Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets


Book Description

The Mandelbrot set has emerged as one of the most recognizable objects in mathematics. While there is no question of its beauty, relatively few people appreciate the fact that the mathematics behind such images is equally beautiful. This book presents lectures delivered during the AMS Short Course entitled Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets, held at the Joint Mathematics Meetings in Cincinnati in January 1994. The lectures cover a wide range of topics, including the classical work of Julia and Fatou on local dynamics of analytic maps as well as recent work on the dynamics of quadratic and cubic polynomials, the geometry of Julia sets, and the structure of various parameter spaces. Among the other topics are recent results on Yoccoz puzzles and tableaux, limiting dynamics near parabolic points, the spider algorithm, extensions of the theory to rational maps, Newton's method, and entire transcendental functions. Much of the book is accessible to anyone with a background in the basics of dynamical systems and complex analysis.




The Beauty of Fractals


Book Description

Now approaching its tenth year, this hugely successful book presents an unusual attempt to publicise the field of Complex Dynamics. The text was originally conceived as a supplemented catalogue to the exhibition "Frontiers of Chaos", seen in Europe and the United States, and describes the context and meaning of these fascinating images. A total of 184 illustrations - including 88 full-colour pictures of Julia sets - are suggestive of a coffee-table book. However, the invited contributions which round off the book lend the text the required formality. Benoit Mandelbrot gives a very personal account, in his idiosyncratic self-centred style, of his discovery of the fractals named after him and Adrien Douady explains the solved and unsolved problems relating to this amusingly complex set.




The Mandelbrot Set, Theme and Variations


Book Description

The Mandelbrot set is a fractal shape that classifies the dynamics of quadratic polynomials. It has a remarkably rich geometric and combinatorial structure. This volume provides a systematic exposition of current knowledge about the Mandelbrot set and presents the latest research in complex dynamics. Topics discussed include the universality and the local connectivity of the Mandelbrot set, parabolic bifurcations, critical circle homeomorphisms, absolutely continuous invariant measures and matings of polynomials, along with the geometry, dimension and local connectivity of Julia sets. In addition to presenting new work, this collection documents important results hitherto unpublished or difficult to find in the literature. This book will be of interest to graduate students in mathematics, physics and mathematical biology, as well as researchers in dynamical systems and Kleinian groups.




Fractals and Chaos


Book Description

Just 23 years ago Benoit Mandelbrot published his famous picture of the Mandelbrot set, but that picture has changed our view of the mathematical and physical universe. In this text, Mandelbrot offers 25 papers from the past 25 years, many related to the famous inkblot figure. Of historical interest are some early images of this fractal object produced with a crude dot-matrix printer. The text includes some items not previously published.




Iteration of Rational Functions


Book Description

This book focuses on complex analytic dynamics, which dates from 1916 and is currently attracting considerable interest. The text provides a comprehensive, well-organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The coverage extends from early memoirs of Fatou and Julia to important recent results and methods of Sullivan and Shishikura. Many details of the proofs have not appeared in print before.




Computability of Julia Sets


Book Description

Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content. Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computer screen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking - it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized. The book summarizes the present knowledge (most of it from the authors' own work) about the computational properties of Julia sets in a self-contained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems.




Science at the Frontier


Book Description

Science at the Frontier takes you on a journey through the minds of some of the nation's leading young scientists as they explore the most exciting areas of discovery today. Based on the second Frontiers of Science symposium sponsored by the National Academy of Sciences, this book describes recent accomplishments and new directions in ten basic fields, represented by outstanding scientists convening to discuss their research. It captures the excitement and personal quality of these exchanges, sometimes pointing to surprising connections spanning the boundaries of traditional disciplines, while providing a context for the reader that explains the basic scientific framework for the fields under discussion. The volume explores: New modifications to scientific theory as geologists probe deep inside the earth and astrophysicists reach to the limits of the observable universe for answers to some of nature's most fundamental and vexing questions. The influence of research in smog formation on the public debate about how to effectively control air pollution. The increasing use of computer modeling in science, from describing the evolution of cellular automata to revealing the workings of the human brain via neural networks. The rise of dynamical systems (the study of chaotic behavior in nature) to a full-fledged science. The search to understand the regulation of gene activity and the many biological problems-such as the onset of cancer-to which it applies. Recent progress in the quest to transform what we know about photosynthesis into functional, efficient systems to tap the sun's energy. Current developments in magnetic resonance imaging and its promise for new breakthroughs in medical diagnosis. Throughout this work the reader is witness to scientific discovery and debate centered on such common concerns as the dramatic and transforming effect of computers on scientists' thinking and research; the development of more cross-disciplinary perspectives; and the very nature of the scientific enterprise itself-what it is to be part of it, and its significance for society. Science at the Frontier is must reading for informed lay readers, scientists interested in fields other than their own, and science students considering a future specialization.




An Introduction To Chaotic Dynamical Systems


Book Description

The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry. Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas.




A First Course in Discrete Dynamical Systems


Book Description

Given the ease with which computers can do iteration it is now possible for almost anyone to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. Mathematica programs that illustrate the dynamics are included in an appendix.




Chaos and Fractals: The Mathematics Behind the Computer Graphics


Book Description

The terms chaos and fractals have received widespread attention in recent years. The alluring computer graphics images associated with these terms have heightened interest among scientists in these ideas. This volume contains the introductory survey lectures delivered in the American Mathematical Society Short Course, Chaos and Fractals: The Mathematics Behind the Computer Graphics, on August 6-7, 1988, given in conjunction with the AMS Centennial Meeting in Providence, Rhode Island. In his overview, Robert L. Devaney introduces such key topics as hyperbolicity, the period doubling route to chaos, chaotic dynamics, symbolic dynamics and the horseshoe, and the appearance of fractals as the chaotic set for a dynamical system. Linda Keen and Bodil Branner discuss the Mandelbrot set and Julia sets associated to the complex quadratic family z -> z2 + c. Kathleen T. Alligood, James A. Yorke, and Philip J. Holmes discuss some of these topics in higher dimensional settings, including the Smale horseshoe and strange attractors. Jenny Harrison and Michael F. Barnsley give an overview of fractal geometry and its applications. -- from dust jacket.