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Lectures on Fractal Geometry and Dynamical Systems

Yakov Pesin Pennsylvania State University, University Park, PA
Vaughn Climenhaga Pennsylvania State University, University Park, PA
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Softcover ISBN: 978-0-8218-4889-0
Product Code: STML/52
List Price: $55.00 Individual Price:$44.00
Electronic ISBN: 978-1-4704-1221-0
Product Code: STML/52.E
List Price: $51.00 Individual Price:$40.80
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List Price: $82.50 Click above image for expanded view Lectures on Fractal Geometry and Dynamical Systems Yakov Pesin Pennsylvania State University, University Park, PA Vaughn Climenhaga Pennsylvania State University, University Park, PA Available Formats:  Softcover ISBN: 978-0-8218-4889-0 Product Code: STML/52  List Price:$55.00 Individual Price: $44.00  Electronic ISBN: 978-1-4704-1221-0 Product Code: STML/52.E  List Price:$51.00 Individual Price: $40.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$82.50
• Book Details

Student Mathematical Library
Volume: 522009; 314 pp
MSC: Primary 37;

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.

This book is published in cooperation with Mathematics Advanced Study Semesters.

• Chapters
• Chapter 1. Basic concepts and examples
• Chapter 2. Fundamentals of dimension theory
• Chapter 3. Measures: Definitions and examples
• Chapter 4. Measures and dimensions
• Chapter 5. Discrete-time systems: The FitzHugh–Nagumo model
• Chapter 6. The bifurcation diagram for the logistic map
• Chapter 7. Chaotic attractors and persistent chaos
• Chapter 8. Horseshoes and intermittent chaos
• Chapter 9. Continuous-time systems: The Lorenz model
• Appendix
• Hints to selected exercises

• Reviews

• [F]or a student with a reasonable background in topology and measure theory this is a very useful book covering many of the main ideas in fractal geometry and dynamical systems in an accessible way, with a particular emphasis on dynamically-defined fractals.

Ian Melbourne, Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 522009; 314 pp
MSC: Primary 37;

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.

This book is published in cooperation with Mathematics Advanced Study Semesters.

• Chapters
• Chapter 1. Basic concepts and examples
• Chapter 2. Fundamentals of dimension theory
• Chapter 3. Measures: Definitions and examples
• Chapter 4. Measures and dimensions
• Chapter 5. Discrete-time systems: The FitzHugh–Nagumo model
• Chapter 6. The bifurcation diagram for the logistic map
• Chapter 7. Chaotic attractors and persistent chaos
• Chapter 8. Horseshoes and intermittent chaos
• Chapter 9. Continuous-time systems: The Lorenz model
• Appendix
• Hints to selected exercises
• [F]or a student with a reasonable background in topology and measure theory this is a very useful book covering many of the main ideas in fractal geometry and dynamical systems in an accessible way, with a particular emphasis on dynamically-defined fractals.

Ian Melbourne, Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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