You are here

Fractals in Probability and Analysis

Christopher J. Bishop and Yuval Peres
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics 162
[Reviewed by
Tushar Das
, on

Bishop and Peres, experts working in a variety of areas lying at the interface of geometric measure theory and probability theory, have written a text that provides students and beginning researchers in analysis (broadly construed) with an excellent introduction to both the rudiments of fractal geometry and to a number of research directions that are under active investigation today. The book could be profitably read by graduate students and interested non-specialists who have acquired some familiarity with measure theory and probability at the level of a first graduate course, e.g. the material contained in Folland’s excellent Real Analysis (2nd ed., Wiley, 1999).

There currently exist a number of successful texts written by well-established experts on fractal geometry at both the advanced undergraduate level (e.g., Measure, Topology, and Fractal Geometry by G.A. Edgar (Springer 1990) and Fractal Geometry: Mathematical Foundations and Applications, by K. J. Falconer (4th ed., Wiley 2014)), as well as at the graduate level and beyond (e.g., The Geometry of Fractal Sets by K. J. Falconer (Cambridge 1985), Geometry of sets and measures in Euclidean spaces by P. Mattila (Cambridge 1995), and Fourier Analysis and Hausdorff Dimension by P. Mattila (Cambridge 2016)).

There are at least two outstanding features of Bishop-Peres’s new textbook that will help it stand with self-assurance alongside such august company. The first feature is the remarkable clarity of exposition. The proofs are beautifully presented, with a stress on communicating ideas and methods (over technicalities). This leads the authors to study the simplest cases of problems/results that already contain the most important ideas. The second feature, which moves this text into its own class among existing graduate texts on the subject, is an exceptional list of 378 exercises. Some of the more challenging problems among these have invitations to the literature, and there are also hints to some of the problems collected in a 19 page appendix. The text results from a number of courses taught by both authors, and their cumulative expertise gleaned from such experiences shines throughout the production.

At times the writing switches to a conversational tone, and the reader is led away from the mathematics per se, to some related recollection that lends intimacy and amusement. To give a flavor of this timbre, which is perhaps not entirely unfamiliar to those who have attended memorable graduate classes or seminars, we quote from the text:

The second author heard the following story from Shizuo Kakutani about the paper Dvoretzky et al. (1961). Erdős was staying at Dvoretszky’s apartment in New York City and Kakutani drove down from Yale and the three of them constructed a proof that Brownian motion has points of increase. Satisfied with the proof, Kakutani left to go home, but, despite repeated attempts, could not get his car to start. It was too late to find a mechanic, so he spent the night at Dvoretszky’s apartment with the others, filling in the remaining details of the existence proof until it turned into a non-existence proof around 2am. In the morning the car started perfectly on the first attempt (but if it hadn’t, we might not have Exercise 5.6).

The first couple of chapters study Minkowski, Hausdorff and packing dimensions and the now classic setting of self-similar iterated function systems (IFSes), though the authors do not use this terminology. The next chapter is on Frostman’s Lemma (proved here using Ford-Fulkerson’s classic MaxFlow-MinCut Theorem) and applications to the behavior of dimension under products, projections and slices.

The fourth chapter studies a class of self-affine fractals that are invariant under affine maps of the form \( (x,y) \mapsto (nx,my) \) for integers \(n\) and \(m\). The student will notice the sharp increase in difficulty when computing the Minkowski, Hausdorff and packing dimensions of such fractals, in contrast to their self-similar counterparts. One wishes there were a more thorough and uniform exposition of self-affine sets, especially covering the extensions considered by Lalley and Gatzouras (1992) and Barański (2007). The authors might also have highlighted the fact that much of the theory for self-affine fractals is still nascent. Though there have been recent breakthroughs (e.g., Bárány-Käenmäki’s “Ledrappier-Young formula and exact dimensionality of self-affine measures” Adv. Math., 318 (2017), 88–129), there are plenty of lacunae in dimensions greater than 2.

In Chapter 5 on “Graphs of continuous functions” the authors introduce the intensively studied class of nowhere differentiable Weierstrass functions as \[ f_{\alpha,b}(x) := \sum_{n=1}^{\infty} b^{-n\alpha} \cos(b^n x) \] with \(b\) an integer larger than \(1\) and \(0 < \alpha \leq 1\). They mention that it “had been long conjectured that the Hausdorff dimension of the Weierstrass graph was \(2-\alpha\)”, and that “Shen (2015) showed this holds for all integers \(b \geq 2\) and all \(0 < \alpha < 1\)”. Reading this may give the reader the impression that this long standing open problem had been finally resolved (perhaps modulo a boundary case).

The conjecture the authors appear to refer to was made by Mandelbrot in his Fractals: form, chance, and dimension (Freeman 1977). After Hardy’s “Weierstrass’s Non-Differentiable Function” (Trans. AMS 17(3), (1916), 301–325), Besicovitch and Ursell (J. Lond. Math. Soc. (1937) 1(1), 18–25) began the study of the fractal properties of the class of Weierstrass functions, and this line of research was taken up by various authors since that time. Therefore, it would not be too surprising if such a statement was conjectured by experts in geometric measure theory prior to Mandelbrot’s statement of the conjecture in 1977.

The crucial point that the authors fail to mention is that conjecture was made for a much larger class of functions, viz. where the parameter \(b > 1\) is allowed to be real (Hardy proved that all such functions are nowhere differentiable in his paper mentioned earlier). Shen’s powerful theorem improves the work of several mathematicians, and though it does go a long way towards the Mandelbrot conjecture, it also falls short of it. Shen’s paper distinguishes between the original conjecture and his contribution. The crucial assumption that \(b\) is an integer allows one to transfer the problem to a dynamical setting by interpreting the graph as a repeller for a certain two-dimensional expanding dynamical system. However, it should be emphasized that, even post-Shen, it is not at all clear (at least to this reviewer) how one would proceed towards resolving the general case, i.e. where \(b>1\) is real.

The next couple of chapters study Brownian motion, starting with Lévy’s construction. Highlights include a proof of the law of the iterated logarithm for Brownian motion, as well as Donsker’s Invariance Principle, which implies Brownian motion is the limit of a wide variety of i.i.d. random walks. The suite ends with classical connections between Brownian motion and potential theory (e.g. the Dirichlet problem and recurrence/transience properties of Brownian motion in \(\mathbb{R}^d\)). This is followed by a brief chapter on discrete analogues of random processes (discrete Markov processes, simple random walk on \(\mathbb{Z}^d\) and percolation on trees).

The penultimate chapter is on “Besicovitch-Kakeya sets”, subsets of \(\mathbb{R}^d\) with zero \(d\)-dimensional measure that contain a line segment in every direction. A number of conceptually distinct constructions of Besicovitch sets are presented, and Körner’s beautiful proof that topologically generic Besicovitch sets have measure zero (which is sadly not accessible via the arxiv or even the author’s website) is outlined in exercise 9.19. The infamous Kakeya conjecture states that the dimension of a Besicovitch-Kakeya set in \(\mathbb{R}^d\) is \(d\) for \(d > 2\) (the planar case was proved by Davies in 1971). The reader interested in the suite of conjectures surrounding Kakeya conjecture in harmonic analysis may also refer to “Part IV: Fourier restriction and Kakeya type problems” in Mattila’s Fourier Analysis and Hausdorff Dimension (Cambridge 2016). The final chapter is devoted to Peter Jones’s \(\beta\)-numbers and a novel proof (due to Kate Okikiolu) of his eponymous Traveling Salesman Theorem.

Minor quibbles may all too easily surface for a text that aims to introduce such a wide scope of material, e.g., with regard to missing attributions and further pointers to the literature. We present some sample marginalia.

Though there is a good introduction to self-similar iterated function systems (IFSes), the authors missed the opportunity to point readers to both the more general class of conformal IFSes, and the more general setting of countable alphabet IFSes. The proof of Schief’s theorem (on the equivalence of the open-set condition and positivity of the Hausdorff measure for finite alphabet self-similar IFSes) may have been followed up with references in the end-of-chapter notes to “Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets” (Proc. Amer. Math. Soc. 129 (2001), 2689–2699) by Y. Peres, M. Rams, K. Simon, and B. Solomyak; as well as to “The OSC does not imply the SOSC for infinite iterated function systems” (Proc. Amer. Math. Soc. 133 (2005), 437–440) by T. Szarek and S. Wedrychowicz.

Infinite alphabet conformal IFSes, now studied for over a quarter century in earnest, show up naturally in the study of continued fractions, of infinitely generated Kleinian groups, and of meromorphic dynamical systems. For instance, “Dimensions and measures in infinite iterated function systems” (Proc. London Math. Soc. (3) 73 (1996), 105–154) and the subsequent Graph directed Markov systems (Vol. 148 of the Cambridge Tracts in Mathematics, 2003), both by Daniel Mauldin and Mariusz Urbański, are key references for the study of such dynamical systems. Without such pointers, a neophyte reader would be quite unaware of the extremely rich seam of mathematics surrounding the thermodynamic formalism (a conglomeration of techniques due originally to Rufus Bowen, David Ruelle and Yakov Sinai) and its various applications. For the reader interested in learning the basics, see e.g., Rufus Bowen’s Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, (Springer 2008). Applications abound, for recent examples see “An introduction to pressure metrics on higher Teichmüller spaces" by Bridgeman, Canary and Sambarino (ETDS, forthcoming) and “On Zaremba’s Conjecture” by Bourgain and Kontorovich (Ann. of Math. 2014).

There is sadly scant discussion of Bowen’s beautiful dichotomy, viz. that “a connected limit set of a co-compact Kleinian group is either a circle (or line) or has dimension strictly greater than unity”, that makes precise a certain property first studied by Poincaré (in the setting of quasi-Fuchsian groups) in the late 19th century. I would recommend Dennis Sullivan’s exciting survey “Discrete conformal groups and measurable dynamics” (Bull. AMS 6 (1982), 57-73), and perhaps the first couple of sections of “Dimension rigidity in conformal structures” (Adv. Math., Vol. 308 (2017), 1127–1186) by David Simmons, Mariusz Urbański and the reviewer. It would have been gratifying to have seen a self-contained exposition of some of Bishop’s own beautiful results regarding Kleinian groups (e.g., Acta Math. (1997) 179(1), 1–39, or Ann. of Math. (2001) 154(1), 205–217), as well as his recent construction of a transcendental Julia set of dimension \(1\) — but such a wish list could easily grow out of bounds.

Chapter 4 on “Self-affine sets” includes results of Peres and his coauthors from the 1990s that perhaps find themselves in textbook form for the first time. Some of these results, though extremely difficult to improve upon, continue to intrigue researchers and still await their final formulation. The end-of-chapter notes point to useful surveys of self-affine sets by Peres and Solomyak (2000) and Falconer (2013), which do include questions left open in the wake of such results and much more beyond. However, it would be extremely useful (especially for students and younger researchers) to have updates to the problems contained in such surveys. The inclusion of a (even partial, modest) list of unresolved problems related to the material in each chapter would be a Herculean task, but one that would be graciously received by the community.

Two further wish-list entries: It would have been nice to have seen an exposition of the Ledrappier-Young theory in the basic setting of self-affine IFSes, which does not currently exist in textbook form. It would also have been useful to have a chapter on Bernoulli convolutions, especially one that included an exposition of the recent work of Hochman (Ann. of Math. (2014), 180(2), 773–822) and Shmerkin (Geom. Funct. Anal. (2014), 24(3), 946–958). In this regard, the authors refer to the following survey paper, but forget to include the same in the bibliography: “Sixty years of Bernoulli convolutions” by Y. Peres, W. Schlag and B. Solomyak (in Fractals and Stochastics II, Proceedings of the Greifswald 1998 Conference, (ed. Bandt, Graf, and Zähle), Springer 1999).

As an example of a careless slip, the authors forget to give a precise definition of a homogeneous set in Section 2.3 titled “Homogeneous sets”, though the section does start with an intuitive description (“In this section we will consider sets that, in some sense, look the same at all scales. In particular, we will present criteria that ensure that the Hausdorff and Minkowski dimensions agree.”), and the index entry for the term returns the reader to the guilty section. The student reading Exercise 2.18 — Is \(K = \{0\} \cup \{1, \frac12, \frac13, \dots \} \) homogeneous? — may be induced to guess what was meant by the authors from the context of what was proved in Section 2.3. The missing definition is perhaps due to Fursternberg originally, and may be found for instance in his Porter lectures (Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton 1981): Given a continuous map \( T \) from a compact space \( X \) to itself, a closed subset \( A \) of \( X \) is called homogeneous with respect to \( T \) if there exists a group \( G \) of homeomorphisms of \( X \) each of which commutes with \( T \) and leaves \( A \) invariant, and such that the dynamical system \( (A,G) \) is minimal (i.e. no proper closed subset of \( A \) is invariant under the action of \( G \)).

One could continue with further quibbling and suggestions, but such should be taken as marginal recommendations for improving what is already an excellent textbook. One that will certainly inspire students to dive deeper into both fertile and yet burgeoning mathematical areas.

With a view to future readers of their text the authors might maintain a website with an errata, further updates and embellishments to their end-of-chapter notes, and hopefully even more problems and hints to their solution! Such addenda/updates could easily be added to newer editions of the text, or possibly kept somewhere relatively stable and easily accessible on the internet.

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

1. Minkowski and Hausdorff dimensions
2. Self-similarity and packing dimension
3. Frostman's theory and capacity
4. Self-affine sets
5. Graphs of continuous functions
6. Brownian motion, part I
7. Brownian motion, part II
8. Random walks, Markov chains and capacity
9. Besicovitch–Kakeya sets
10. The traveling salesman theorem
Appendix A. Banach's fixed-point theorem
Appendix B. Frostman's lemma for analytic sets
Appendix C. Hints and solutions to selected exercises