**Student Mathematical Library**

Volume: 52;
2009;
314 pp;
Softcover

MSC: Primary 37;

Print ISBN: 978-0-8218-4889-0

Product Code: STML/52

List Price: $51.00

Individual Price: $40.80

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**Electronic ISBN: 978-1-4704-1221-0
Product Code: STML/52.E**

List Price: $51.00

Individual Price: $40.80

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#### Supplemental Materials

# Lectures on Fractal Geometry and Dynamical Systems

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*Yakov Pesin; Vaughn Climenhaga*

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

#### Table of Contents

# Table of Contents

## Lectures on Fractal Geometry and Dynamical Systems

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Foreword: MASS and REU at Penn State University xi12 free
- Preface xiii14 free
- Basic concepts and examples 118 free
- Fundamentals of dimension theory 5370
- Measures: definitions and examples 101118
- Measures and dimensions 123140
- Discrete-time systems: the FitzHugh–Nagumo model 159176
- The bifurcation diagram for the logistic map 191208
- Chaotic attractors and persistent chaos 209226
- Horseshoes and intermittent chaos 233250
- Continuous-time systems: the Lorenz model 253270
- Appendix 289306
- Hints to selected exercises 295312
- Suggested reading 299316
- Bibliography 305322
- Index 309326 free
- Back Cover Back Cover1334

#### Readership

Undergraduate and graduate students interested in dynamical systems and fractal geometry.

#### 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