One-Dimensional Dynamics
Title | One-Dimensional Dynamics PDF eBook |
Author | Welington de Melo |
Publisher | Springer Science & Business Media |
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).
One-Dimensional Dynamical Systems
Title | One-Dimensional Dynamical Systems PDF eBook |
Author | Ana Rodrigues |
Publisher | CRC Press |
Pages | 119 |
Release | 2021-08-10 |
Genre | Mathematics |
ISBN | 1000427978 |
• Example-driven approach • Suitable as supplementary reading for a graduate or advanced undergraduate course in dynamical systems
Dynamics of One-Dimensional Maps
Title | Dynamics of One-Dimensional Maps PDF eBook |
Author | A.N. Sharkovsky |
Publisher | Springer Science & Business Media |
Pages | 268 |
Release | 2013-06-29 |
Genre | Mathematics |
ISBN | 940158897X |
maps whose topological entropy is equal to zero (i.e., maps that have only cyeles of pe 2 riods 1,2,2 , ... ) are studied in detail and elassified. Various topological aspects of the dynamics of unimodal maps are studied in Chap ter 5. We analyze the distinctive features of the limiting behavior of trajectories of smooth maps. In particular, for some elasses of smooth maps, we establish theorems on the number of sinks and study the problem of existence of wandering intervals. In Chapter 6, for a broad elass of maps, we prove that almost all points (with respect to the Lebesgue measure) are attracted by the same sink. Our attention is mainly focused on the problem of existence of an invariant measure absolutely continuous with respect to the Lebesgue measure. We also study the problem of Lyapunov stability of dynamical systems and determine the measures of repelling and attracting invariant sets. The problem of stability of separate trajectories under perturbations of maps and the problem of structural stability of dynamical systems as a whole are discussed in Chap ter 7. In Chapter 8, we study one-parameter families of maps. We analyze bifurcations of periodic trajectories and properties of the set of bifurcation values of the parameter, in eluding universal properties such as Feigenbaum universality.
Topics from One-Dimensional Dynamics
Title | Topics from One-Dimensional Dynamics PDF eBook |
Author | Karen M. Brucks |
Publisher | Cambridge University Press |
Pages | 312 |
Release | 2004-07-12 |
Genre | Mathematics |
ISBN | 9780521838962 |
One-dimensional dynamics has generated many results, and avenues of active mathematical research with numerous inroads to this research remain to be pursued by the advanced undergraduate or beginning graduate student. While much of the material in this book is not covered elsewhere, some aspects present new research topics whose connections are drawn to other research areas from the text. Although the material presented is not meant to be approached in a linear fashion, anybody with an interest in dynamics will find many topics of interest.
One-Dimensional Dynamics
Title | One-Dimensional Dynamics PDF eBook |
Author | Welington de Melo |
Publisher | Springer |
Pages | 606 |
Release | 2011-12-16 |
Genre | Mathematics |
ISBN | 9783642780455 |
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).
Grammatical Complexity And One-dimensional Dynamical Systems
Title | Grammatical Complexity And One-dimensional Dynamical Systems PDF eBook |
Author | Huimin Xie |
Publisher | World Scientific |
Pages | 290 |
Release | 1996-11-23 |
Genre | Science |
ISBN | 9814499897 |
A combinatorial method is developed in this book to explore the mysteries of chaos, which has became a topic of science since 1975. Using tools from theoretical computer science, formal languages and automata, the complexity of symbolic behaviors of dynamical systems is classified and analysed thoroughly. This book is mainly devoted to explanation of this method and apply it to one-dimensional dynamical systems, including the circle and interval maps, which are typical in exhibiting complex behavior through simple iterated calculations. The knowledge for reading it is self-contained in the book.
Mathematical Tools for One-Dimensional Dynamics
Title | Mathematical Tools for One-Dimensional Dynamics PDF eBook |
Author | Edson de Faria |
Publisher | Cambridge University Press |
Pages | 192 |
Release | 2008-10-02 |
Genre | Mathematics |
ISBN | 1139474847 |
Originating with the pioneering works of P. Fatou and G. Julia, the subject of complex dynamics has seen great advances in recent years. Complex dynamical systems often exhibit rich, chaotic behavior, which yields attractive computer generated pictures, for example the Mandelbrot and Julia sets, which have done much to renew interest in the subject. This self-contained book discusses the major mathematical tools necessary for the study of complex dynamics at an advanced level. Complete proofs of some of the major tools are presented; some, such as the Bers-Royden theorem on holomorphic motions, appear for the very first time in book format. An appendix considers Riemann surfaces and Teichmüller theory. Detailing the very latest research, the book will appeal to graduate students and researchers working in dynamical systems and related fields. Carefully chosen exercises aid understanding and provide a glimpse of further developments in real and complex one-dimensional dynamics.