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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions.
Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines.
Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences.
- Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals
- Carefully explains each topic using illustrative examples and diagrams
- Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics
- Features a wide range of exercises, enabling readers to consolidate their understanding
- Supported by a website with solutions to exercises and additional material www.wileyeurope.com/fractal
Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)
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Veröffentlichungsjahr: 2013
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Contents
Cover
Preface to the first edition
Preface to the second edition
Preface to the third edition
Course suggestions
Introduction
Notes and references
Part I: Foundations
Chapter 1: Mathematical background
1.1 Basic set theory
1.2 Functions and limits
1.3 Measures and mass distributions
1.4 Notes on probability theory
1.5 Notes and references
Exercises
Chapter 2: Box-counting dimension
2.1 Box-counting dimensions
2.2 Properties and problems of box-counting dimension
*2.3 Modified box-counting dimensions
2.4 Some other definitions of dimension
2.5 Notes and references
Exercises
Chapter 3: Hausdorff and packing measures and dimensions
3.1 Hausdorff measure
3.2 Hausdorff dimension
3.3 Calculation of Hausdorff dimension—simple examples
3.4 Equivalent definitions of Hausdorff dimension
*3.5 Packing measure and dimensions
*3.6 Finer definitions of dimension
*3.7 Dimension prints
*3.8 Porosity
3.9 Notes and references
Exercises
Chapter 4: Techniques for calculating dimensions
4.1 Basic methods
4.2 Subsets of finite measure
4.3 Potential theoretic methods
*4.4 Fourier transform methods
4.5 Notes and references
Exercises
Chapter 5: Local structure of fractals
5.1 Densities
5.2 Structure of 1-sets
5.3 Tangents to
-sets
5.4 Notes and references
Exercises
Chapter 6: Projections of fractals
6.1 Projections of arbitrary sets
6.2 Projections of
s
-sets of integral dimension
6.3 Projections of arbitrary sets of integral dimension
6.4 Notes and references
Exercises
Chapter 7: Products of fractals
7.1 Product formulae
7.2 Notes and references
Exercises
Chapter 8: Intersections of fractals
8.1 Intersection formulae for fractals
8.2 Sets with large intersection
*8.3 Notes and references
Exercises
Part II: Applications and Examples
Chapter 9: Iterated function systems—self-similar and self-affine sets
9.1 Iterated function systems
9.2 Dimensions of self-similar sets
9.3 Some variations
9.4 Self-affine sets
9.5 Applications to encoding images
*9.6 Zeta functions and complex dimensions
9.7 Notes and references
Exercises
Chapter 10: Examples from number theory
10.1 Distribution of digits of numbers
10.2 Continued fractions
10.3 Diophantine approximation
10.4 Notes and references
Exercises
Chapter 11: Graphs of functions
11.1 Dimensions of graphs
*11.2 Autocorrelation of fractal functions
11.3 Notes and references
Exercises
Chapter 12: Examples from pure mathematics
12.1 Duality and the Kakeya problem
12.2 Vitushkin's conjecture
12.3 Convex functions
12.4 Fractal groups and rings
12.5 Notes and references
Exercises
Chapter 13: Dynamical systems
13.1 Repellers and iterated function systems
13.2 The logistic map
13.3 Stretching and folding transformations
13.4 The solenoid
13.5 Continuous dynamical systems
*13.6 Small divisor theory
*13.7 Lyapunov exponents and entropies
13.8 Notes and references
Exercises
Chapter 14: Iteration of complex functions—Julia sets and the Mandelbrot set
14.1 General theory of Julia sets
14.2 Quadratic functions—the Mandelbrot set
14.3 Julia sets of quadratic functions
14.4 Characterisation of quasi-circles by dimension
14.5 Newton's method for solving polynomial equations
14.6 Notes and references
Exercises
Chapter 15: Random fractals
15.1 A random Cantor set
15.2 Fractal percolation
15.3 Notes and references
Exercises
Chapter 16: Brownian motion and Brownian surfaces
16.1 Brownian motion in
16.2 Brownian motion in
16.3 Fractional Brownian motion
16.4 Fractional Brownian surfaces
16.5 Lévy stable processes
16.6 Notes and references
Exercises
Chapter 17: Multifractal measures
17.1 Coarse multifractal analysis
17.2 Fine multifractal analysis
17.3 Self-similar multifractals
17.4 Notes and references
Exercises
Chapter 18: Physical applications
18.1 Fractal fingering
18.2 Singularities of electrostatic and gravitational potentials
18.3 Fluid dynamics and turbulence
18.4 Fractal antennas
18.5 Fractals in finance
18.6 Notes and references
Exercises
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Introduction
Part I
Chapter 1
List of Illustrations
Figure 0.1
Figure 0.2
Figure 0.3
Figure 0.4
Figure 0.5
Figure 0.6
Figure 0.7
Figure 0.8
Figure 0.9
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure 1.5
Figure 2.1
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 4.1
Figure 4.2
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 8.1
Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4
Figure 9.5
Figure 9.6
Figure 9.7
Figure 9.8
Figure 9.9
Figure 9.11
Figure 9.10
Figure 9.12
Figure 9.13
Figure 9.14
Figure 11.1
Figure 11.2
Figure 11.3
Figure 11.4
Figure 11.5
Figure 12.1
Figure 12.2
Figure 12.3
Figure 13.1
Figure 13.2
Figure 13.3
Figure 13.4
Figure 13.5
Figure 13.6
Figure 13.7
Figure 13.8
Figure 13.9
Figure 13.10
Figure 13.11
Figure 13.12
Figure 13.13
Figure 13.14
Figure 13.15
Figure 13.16
Figure 13.17
Figure 14.2
Figure 14.3
Figure 14.4
Figure 14.5
Figure 14.6
Figure 14.7
Figure 14.1
Figure 14.8
Figure 14.9
Figure 14.10
Figure 15.1
Figure 15.2
Figure 15.3
Figure 15.4
Figure 15.5
Figure 16.1
Figure 16.2
Figure 16.3
Figure 16.4
Figure 17.1
Figure 17.2
Figure 17.3
Figure 17.4
Figure 18.1
Figure 18.2
Figure 18.3
Figure 18.4
Figure 18.5
Figure 18.6
Fractal Geometry
Mathematical Foundations and Applications
Third Edition
Kenneth Falconer
University of St Andrews, UK
This edition first published 2014
© 2014 John Wiley & Sons, Ltd
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A catalogue record for this book is available from the British Library.
ISBN: 978-1-119-94239-9
Preface to the first edition
I am frequently asked questions such as ‘What are fractals?’, ‘What is fractal dimension?’, ‘How can one find the dimension of a fractal and what does it tell us anyway?’ or ‘How can mathematics be applied to fractals?’ This book endeavours to answer some of these questions.
The main aim of the book is to provide a treatment of the mathematics associated with fractals and dimensions at a level which is reasonably accessible to those who encounter fractals in mathematics or science. Although basically a mathematics book, it attempts to provide an intuitive as well as a mathematical insight into the subject.
The book falls naturally into two parts. Part I is concerned with the general theory of fractals and their geometry. Firstly, various notions of dimension and methods for their calculation are introduced. Then geometrical properties of fractals are investigated in much the same way as one might study the geometry of classical figures such as circles or ellipses: locally, a circle may be approximated by a line segment, the projection or ‘shadow’ of a circle is generally an ellipse, a circle typically intersects a straight line segment in two points (if at all) and so on. There are fractal analogues of such properties, usually with dimension playing a key role. Thus, we consider, for example, the local form of fractals and projections and intersections of fractals.
Part II of the book contains examples of fractals, to which the theory of the first part may be applied, drawn from a wide variety of areas of mathematics and physics. Topics include self-similar and self-affine sets, graphs of functions, examples from number theory and pure mathematics, dynamical systems, Julia sets, random fractals and some physical applications.
There are many diagrams in the text and frequent illustrative examples. Computer drawings of a variety of fractals are included, and it is hoped that enough information is provided to enable readers with a knowledge of programming to produce further drawings for themselves.
It is hoped that the book will be a useful reference for researchers, by providing an accessible development of the mathematics underlying fractals and showing how it may be applied in particular cases. The book covers a wide variety of mathematical ideas that may be related to fractals and, particularly in Part II, provides a flavour of what is available rather than exploring any one subject in too much detail. The selection of topics is to some extent at the author's whim—there are certainly some important applications that are not included. Some of the material dates back to early twentieth century, whilst some is very recent.
Notes and references are provided at the end of each chapter. The references are by no means exhaustive; indeed, complete references on the variety of topics covered would fill a large volume. However, it is hoped that enough information is included to enable those who wish to do so to pursue any topic further.
It would be possible to use the book as a basis for a course on the mathematics of fractals, at postgraduate or, perhaps, final-year undergraduate level, and exercises are included at the end of each chapter to facilitate this. Harder sections and proofs are marked with an asterisk and may be omitted without interrupting the development.
An effort has been made to keep the mathematics to a level that can be understood by a mathematics or physics graduate and, for the most part, by a diligent final-year undergraduate. In particular, measure theoretic ideas have been kept to a minimum, and the reader is encouraged to think of measures as ‘mass distributions’ on sets. Provided that it is accepted that measures with certain (intuitively almost obvious) properties exist, there is little need for technical measure theory in our development.
Results are always stated precisely to avoid the confusion which would otherwise result. Our approach is generally rigorous, but some of the harder or more technical proofs are either just sketched or omitted altogether. (However, a few harder proofs that are not available in that form elsewhere have been included, in particular those on sets with large intersection and on random fractals.) Suitable diagrams can be a help in understanding the proofs, many of which are of a geometric nature. Some diagrams are included in the book; the reader may find it helpful to draw others.
Chapter 1 begins with a rapid survey of some basic mathematical concepts and notation, for example, from the theory of sets and functions, which are used throughout the book. It also includes an introductory section on measure theory and mass distributions which, it is hoped, will be found adequate. The section on probability theory may be helpful for the chapters on random fractals and Brownian motion.
With the wide variety of topics covered, it is impossible to be entirely consistent in the use of notation, and inevitably, sometimes, there has to be a compromise between consistency within the book and standard usage.
In the past few years, fractals have become enormously popular as an art form, with the advent of computer graphics, and as a model of a wide variety of physical phenomena. Whilst it is possible in some ways to appreciate fractals with little or no knowledge of their mathematics, an understanding of the mathematics that can be applied to such a diversity of objects certainly enhances one's appreciation. The phrase ‘the beauty of fractals’ is often heard—it is the author's belief that much of their beauty is to be found in their mathematics.
It is a pleasure to acknowledge those who have assisted in the preparation of this book. Philip Drazin and Geoffrey Grimmett provided helpful comments on parts of the manuscript. Peter Shiarly gave valuable help with the computer drawings, and Aidan Foss produced some diagrams. I am indebted to Charlotte Farmer, Jackie Cowling and Stuart Gale of John Wiley & Sons for overseeing the production of the book.
Special thanks are due to David Marsh—not only did he make many useful comments on the manuscript and produce many of the computer pictures but he also typed the entire manuscript in a most expert way.
Finally, I would like to thank my wife Isobel for her support and encouragement, which extended to reading various drafts of the book.
Kenneth J. FalconerBristolApril 1989
Preface to the second edition
It is 13 years since Fractal Geometry—Mathematical Foundations and Applications was first published. In the meantime, the mathematics and applications of fractals have advanced enormously, with an ever-widening interest in the subject at all levels. The book was originally written for those working in mathematics and science who wished to know more about fractal mathematics. Over the past few years, with changing interests and approaches to mathematics teaching, many universities have introduced undergraduate and postgraduate courses on fractal geometry and a considerable number have been based on parts of this book.
Thus, this edition has two main aims. Firstly, it indicates some recent developments in the subject, with updated notes and suggestions for further reading. Secondly, more attention is given to the needs of students using the book as a course text, with extra details to help understanding, along with the inclusion of further exercises.
Parts of the book have been rewritten. In particular, multifractal theory has advanced considerably since the first edition was published, so the chapter on ‘Multifractal Measures’ has been completely rewritten. The notes and references have been updated. Numerous minor changes, corrections and additions have been incorporated, and some of the notation and terminology has been changed to conform with what has become standard usage. Many of the diagrams have been replaced to take advantage of the more sophisticated computer technology now available. Where possible, the numbering of sections, equations and figures has been left as in the first edition, so that earlier references to the book remain valid.
Further exercises have been added at the end of the chapters. Solutions to these exercises and additional supplementary material may be found on the World Wide Web at http://www.wileyeurope.com/fractal.
In 1997, a sequel, Techniques in Fractal Geometry, was published, presenting a variety of techniques and ideas current in fractal research. Readers wishing to study fractal mathematics beyond the bounds of this book may find the sequel helpful.
I am most grateful to all who have made constructive suggestions on the text. In particular, I am indebted to Carmen Fernández, Gwyneth Stallard and Alex Cain for their help with this revision. I am also very grateful for the continuing support given to the book by the staff of John Wiley & Sons; and in particular, to Rob Calver and Lucy Bryan, for overseeing the production of this second edition and to John O'Connor and Louise Page for the cover design.
Kenneth J. FalconerSt AndrewsJanuary 2003
Preface to the third edition
It is now 23 years since Fractal Geometry—Mathematical Foundations and Applications was first published and 10 years since the second edition. During those years, interest in the mathematics and applications of fractals has seen a phenomenal increase at all levels, with many mathematicians and scientists now involved in fractal-related topics. The book was originally written for researchers wanting to know more about fractals and their mathematics. However, many universities now present undergraduate and postgraduate courses on fractal geometry, often based on parts of this book, and the needs of students have been very much in mind during the revision. I am continually surprised by the number of researchers whom I meet who tell me that they first learnt about fractal geometry from this book.
This edition incorporates substantial changes from its predecessor with parts rewritten and new sections added. Student courses on fractal geometry usually present the simpler box-counting dimension before the more sophisticated Hausdorff dimension, so Chapters 2 and 3 have been reorganised in this way. The chapter on Brownian motion has been largely rewritten, and there are numerous minor changes and additions throughout the text. Some new sections have been added to give a glimpse of some of the recent ideas and directions, such as porosity and complex dimensions, that have evolved in fractal geometry.
When the first edition was written, the literature on fractals was comparatively limited and it was possible to include a reasonably comprehensive bibliography. In recent years, there has been an explosion in the number of research papers in the area and only a tiny proportion can be listed. Thus, the bibliography now focusses on papers that have a historical or innovative significance together with books and survey articles that provide overviews and many further references. The notes at the end of each chapter are a pointer to where next to go to find out more about a topic.
Some exercises at the end of the chapters have been modified and some more have been added, and, as before, solutions and other supplementary material may be found on the website http://www.wiley.com/go/fractal.
Those wishing to study fractals further may find helpful the sequel, Techniques in Fractal Geometry published in 1997, which is a natural continuation of this book and includes many ideas in use in current research.
Once again, I express my gratitude to the support given to the book by the staff of John Wiley & Sons and, in particular, to Richard Davies, Prachi Sinha-Sahay and Debbie Jupe for overseeing the production of this third edition. I am also very grateful to Ben Falconer for the new cover picture.
Kenneth J. FalconerSt AndrewsJune 2013
Course suggestions
There is far too much material in this book for a standard length course on fractal geometry. Depending on the emphasis required, appropriate sections may be selected as a basis for an undergraduate or a postgraduate course.
A course for mathematics students could be based on the following sections.
Mathematical background
1.1
Basic set theory;
1.2
Functions and limits;
1.3
Measures and mass distributions.
Box-counting dimension
2.1
Box-counting dimensions;
2.2
Properties of box-counting dimensions.
Hausdorff measures and dimension
3.1
Hausdorff measure;
3.2
Hausdorff dimension;
3.3
Calculation of Hausdorff dimension;
4.1
Basic methods of calculating dimensions.
Iterated function systems
9.1
Iterated function systems;
9.2
Dimensions of self-similar sets;
9.3
Some variations;
10.2
Continued fraction examples.
Graphs of functions
11.1
Dimensions of graphs, the Weierstrass function and self-affine graphs.
Dynamical systems
13.1
Repellers and iterated function systems;
13.2
The logistic map.
Iteration of complex functions
14.1
Sketch of general theory of Julia sets;
14.2
The Mandelbrot set;
14.3
Julia sets of quadratic functions.
Introduction
In the past, mathematics has been concerned largely with sets and functions to which the methods of classical calculus can be applied. Sets or functions that are not sufficiently smooth or regular have tended to be ignored as ‘pathological’ and not worthy of study. Certainly, they were regarded as individual curiosities and only rarely were thought of as a class to which a general theory might be applicable.
In recent years, this attitude has changed. It has been realised that a great deal can be said, and is worth saying, about the mathematics of non-smooth objects. Moreover, irregular sets provide a much better representation of many natural phenomena than do the figures of classical geometry. Fractal geometry provides a general framework for the study of such irregular sets.
We begin by looking briefly at a number of simple examples of fractals, and note some of their features.
The middle third Cantor set is one of the best known and most easily constructed fractals; nevertheless, it displays many typical fractal characteristics. It is constructed from a unit interval by a sequence of deletion operations (see Figure 0.1). Let be the interval [0, 1]. (Recall that denotes the set of real numbers such that .) Let be the set obtained by deleting the middle third of , so that consists of the two intervals and . Deleting the middle thirds of these intervals gives ; thus, comprises the four intervals . We continue in this way, with obtained by deleting the middle third of each interval in . Thus, consists of intervals each of length . The middle third Cantor set F consists of the numbers that are in for all k; mathematically, is the intersection . The Cantor set may be thought of as the limit of the sequence of sets as k tends to infinity. It is obviously impossible to draw the set itself, with its infinitesimal detail, so ‘pictures of ’ tend to be pictures of one of the , which are a good approximation to when k is reasonably large (see Figure 0.1).
Figure 0.1 Construction of the middle third Cantor set , by repeated removal of the middle third of intervals. Note that and , the left and right parts of , are copies of scaled by a factor .
At first glance, it might appear that we have removed so much of the interval [0, 1] during the construction of , that nothing remains. In fact, is an infinite (and indeed uncountable) set, which contains infinitely many numbers in every neighbourhood of each of its points. The middle third Cantor set consists precisely of those numbers in [0, 1] whose base-3 expansion does not contain the digit 1, that is, all numbers with or 2 for each i. To see this, note that to get from , we remove those numbers with ; to get from , we remove those numbers with and so on.
Figure 0.2 (a) Construction of the von Koch curve . At each stage, the middle third of each interval is replaced by the other two sides of an equilateral triangle. (b) Three von Koch curves fitted together to form a snowflake curve.
We list some of the features of the middle third Cantor set ; as we shall see, similar features are found in many fractals.
is self-similar. It is clear that the part of
in the interval
and the part of
in
are both geometrically similar to
, scaled by a factor
. Again, the parts of
in each of the four intervals of
are similar to
but scaled by a factor
, and so on. The Cantor set contains copies of itself at many different scales.
The set
has a ‘fine structure’; that is, it contains detail at arbitrarily small scales. The more we enlarge the picture of the Cantor set, the more gaps become apparent to the eye.
Although
has an intricately detailed structure, the actual definition of
is very straightforward.
is obtained by a recursive procedure. Our construction consisted of repeatedly removing the middle thirds of intervals. Successive steps give increasingly good approximations
to the set
.
The geometry of
is not easily described in classical terms: neither is it the locus of the points that satisfy some simple geometric condition nor is it the set of solutions of any simple equation.
It is awkward to describe the local geometry of
—near each of its points are a large number of other points, separated by gaps of varying lengths.
Although
is in some ways quite a large set (it is uncountably infinite), its size is not quantified by the usual measures such as length—by any reasonable definition
has length zero.
Our second example, the von Koch curve, will also be familiar to many readers (see Figure 0.2). We let be a line segment of unit length. The set consists of the four segments obtained by removing the middle third of and replacing it by the other two sides of the equilateral triangle based on the removed segment. We construct by applying the same procedure to each of the segments in and so on. Thus, comes from replacing the middle third of each straight line segment of by the other two sides of an equilateral triangle. When k is large, the curves and differ only in fine detail and as k tends to be infinity, the sequence of polygonal curves approaches a limiting curve , called the von Koch curve.
The von Koch curve has features in many ways similar to those listed for the middle third Cantor set. It is made up of four ‘quarters’ each similar to the whole, but scaled by a factor . The fine structure is reflected in the irregularities at all scales; nevertheless, this intricate structure stems from a basically simple construction. Whilst it is reasonable to call F a curve, it is much too irregular to have tangents in the classical sense. A simple calculation shows that is of length ; letting k tend to infinity implies that F has infinite length. On the other hand, F occupies zero area in the plane, so neither length nor area provides a very useful description of the size of F.
Figure 0.3 Construction of the Sierpiski triangle .
Many other sets may be constructed using such recursive procedures. For example, the Sierpiski triangle or gasket is obtained by repeatedly removing (inverted) equilateral triangles from an initial equilateral triangle of unit side length (see Figure 0.3). (For many purposes, it is better to think of this procedure as repeatedly replacing an equilateral triangle by three triangles of half the height.) A plane analogue of the Cantor set, a ‘Cantor dust’, is illustrated in Figure 0.4. At each stage, each remaining square is divided into 16 smaller squares of which four are kept and the rest discarded. (Of course, other arrangements or numbers of squares could be used to get different sets.) It should be clear that such examples have properties similar to those mentioned in connection with the Cantor set and the von Koch curve. The example depicted in Figure 0.5 is constructed using two different similarity ratios.
Figure 0.4 Construction of a ‘Cantor dust’ .
Figure 0.5 Construction of a self-similar fractal with two different similarity ratios.
Figure 0.6 A Julia set.
There are many other types of construction, some of which will be discussed in detail later in the book, that also lead to sets with these sorts of properties. The highly intricate structure of the Julia set illustrated in Figure 0.6 stems from the single quadratic function for a suitable constant . Although the set is not strictly self-similar in the sense that the Cantor set and von Koch curve are, it is ‘quasi-self-similar’ in that arbitrarily small portions of the set can be magnified and then distorted smoothly to coincide with a large part of the set.
Figure 0.7 shows the graph of the function ; the infinite summation leads to the graph having a fine structure, rather than being a smooth curve to which classical calculus is applicable.
Some of these constructions may be ‘randomised’. Figure 0.8 shows a ‘random von Koch curve’—a coin was tossed at each step in the construction to determine on which side of the curve to place the new pair of line segments. This random curve certainly has a fine structure, but the strict self-similarity of the von Koch curve has been replaced by a ‘statistical self-similarity’.
Figure 0.7 Graph of .
These are all examples of sets that are commonly referred to as fractals. (The word ‘fractal’ was coined by Mandelbrot in his fundamental essay from the Latin fractus, meaning broken, to describe objects that were too irregular to fit into a traditional geometrical setting.) Properties such as those listed for the Cantor set are characteristic of fractals, and it is sets with such properties that we will have in mind throughout the book. Certainly, any fractal worthy of the name will have a fine structure, that is, detail at all scales. Many fractals have some degree of self-similarity—they are made up of parts that resemble the whole in some way. Sometimes, the resemblance may be weaker than strict geometrical similarity; for example, the similarity may be approximate or statistical.
Methods of classical geometry and calculus are unsuited to study fractals and we need alternative techniques. The main tool of fractal geometry is dimension in its many forms. We are familiar enough with the idea that a (smooth) curve is a 1-dimensional object and a surface is 2-dimensional. It is less clear that, for many purposes, the Cantor set should be regarded as having dimension and the von Koch curve as having dimension . This latter number is, at least, consistent with the von Koch curve being ‘larger than 1-dimensional’ (having infinite length) and ‘smaller than 2-dimensional’ (having zero area).
Figure 0.8 A random version of the von Koch curve.
Figure 0.9 Division of certain sets into four parts. The parts are similar to the whole with ratios: (a) for line segment; (b) for square; (c) for middle third Cantor set; (d) for von Koch curve.
The following argument gives one (rather crude) interpretation of the meaning of these ‘dimensions’ indicating how they reflect scaling properties and self-similarity. As Figure 0.9 indicates, a line segment is made up of four copies of itself, scaled by a factor . The segment has dimension . A square, however, is made up of four copies of itself scaled by a factor (i.e. with half the side length) and has dimension . In the same way, the von Koch curve is made up of four copies of itself scaled by a factor , and has dimension , and the Cantor set may be regarded as comprising four copies of itself scaled by a factor and having dimension . In general, a set made up of m copies of itself scaled by a factor r might be thought of as having dimension . The number obtained in this way is usually referred to as the similarity dimension of the set.
Unfortunately, similarity dimension is meaningful only for a relatively small class of strictly self-similar sets. Nevertheless, there are other definitions of dimension that are much more widely applicable. For example, Hausdorff dimension and the box-counting dimensions may be defined for any sets and, in these four examples, may be shown to equal the similarity dimension. The early chapters of the book are concerned with the definition and properties of Hausdorff and box dimensions, along with methods for their calculation. Very roughly, a dimension provides a description of how much space a set fills. It is a measure of the prominence of the irregularities of a set when viewed at very small scales. A dimension contains much information about the geometrical properties of a set.
A word of warning is appropriate at this point. It is possible to define the ‘dimension’ of a set in many ways, some satisfactory and others less so. It is important to realise that different definitions may give different values of dimension for the same set and may also have very different properties. Inconsistent usage has sometimes led to considerable confusion. In particular, warning lights flash in my mind (as in the minds of other mathematicians) whenever the term ‘fractal dimension’ is seen. Although some authors attach a precise meaning to this, I have known others interpret it inconsistently in a single piece of work. The reader should always be aware of the definition in use in any discussion.
In his original essay, Mandelbrot defined a fractal to be a set with Hausdorff dimension strictly greater than its topological dimension. (The topological dimension of a set is always an integer and is 0 if it is totally disconnected, 1 if each point has arbitrarily small neighbourhoods with boundary of dimension 0 and so on.) This definition proved to be unsatisfactory in that it excluded a number of sets that clearly ought to be regarded as fractals. Various other definitions have been proposed, but they all seem to have this same drawback.
My personal feeling is that the definition of a ‘fractal’ should be regarded in the same way as a biologist regards the definition of ‘life’. There is no hard-and-fast definition but just a list of properties characteristic of a living thing, such as the ability to reproduce or to move or to exist to some extent independently of the environment. Most living things have most of the characteristics on the list, although there are living objects that are exceptions to each of them. In the same way, it seems best to regard a fractal as a set that has properties such as those listed below, rather than to look for a precise definition which will almost certainly exclude some interesting cases. From the mathematician's point of view, this approach is no bad thing. It is difficult to avoid developing properties of dimension other than in a way that applies to ‘fractal’ and ‘non-fractal’ sets alike. For ‘non-fractals’, however, such properties are of little interest—they are generally almost obvious and could be obtained more easily by other methods.
Therefore, when we refer to a set as a fractal, we will typically have the following in mind.
has a fine structure, that is, detail on arbitrarily small scales.
is too irregular to be described in traditional geometrical language, both locally and globally.
Often
has some form of self-similarity, perhaps approximate or statistical.
Usually, the ‘fractal dimension’ of
(defined in some way) is greater than its topological dimension.
In most cases of interest,
is defined in a very simple way, perhaps recursively.
What can we say about the geometry of as diverse a class of objects as fractals? Classical geometry gives us a clue. In Part I of this book, we study certain analogues of familiar geometrical properties in the fractal situation. The orthogonal projection or ‘shadow’ of a circle in space onto a plane is, in general, an ellipse. The fractal projection theorems tell us about the ‘shadows’ of a fractal. For many purposes, a tangent provides a good local approximation to a circle. Although fractals tend not to have tangents in any sense, it is often possible to say a surprising amount about their local form. Two circles in the plane in ‘general position’ either intersect in two points or not at all (we regard the case of mutual tangents as ‘exceptional’). Using dimension, we can make similar statements about the intersection of fractals. Moving a circle perpendicular to its plane sweeps out a cylinder, with properties that are related to those of the original circle. Similar, and indeed more general, constructions are possible with fractals.
Although classical geometry is of considerable intrinsic interest, it is also called upon widely in other areas of mathematics. For example, circles or parabolae occur as the solution curves of certain differential equations, and a knowledge of the geometrical properties of such curves aids our understanding of the differential equations. In the same way, thegeneral theory of fractal geometry can be applied to the many branches of mathematics in which fractals occur. Various examples of this are given in Part II of the book.
Historically, interest in geometry has been stimulated by its applications to nature. The ellipse assumed importance as the shape of planetary orbits, as did the sphere as the shape of the earth. The geometry of the ellipse and sphere can be applied to these physical situations. Of course, orbits are not quite elliptical, and the earth is not actually spherical, but for many purposes, such as the prediction of planetary motion or the study of the earth's gravitational field, these approximations may be perfectly adequate.
A similar situation pertains to fractals. A glance at the recent physics literature shows the variety of natural objects that are described as fractals—cloud boundaries, topographical surfaces, coastlines, turbulence in fluids and so on. None of these are actual fractals—their fractal features disappear if they are viewed at sufficiently small scales. Nevertheless, over certain ranges of scale, they appear very much like fractals, and at such scales may usefully be regarded as such. The distinction between ‘natural fractals’ and the mathematical ‘fractal sets’ that might be used to describe them was emphasised in Mandelbrot's original essay, but this distinction seems to have become somewhat blurred. There are no true fractals in nature. (There are no true straight lines or circles either!)
If the mathematics of fractal geometry is to be really worthwhile, then it should be applicable to physical situations. Considerable progress is being made in this direction and some examples are given towards the end of this book. Although there are natural phenomena that have been explained in terms of fractal mathematics (Brownian motion is a good example), many applications tend to be descriptive rather than predictive. Much of the basic mathematics used in the study of fractals is not particularly new, although much recent mathematics has been specifically geared to address fractals. For further progress to be made, development and application of appropriate mathematics remain a high priority.
Notes and references
Unlike the rest of the book, which consists of fairly solid mathematics, this introduction contains some of the author's opinions and prejudices, which may well not be shared by other workers on fractals. Caveat emptor!
The foundational treatise on fractals, which may be appreciated at many levels, is the scientific, philosophical and pictorial essay of Mandelbrot (1982) (developed from an earlier version, Mandelbrot (1977)), containing a great diversity of natural and mathematical examples. This essay has been the inspiration for much of the work that has been done on fractals.
Many books and papers have been written on diverse aspects of fractals, and appropriate references are cited at the end of each chapter. Here, we mention a selection of books with a broad coverage. There are short overviews by Falconer (2013) and Lesmoir-Gordon, Rood and Edney (2009) aimed at the non-specialist. Introductory mathematical treatments include those by Addison (1997) and Schroeder (2009). The books by Barnsley (2006, 2012); Edgar (1998, 2008) and Peitgen, Jürgens, and Saupe (2004) provide other mathematical treatments at a level that is similar to this. Falconer (1985a); Mattila (1999); Federer (1996) and Morgan (2008) go into more detail on the geometric measure theory side and Rogers (1998) addresses the general theory of Hausdorff measures. Books with a computational emphasis include Devaney and Keen (2006); Hoggar (1993); Baumann (2005); Peitgen and Saupe (2011) and Pickover (2012). The sequel to this book, Falconer (1997), contains more advanced mathematical techniques for studying fractals.
Whilst this book and much of the literature concentrates on fractals in Euclidean spaces, much of the mathematics may be developed in more general settings; see, for example, the books by David and Semmes (1997); Rogers (1998) and Semmes (2000).
Part IFoundations
Chapter 1Mathematical background
This chapter reviews some of the basic mathematical ideas and notations that are used throughout the book. Section 1.1 on set theory and Section 1.2 on functions are rather concise; readers unfamiliar with this type of material are advised to consult a more detailed text on mathematical analysis. Measures and mass distributions play an important part in the theory of fractals and a treatment adequate for our needs is given in Section 1.3. By asking the reader to take on trust the existence of certain measures, we can avoid many of the technical difficulties usually associated with measure theory. Some notes on probability theory are given in Section 1.4; this is needed in Chapters 15 and 16.
1.1 Basic set theory
In this section, we recall some basic notions from set theory and point set topology.
We generally work in n-dimensional Euclidean space, , where is just the set of real numbers or the ‘real line’, and is the (Euclidean) plane. Points in will generally be denoted by lower case letters , , and so on, and we will occasionally use the coordinate form . Addition and scalar multiplication are defined in the usual manner, so that and , where is a real scalar. We use the usual Euclidean distance or metric on so if and are points of , the distance between them is . In particular, the triangle inequality , the reverse triangle inequality and the metric triangle inequality hold for all .
Sets, which will generally be subsets of , are denoted by capital letters , , , and so on. In the usual way, means that the point belongs to the set , and means that is a subset of the set . We write for the set of for which ‘condition’ is true. Certain frequently occurring sets have a special notation. The empty set, which contains no elements, is written as ∅. The integers are denoted by , and the rational numbers by . We use a superscript to denote the positive elements of a set; thus, are the positive real numbers, and are the positive integers. Sometimes we refer to the complex numbers , which for many purposes may be identified with the plane , with corresponding to the point .
The closed ball of centre and radius is defined by . Similarly, the open ball is . Thus, the closed ball contains its bounding sphere, but the open ball does not. Of course, in , a ball is a disc and in a ball is just an interval.If , we write for the closed interval and for the open interval. Similarly, denotes the half-open interval , and so on.
The coordinate cube of side 2 and centre is the set for all . (A cube in is just a square and in is an interval.)
From time to time we refer to the -neighbourhood or -parallel body, , of a set , that is, the set of points within distance of ; thus, (see Figure 1.1).
Figure 1.1 A set and its -neighbourhood .
We write for the union of the sets and , that is, the set of points belonging to either or , or both. Similarly, we write for their intersection, the points in both and . More generally, denotes the union of an arbitrary collection of sets , that is, those points in at least one of the sets , and denotes their intersection, consisting of the set of points common to all of the . A collection of sets is disjoint if the intersection of any pair is the empty set. The difference of and consists of the points in but not . The set is termed the complement of .
The set of all ordered pairs is called the (Cartesian) product of and and is denoted by . If and , then .
If and are subsets of and is a real number, we define the vector sum of the sets as and we define the scalar multiple.
An infinite set is countable if its elements can be listed in the form with every element of appearing at a specific place in the list; otherwise, the set is uncountable. The sets and are countable but is uncountable. Note that a countable union of countable sets is countable.
If is any non-empty set of real numbers, then its supremum is the least number such that for every in or is if no such number exists. Similarly, the infimum is the greatest number such that for all in or is . Intuitively, the supremum and infimum are thought of as the maximum and minimum of the set, although it is important to realise that and need not be members of the set itself. For example, , but . We write for the supremum of the quantity in brackets, which may depend on , as ranges over the set .
We define the diameter of a non-empty subset of as the greatest distance apart of pairs of points in . Thus, . In , a ball of radius has diameter 2, and a cube of side length has diameter . A set is bounded if it has finite diameter or, equivalently, if is contained in some (sufficiently large) ball.
Convergence of sequences is defined in the usual way. A sequence in converges to a point of as if, given , there exists a number such that whenever , that is, if converges to 0. The number is called the limit of the sequence, and we write or .
The ideas of ‘open’ and ‘closed’ that have been mentioned in connection with balls apply to much more general sets. Intuitively, a set is closed if itcontains its boundary and open if it contains none of its boundary points. More precisely, a subset of is open if, for all points in , there is some ball , centred at and is of positive radius that is contained in . A set is closed if whenever is a sequence of points of converging to a point of , then is in (see Figure 1.2). The empty set ∅ and are regarded as both open and closed.
Figure 1.2 (a) An open set—there is a ball contained in the set centred at each point of the set. (b) A closed set—the limit of any convergent sequence of points from the set lies in the set. (c) The boundary of the set in (a) or (b).
It may be shown that a set is open if and only if its complement is closed. The union of any collection of open sets is open, as is the intersection of any finite number of open sets. The intersection of any collection of closed sets is closed, as is the union of any finite number of closed sets (see Exercise 1.6).
A set is called a neighbourhood of a point if there is some (small) ball centred at and contained in .
The intersection of all the closed sets containing a set is called the closure of , written . The union of all the open sets contained in is the interior of . The closure of is thought of as the smallest closed set containing , and the interior as the largest open set contained in . The boundary of is given by , thus if and only if the ball intersects both and its complement for all .
A set is a dense in if , that is, if there are points of arbitrarily close to each point of .
A set is compact if any collection of open sets that covers (i.e. with union containing ) has a finite subcollection which also covers . Technically, compactness is an extremely useful property that enables infinite sets of conditions to be reduced to finitely many. However, as far as most of this book is concerned, it is enough to take the definition of a compact subset of as one that is both closed and bounded.
The intersection of any collection of compact sets is compact. It may be shown that if is a decreasing sequence of compact sets, then the intersection is non-empty (see Exercise 1.7). Moreover, if is contained in for some open set , then the finite intersection is contained in for some .
A subset of is connected if there do not exist open sets and such that contains
