addresses symbolic extension entropy. It offers deep insight into the theory of entropy structure and explains the role of zero-dimensional dynamics as a bridge between measurable and topological dynamics. Part III explains how both measure-theoretic and topological entropy can be extended to operators on relevant function spaces. Intuitive explanations, examples, exercises and open problems make this an ideal text for a graduate course on entropy theory. More experienced researchers can also find inspiration for further research.
| Author | : Robert W. Ghrist |
| Publisher | : Springer |
| Total Pages | : 218 |
| Release | : 2006-11-14 |
| Genre | : Mathematics |
| ISBN | : 354068347X |
The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed.
| Author | : Manuel Ritoré |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 113 |
| Release | : 2010-01-01 |
| Genre | : Mathematics |
| ISBN | : 3034602138 |
Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.
