Lectures On Fractal Geometry And Dynamical Systems

Author : Ya. B. Pesin,Vaughn Climenhaga
Publisher : American Mathematical Soc.
Page : 334 pages
File Size : 55,7 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821848890

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Lectures on Fractal Geometry and Dynamical Systems by Ya. B. Pesin,Vaughn Climenhaga Pdf

Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular 'chaotic' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory - Cantor sets, Hausdorff dimension, box dimension - using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science - the FitzHugh - Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.

Author : Martina Zaehle
Publisher : World Scientific
Page : 141 pages
File Size : 52,9 Mb
Release : 2023-12-27
Category : Mathematics
ISBN : 9789811283352

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Lectures On Fractal Geometry by Martina Zaehle Pdf

This book is based on a series of lectures at the Mathematics Department of the University of Jena, developed in the period from 1995 up to 2015. It is completed by additional material and extensions of some basic results from the literature to more general metric spaces.This book provides a clear introduction to classical fields of fractal geometry, which provide some background for modern topics of research and applications. Some basic knowledge on general measure theory and on topological notions in metric spaces is presumed.

Author : Benoit B. Mandelbrot
Publisher : Unknown
Page : 384 pages
File Size : 53,6 Mb
Release : 2013
Category : Electronic books
ISBN : 1470410834

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Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II by Benoit B. Mandelbrot Pdf

Author : Heinz-Otto Peitgen,Peter H. Richter
Publisher : Springer Science & Business Media
Page : 214 pages
File Size : 55,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9783642617171

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The Beauty of Fractals by Heinz-Otto Peitgen,Peter H. Richter Pdf

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.

Author : David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 40,5 Mb
Release : 2013-10-22
Category : Mathematics
ISBN : 9780821891476

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Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics by David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen Pdf

This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoit Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry (and some aspects of dynamical systems) in pure mathematics. Also included are articles discussing a variety of connections of fractal geometry with other fields of mathematics, including probability theory, number theory, geometric measure theory, partial differential equations, global analysis on non-smooth spaces, harmonic analysis and spectral geometry. The companion volume (Contemporary Mathematics, Volume 601) focuses on applications of fractal geometry and dynamical systems to other sciences, including physics, engineering, computer science, economics, and finance.

Author : David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 55,5 Mb
Release : 2013-10-24
Category : Mathematics
ISBN : 9780821891483

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Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II by David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen Pdf

This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoît Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry and various aspects of dynamical systems in applied mathematics and the applications to other sciences. Also included are articles discussing a variety of connections between these subjects and various areas of physics, engineering, computer science, technology, economics and finance, as well as of mathematics (including probability theory in relation with statistical physics and heat kernel estimates, geometric measure theory, partial differential equations in relation with condensed matter physics, global analysis on non-smooth spaces, the theory of billiards, harmonic analysis and spectral geometry). The companion volume (Contemporary Mathematics, Volume 600) focuses on the more mathematical aspects of fractal geometry and dynamical systems.

Author : Michel Laurent Lapidus,Machiel Van Frankenhuysen
Publisher : American Mathematical Soc.
Page : 534 pages
File Size : 49,6 Mb
Release : 2004
Category : Ergodic theory
ISBN : 9780821836378

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Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot by Michel Laurent Lapidus,Machiel Van Frankenhuysen Pdf

This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry. In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications. This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.

Author : Robert L. Devaney,Kathleen T. Alligood
Publisher : American Mathematical Soc.
Page : 176 pages
File Size : 55,5 Mb
Release : 1989
Category : Mathematics
ISBN : 9780821801376

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Chaos and Fractals: The Mathematics Behind the Computer Graphics by Robert L. Devaney,Kathleen T. Alligood Pdf

"Robert Devaney communicates his deep understanding as well as his enthusiasm for chaos, fractals, and dynamical systems. Starting at a level suitable for well-prepared high school students, he tells the mathematical story behind these fascinating topics. Equations and graphs are clearly shown with computer-generated characters, and Devaney's explanations are lucid and instructive. Illustrating the mathematics are forays into the colorful, unpredictable world of fractals and Julia sets. Devaney explains how the computer is used to generate the pictures and shows how the various colors are chosen for graphical representations ... Though the mathematical background required is elementary, those at the collegiate level and beyond will appreciate ... the clarity of exposition and the sheer beauty of the graphics."--Container.

Author : Sangita Jha,Mrinal Kanti Roychowdhury,Saurabh Verma
Publisher : American Mathematical Society
Page : 270 pages
File Size : 48,9 Mb
Release : 2024-04-18
Category : Mathematics
ISBN : 9781470472160

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Recent Developments in Fractal Geometry and Dynamical Systems by Sangita Jha,Mrinal Kanti Roychowdhury,Saurabh Verma Pdf

This volume contains the proceedings of the virtual AMS Special Session on Fractal Geometry and Dynamical Systems, held from May 14–15, 2022. The content covers a wide range of topics. It includes nonautonomous dynamics of complex polynomials, theory and applications of polymorphisms, topological and geometric problems related to dynamical systems, and also covers fractal dimensions, including the Hausdorff dimension of fractal interpolation functions. Furthermore, the book contains a discussion of self-similar measures as well as the theory of IFS measures associated with Bratteli diagrams. This book is suitable for graduate students interested in fractal theory, researchers interested in fractal geometry and dynamical systems, and anyone interested in the application of fractals in science and engineering. This book also offers a valuable resource for researchers working on applications of fractals in different fields.

Author : Michael F. Barnsley,Stephen G. Demko
Publisher : Academic Press
Page : 305 pages
File Size : 53,8 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483269085

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Chaotic Dynamics and Fractals by Michael F. Barnsley,Stephen G. Demko Pdf

Chaotic Dynamics and Fractals covers the proceedings of the 1985 Conference on Chaotic Dynamics, held at the Georgia Institute of Technology. This conference deals with the research area of chaos, dynamical systems, and fractal geometry. This text is organized into three parts encompassing 16 chapters. The first part describes the nature of chaos and fractals, the geometric tool for some strange attractors, and other complicated sets of data associated with chaotic systems. This part also considers the Henon-Hiles Hamiltonian with complex time, a Henon family of maps from C2 into itself, and the idea of turbulent maps in the course of presenting results on iteration of continuous maps from the unit interval to itself. The second part discusses complex analytic dynamics and associated fractal geometry, specifically the bursts into chaos, algorithms for obtaining geometrical and combinatorial information, and the parameter space for iterated cubic polynomials. This part also examines the differentiation of Julia sets with respects to a parameter in the associated rational map, permitting the formulation of Taylor series expansion for the sets. The third part highlights the applications of chaotic dynamics and fractals. This book will prove useful to mathematicians, physicists, and other scientists working in, or introducing themselves to, the field.

Author : Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily
Publisher : Springer Nature
Page : 226 pages
File Size : 44,7 Mb
Release : 2020-01-01
Category : Mathematics
ISBN : 9783030358549

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Dynamics with Chaos and Fractals by Marat Akhmet,Mehmet Onur Fen,Ejaily Milad Alejaily Pdf

The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.

Author : Christoph Bandt,Siegfried Graf,Martina Zähle
Publisher : Birkhäuser
Page : 286 pages
File Size : 50,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883801

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Fractal Geometry and Stochastics II by Christoph Bandt,Siegfried Graf,Martina Zähle Pdf

A collection of contributions by outstanding mathematicians, highlighting the principal directions of research on the combination of fractal geometry and stochastic methods. Clear expositions introduce the most recent results and problems on these subjects and give an overview of their historical development.

Author : Jacques Bélair,Serge Dubuc
Publisher : Springer Science & Business Media
Page : 485 pages
File Size : 41,9 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9789401579315

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Fractal Geometry and Analysis by Jacques Bélair,Serge Dubuc Pdf

This ASI- which was also the 28th session of the Seminaire de mathematiques superieures of the Universite de Montreal - was devoted to Fractal Geometry and Analysis. The present volume is the fruit of the work of this Advanced Study Institute. We were fortunate to have with us Prof. Benoit Mandelbrot - the creator of numerous concepts in Fractal Geometry - who gave a series of lectures on multifractals, iteration of analytic functions, and various kinds of fractal stochastic processes. Different foundational contributions for Fractal Geometry like measure theory, dy namical systems, iteration theory, branching processes are recognized. The geometry of fractal sets and the analytical tools used to investigate them provide a unifying theme of this book. The main topics that are covered are then as follows. Dimension Theory. Many definitions of fractional dimension have been proposed, all of which coincide on "regular" objects, but often take different values for a given fractal set. There is ample discussion on piecewise estimates yielding actual values for the most common dimensions (Hausdorff, box-counting and packing dimensions). The dimension theory is mainly discussed by Mendes-France, Bedford, Falconer, Tricot and Rata. Construction of fractal sets. Scale in variance is a fundamental property of fractal sets.

Author : Hillel Furstenberg
Publisher : American Mathematical Society
Page : 82 pages
File Size : 44,7 Mb
Release : 2014-08-08
Category : Mathematics
ISBN : 9781470410346

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Ergodic Theory and Fractal Geometry by Hillel Furstenberg Pdf

Fractal geometry represents a radical departure from classical geometry, which focuses on smooth objects that "straighten out" under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as "zooming in". This zooming-in process has its parallels in dynamics, and the varying "scenery" corresponds to the evolution of dynamical variables. The present monograph focuses on applications of one branch of dynamics--ergodic theory--to the geometry of fractals. Much attention is given to the all-important notion of fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics. A co-publication of the AMS and CBMS.

Author : Hillel Furstenberg
Publisher : Unknown
Page : 0 pages
File Size : 53,8 Mb
Release : 2017-06-05
Category : Mathematics
ISBN : 1470437260

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Ergodic Theory and Fractal Geometry by Hillel Furstenberg Pdf

fractal Geometry represents a radical departure from classical Geometry, which focuses on smooth objects that straighten out under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as zooming in. this zooming-in process has its parallels in dynamics, and the varying scenery corresponds to the evolution of dynamical variables. the present monograph focuses on applications of one branch of dynamics ergodic theory the Geometry of fractals. Much attention is given to the all-important notion of Fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of Fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics.