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Fractal Geometry: Mathematical Foundations and Applications

Kenneth Falconer
Publication Date: 
Number of Pages: 
[Reviewed by
Tushar Das
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The reviewer should disclose at the outset that he studied fractal geometry as a (second-year) undergraduate at St. Andrews from the second edition of this text … and what a wonderful course that was! One hopes, perhaps idealistically, that every mathematics undergraduate be treated to such a verdant oasis before they graduate. In fact, the curious student could very profitably study Falconer’s text on her own, with possibly some help/guidance necessary at certain technical corners. There is, for such a student, a complete set of solutions to the exercises available online.

The first edition of the text being reviewed was written circa 1989, the second in 2003. In between, Falconer wrote a follow-up text for graduate students and researchers interested in tackling the current literature titled: Techniques in Fractal Geometry (TFG), published by Wiley in 1997. And before he got started on this duo, he had already written what many mathematicians still consider the connoisseur’s choice: The Geometry of Fractal Sets (GFS), published in 1985 as Volume 85 of the Cambridge Tracts in Mathematics. It seems as though Fractal Geometry: Mathematical Foundations and Applications (FGFA) was written to make accessible to beginners the material from his slim (about 180 pages) 1985 tract, as well as to attract researchers from fields beyond mathematics and its often intimidating rigor. Professor Falconer’s expository talents continue to blossom. Last year, in 2013, we were treated to his slimmest and most inclusive book so far: Fractals: A Very Short Introduction (FVSI), published by Oxford.


The book, consisting of eighteen chapters with an average of 20 pages per chapter, is structured in two parts: the first being “Foundations” (eight chapters), and the second “Applications and Examples” (ten chapters).

Foundations: There is a rapid introductory chapter on various elements of necessary mathematical background, and then we are off to build a tool-kit to study and dissect fractals: the box-counting dimension is first introduced, then the Hausdorff and packing measures and dimensions, followed by a chapter on basic techniques for calculating dimensions. The more technical Chapters 5 through 8 give the reader a taste of geometric measure theory: they study the local structure, projections, products and intersections of fractals respectively.

Applications and Examples: After tool-building, the reader is treated to an incredible range of mathematical vistas wherein one may find fractal sets of one sort or the other. Chapter 9 studies self-similar and self-affine Iterated Function Systems (IFSs) that generalize constructions like the Cantor middle-third set the student may have met in real analysis or undergraduate topology. There is a new section on Lapidus and van Frankenhuijsen’s theory of complex dimensions. Chapter 10 studies beautiful examples of fractal subsets that arise from within number theory. More complicated examples like the set of reals in [0,1] whose continued fraction entries are all either 1s or 2s are precluded from the discussion, but the interested reader is led to excellent references in the literature. Chapter 11 studies the dimensions of graphs of functions, where calculating the precise value of the Hausdorff dimension has remained an active area of research. Chapter 12 presents a smorgasbord from “pure” (this word rings false here in the reviewer’s ear!) mathematics, e.g. the Kakeya problem, Vitushkin’s conjecture, and fractal groups/rings.

Chapters 13 and 14, on dynamical systems and holomorphic dynamics, are among the reviewer’s favorites! He remembers these chapters making a strong impression when he was a student, and has continued to study such ever since. To be sure, entire books could and have been written on each of those chapters. Chapters 15 and 16 throw stochastic processes into the mix and study random fractals, Brownian motion and Brownian surfaces. Chapter 17 presents the analysis of multifractals — this is fairly new ground, and it is easy for the reader to get to the edge of contemporary research after reading this chapter and following up with the end-of-chapter references. It seems to the reviewer that a chapter/section on the thermodynamic formalism (see comments below) would also be in order here. The last chapter runs through a gamut of fractals that show up in various physical applications. There are the now-classical examples from the study of turbulence, but also modern applications, e.g. to fractal antennae!

Each chapter ends with both “Notes and references” and “Exercises”, both of which deserve comment. The exercises are well-crafted and have been classroom-tested in various settings. They also give students an invitation to aspects of modern mathematics that go well beyond the standard college fare. There has been considerable effort in making the notes and references up-to-date (i.e., up to 2013) and in many cases these lead the student/researcher to state-of-the-art research literature. No doubt that both these aspects add to the book’s continued success.

A Wishlist

For a $60 textbook with plenty of visually attractive content the paper quality is extremely poor. The book is filled with diagrams which makes for pleasant reading. However, almost all the diagrams show through the page due to the thinness of the paper. This is particularly unfortunate when there are diagrams on both sides of a single page. I’d recommend better printing and paper for the 4th edition! Color images of fractals like the Mandelbrot and Julia sets would also be very welcome.

The book does a wonderful job in taking the reader on a breathtaking, as should be clear from the summary, tour of mathematical vistas. However I was sorry to see both FGFA and FGVSI miss introducing a vital area of beautiful fractal mathematics that historically was the first place in which such intricate objects came into being unannounced. In 1883 Poincaré published, in the first volume of Mittag-Leffler’s freshly minted Acta Mathematica, his investigations regarding Fuchsian and then Kleinian groups (and functions). These are discrete subgroups of the isometry group of two- and three-dimensional hyperbolic space, respectively. It was here while perturbing the generators of a Fuchsian group that Poincaré stumbled upon the first Kleinian (now called quasi-Fuchsian in the literature) groups. Their limit sets were extremely intricate curves, in Poincaré’s words:

Ces domaines sont séparés par une ligne L, si l’on peut appeler ça une ligne. … De plus j’ai tout lieu de croire qu’il n’y a pas de tangente aux points de L qui ne font pas partie de P.

“Mémoire sur les groupes kleinéens”, Acta Math, 3 (1883), 49–92.

Poincaré’s work on Fuchsian and Kleinian groups/functions has been beautifully exposed in a number of places, e.g. see Chapter 3 of Jeremy Gray’s magisterial scientific biography. The reviewer is currently at work on an exposition of this history with a view to carefully describing Poincaré’s discovery, its reception and subsequent mathematical developments right up to Sullivan’s revival in the early 1980s with his famous proof of the no wandering domains theorem. In the process of his investigations, Sullivan uncovered an incredibly beautiful dictionary relating results/concepts from holomorphic dynamics and Kleinian groups which continues to stimulate research to this day.

Falconer does have an excellent chapter (14) on the former, but is missing what would be a useful “dual” chapter on Fuchsian and Kleinian groups. This would also provide a nice complement to the self-similar and self-affine IFSs from Chapter 9. In fact, it would be a good idea to add an extra section to Chapter 9 introducing non-linear generalizations, e.g. conformal IFSs à la Mauldin-Urbański, or even the simpler cookie-cutter sets as described in Chapter 4 (Cookie-cutters and Bounded Distortion) of Falconer’s TFG. It would also be nice to include the “heart” of Chapter 5 (Thermodynamic Formalism) from TFG: explaining Bowen’s generalization (to non-linear situations) of the Moran-Hutchinson formula for the Haudorff dimension of the attractor/limit set.

Some amusing typos

A. On p. 75, the last paragraph of Section 4.1 following the proof of Proposition 4.9 ends with what probably was an ignored suggestion that was subsequently missed by the copy-editors. However the mathematical content intended is clear. Here’s how the paragraph should read:

Note that it is immediate from Proposition 4.9 that if \[ \varliminf_{r \to 0} \log \frac{\mu (B(x,r))}{\log r} = s \] for all \(x \in F\), then \(\mathrm{dim}_H F = s\). We omit the details, but related arguments give the dual result that if \[ \varlimsup_{r \to 0} \log \frac{\mu (B(x,r))}{\log r} = s \] for all \(x \in F\), then \(\mathrm{dim}_P F = s\).

It may be of interest to the reader that the results from Section 4.1 hold in greater generality beyond \(\mathbb{R}^n\), e.g. in infinite-dimensional separable Hilbert spaces. For a clean and self-contained exposition see Section 8 (A digression to geometric measure theory) of D. Mauldin, T. Szarek and M. Urbański, “Graph Directed Markov Systems on Hilbert Spaces”, Math. Proc. Cambridge Phil. Soc. 147 (2009), 455-488.

B. Though the surname Urbański is correctly spelt earlier in the bibliography, the entry on p. 356 should have “Urbański M. (1990)” and not “Urbanski C. (1990)”. What perhaps makes at least the misspelling of this Polish surname somewhat excusable, is that the a similar slip occurs on more than a few occasions at the the author’s own web site.


Falconer’s book is excellent in many respects and the reviewer strongly recommends it. May every university library own a copy, or three! And if you’re a student reading this, go check it out today!

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin – La Crosse.

Preface to the first edition ix

Preface to the second edition xiii

Preface to the third edition xv

Course suggestions xvii

Introduction xix


1 Mathematical background 3
1.1 Basic set theory 3
1.2 Functions and limits 7
1.3 Measures and mass distributions 11
1.4 Notes on probability theory 17
1.5 Notes and references 24
Exercises 24

2 Box-counting dimension 27
2.1 Box-counting dimensions 27
2.2 Properties and problems of box-counting dimension 34
*2.3 Modified box-counting dimensions 38
2.4 Some other definitions of dimension 40
2.5 Notes and references 41
Exercises 42

3 Hausdorff and packing measures and dimensions 44
3.1 Hausdorff measure 44
3.2 Hausdorff dimension 47
3.3 Calculation of Hausdorff dimension – simple examples 51
3.4 Equivalent definitions of Hausdorff dimension 53
*3.5 Packing measure and dimensions 54
*3.6 Finer definitions of dimension 57
*3.7 Dimension prints 58
*3.8 Porosity 60
3.9 Notes and references 63
Exercises 64

4 Techniques for calculating dimensions 66
4.1 Basic methods 66
4.2 Subsets of finite measure 75
4.3 Potential theoretic methods 77
*4.4 Fourier transform methods 80
4.5 Notes and references 81
Exercises 81

5 Local structure of fractals 83
5.1 Densities 84
5.2 Structure of 1-sets 87
5.3 Tangents to s-sets 92
5.4 Notes and references 96
Exercises 96

6 Projections of fractals 98
6.1 Projections of arbitrary sets 98
6.2 Projections of s-sets of integral dimension 101
6.3 Projections of arbitrary sets of integral dimension 103
6.4 Notes and references 105
Exercises 106

7 Products of fractals 108
7.1 Product formulae 108
7.2 Notes and references 116
Exercises 116

8 Intersections of fractals 118
8.1 Intersection formulae for fractals 119
*8.2 Sets with large intersection 122
8.3 Notes and references 128
Exercises 128


9 Iterated function systems – self-similar and self-affine sets 133
9.1 Iterated function systems 133
9.2 Dimensions of self-similar sets 139
9.3 Some variations 143
9.4 Self-affine sets 149
9.5 Applications to encoding images 155
*9.6 Zeta functions and complex dimensions 158
9.7 Notes and references 167
Exercises 167

10 Examples from number theory 169
10.1 Distribution of digits of numbers 169
10.2 Continued fractions 171
10.3 Diophantine approximation 172
10.4 Notes and references 176
Exercises 176

11 Graphs of functions 178
11.1 Dimensions of graphs 178
*11.2 Autocorrelation of fractal functions 188
11.3 Notes and references 192
Exercises 192

12 Examples from pure mathematics 195
12.1 Duality and the Kakeya problem 195
12.2 Vitushkin’s conjecture 198
12.3 Convex functions 200
12.4 Fractal groups and rings 201
12.5 Notes and references 204
Exercises 204

13 Dynamical systems 206
13.1 Repellers and iterated function systems 208
13.2 The logistic map 209
13.3 Stretching and folding transformations 213
13.4 The solenoid 217
13.5 Continuous dynamical systems 220
*13.6 Small divisor theory 225
*13.7 Lyapunov exponents and entropies 228
13.8 Notes and references 231
Exercises 232

14 Iteration of complex functions – Julia sets and the Mandelbrot set 235
14.1 General theory of Julia sets 235
14.2 Quadratic functions – the Mandelbrot set 243
14.3 Julia sets of quadratic functions 248
14.4 Characterisation of quasi-circles by dimension 256
14.5 Newton’s method for solving polynomial equations 258
14.6 Notes and references 262
Exercises 262

15 Random fractals 265
15.1 A random Cantor set 266
15.2 Fractal percolation 272
15.3 Notes and references 277
Exercises 277

16 Brownian motion and Brownian surfaces 279
16.1 Brownian motion in \(\mathbb{R}\) 279
16.2 Brownian motion in \(\mathbb{R}^n\) 285
16.3 Fractional Brownian motion 289
16.4 Fractional Brownian surfaces 294
16.5 Lévy stable processes 296
16.6 Notes and references 299
Exercises 299

17 Multifractal measures 301
17.1 Coarse multifractal analysis 302
17.2 Fine multifractal analysis 307
17.3 Self-similar multifractals 310
17.4 Notes and references 320
Exercises 320

18 Physical applications 323
18.1 Fractal fingering 325
18.2 Singularities of electrostatic and gravitational potentials 330
18.3 Fluid dynamics and turbulence 332
18.4 Fractal antennas 334
18.5 Fractals in finance 336
18.6 Notes and references 340
Exercises 341

References 342

Index 357