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Lectures on Fractal Geometry and Dynamical Systems http://dx.doi.org/10.1090/stml/052 Lectures on Fractal Geometry and Dynamical Systems STUDENT MATHEMATICAL LIBRARY Volume 52 Lectures on Fractal Geometry and Dynamical Systems Yakov Pesin Vaughn Climenhaga HEMAT MAT ICS STUDY A DVANCED S EMESTERS American Mathematical Society Mathematics Advanced Study Semesters Editorial Board Gerald B. Folland Brad G. Osgood (Chair) Robin Forman Michael Starbird 2000 Mathematics Subject Classification. Primary 37–01, 37C45; Secondary 37B10, 37D20, 37E05. This work is partially supported by NSF grant 0754911. The photograph of the Australian coastline in Figure 1.2 is courtesy of NASA Earth Observatory: http://earthobservatory.nasa.gov/ NaturalHazards/view.php?id=11595 The map of France in Figure 1.15 is taken from the CIA World Factbook. For additional information and updates on this book, visit www.ams.org/bookpages/stml-52 Library of Congress Cataloging-in-Publication Data Pesin, Ya. B. Lectures on fractal geometry and dynamical systems / Yakov Pesin, Vaughn Climenhaga. p. cm. — (Student mathematical library ; v. 52) Includes bibliographical references and index. ISBN 978-0-8218-4889-0 (alk. paper) 1. Fractals. 2. Dynamics. I. Climenhaga, Vaughn, 1982– II. Title. QA614.86 .P47 2009 2009028324 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294 USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09 Contents Foreword: MASS and REU at Penn State University xi Preface xiii Chapter 1. Basic Concepts and Examples 1 Lecture 1 1 a. A threefold cord: fractals, dynamics, and chaos 1 b. Fractals: intricate geometry and self-similarity 2 c. Dynamics: things that move (or don’t) 7 Lecture 2 10 a. Dynamical systems: terminology and notation 10 b. Population models and the logistic map 13 Lecture 3 19 a. A linear map with chaotic behaviour and the middle-third Cantor set 19 b. The Cantor set and symbolic dynamics 24 Lecture 4 28 a. Some point-set topology 28 b. Metric spaces 31 c. Lebesgue measure 34 Lecture 5 37 a. The topological structure of symbolic space and the Cantor set 37 v vi Contents b. What the coding map doesn’t do 41 c. Geometry of Cantor sets 42 Lecture 6 46 a. More general constructions 46 b. Making sense of it all 50 Chapter 2. Fundamentals of Dimension Theory 53 Lecture 7 53 a. Definition of Hausdorff dimension 53 b. Hausdorff dimension of the middle-third Cantor set 58 c. Alternative definitions of Hausdorff dimension 60 Lecture 8 62 a. Properties of Hausdorff dimension 62 b. Topological dimension 66 Lecture 9 68 a. Comparison of Hausdorff and topological dimension 68 b. Metrics and topologies 71 c. Topology and dimension 74 Lecture 10 75 a. Hausdorff dimension of Cantor sets 75 b. Moran’s Theorem 76 c. Moran constructions 80 d. Dynamical constructions and iterated function systems 82 Lecture 11 85 a. Box dimension: another way of measuring dimension 85 b. Properties of box dimension 88 Lecture 12 90 a. Relationships between the various dimensions 90 b. A counterexample 94 c. Stability and subadditivity 98 Chapter 3. Measures: Definitions and Examples 101 Lecture 13 101 a. A little bit of measure theory 101 b. Lebesgue measure and outer measures 105 Contents vii c. Hausdorff measures 109 Lecture 14 110 a. Choosing a “good” outer measure 110 b. Bernoulli measures on symbolic space 111 c. Measures on Cantor sets 113 d. Markov measures 114 Lecture 15 117 a. The support of a measure 117 b. Subshifts of finite type: One-dimensional Markov maps 120 Chapter 4. Measures and Dimensions 123 Lecture 16 123 a. The Uniform Mass Distribution Principle: Using measures to determine dimension 123 b. Pointwise dimension and the Non-uniform Mass Distribution Principle 125 Lecture 17 128 a. Variable pointwise dimension 128 b. Hausdorff dimension of exact dimensional measures 135 c. Pointwise dimension of Hausdorff measures 137 Lecture 18 138 a. Local entropy 138 b. Kolmogorov–Sinai entropy 142 c. Topological entropy 143 Lecture 19 146 a. Entropy of Markov measures 146 b. Hausdorff dimension of Markov constructions 149 Lecture 20 151 a. Lyapunov exponents 151 b. Fractals within fractals 155 Chapter 5. Discrete-Time Systems: The FitzHugh–Nagumo Model 159 Lecture 21 159 a. The FitzHugh–Nagumo model for neurons 159 viii Contents b. Numerical investigations: From continuous to discrete 164 Lecture 22 167 a. Studying the local map 167 b. Stability of fixed points for general maps 169 Lecture 23 175 a. Stability of fixed points for the FitzHugh–Nagumo model 175 b. Periodic points 177 Lecture 24 181 a. Beyond period-doubling: Down the rabbit hole 181 b. Becoming one-dimensional 185 Chapter 6. The Bifurcation Diagram for the Logistic Map 191 Lecture 25 191 a. Bifurcations of the logistic map 191 b. Classifying bifurcations 194 Lecture 26 198 a. The period-doubling cascade 198 b. Chaos at the end of the bifurcation diagram 199 c. The centre cannot hold: escape to infinity 202 Lecture 27 204 a. Finding the relevant part of phase space: ω-limit sets 204 b. Windows of stability in the bifurcation diagram 206 c. Chaos outside the windows of stability 207 Chapter 7. Chaotic Attractors and Persistent Chaos 209 Lecture 28 209 a. Trapping regions 209 b. Attractors 212 Lecture 29 215 a. The Smale–Williams solenoid 215 b. Uniform hyperbolicity 218 c. Symbolic dynamics 221 Lecture 30 224 a. Dimension of direct products 224 b. Quantifying the attractor 228 Contents ix c. Lyapunov exponents in multiple dimensions 229 d. The non-conformal case 231 e. The attractor for the FitzHugh–Nagumo map 232 Chapter 8. Horseshoes and Intermittent Chaos 233 Lecture 31 233 a. The Smale horseshoe: A trapping region that isn’t 233 b. Hausdorff dimension of the horseshoe 237 c. Symbolic dynamics on the Smale horseshoe 239 Lecture 32 241 a. Variations on a theme: Other horseshoes 241 b. Intermittent chaos vs. persistent chaos 245 c. Homoclinic orbits and horseshoes 247 Chapter 9. Continuous-Time Systems: The Lorenz Model 253 Lecture 33 253 a. Continuous-time systems: Basic concepts 253 b. Fixed points of continuous-time systems 256 Lecture 34 259 a. The pendulum 259 b. Two-dimensional systems 263 Lecture 35 267 a. The Lorenz equations 267 b. Beyond the linear mindset 269 c. Examining the Lorenz system 269 Lecture 36 274 a. Passing to a Poincar´e map 274 b. Horseshoes in the Lorenz system 277 Lecture 37 281 a. The Lorenz attractor 281 b. The geometric Lorenz attractor 281 c. Dimension of the geometric Lorenz attractor 285 d. Back to the Lorenz attractor, and beyond 286 Appendix 289 Hints to selected exercises 295 x Contents Suggested Reading 299 Bibliography 305 Index 309 Foreword: MASS and REU at Penn State University This book is part of a collection published jointly by the American Mathematical Society and the MASS (Mathematics Advanced Study Semesters) program as a part of the Student Mathematical Library series. The books in the collection are based on lecture notes for advanced undergraduate topics courses taught at the MASS and/or Penn State summer REU (Research Experiences for Undergraduates). Each book presents a self-contained exposition of a non-standard mathematical topic, often related to current research areas, which is accessible to undergraduate students familiar with an equivalent of two years of standard college mathematics, and is suitable as a text for an upper division undergraduate course. Started in 1996, MASS is a semester-long program for advanced undergraduate students from across the USA. The program’s curricu- lum amounts to sixteen credit hours. It includes three core courses from the general areas of algebra/number theory, geometry/topology, and analysis/dynamical systems, custom designed every year; an in- terdisciplinary seminar; and a special colloquium. In addition, ev- ery participant completes three research projects, one for each core course. The participants are fully immersed into mathematics, and xi xii Foreword: MASS and REU at Penn State University this, as well as intensive interaction among the students, usually leads to a dramatic increase in their mathematical enthusiasm and achieve- ment. The program is unique for its kind in the United States. The summer mathematical REU program is formally independent of MASS, but there is a significant interaction between the two: about half of the REU participants stay for the MASS semester in the fall. This makes it possible to offer research projects that require more than seven weeks (the length of the REU program) for completion. The summer program includes the MASS Fest, a two- to three-day conference at the end of the REU at which the participants present their research and that also serves as a MASS alumni reunion.
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