Measure, topology, and fractal geometry

292 Pages·2008·5.654 MB·English

by  Gerald Edgar

Undergraduate Texts in Mathematics
Editors
S. Axler
K.A. Ribet
Undergraduate Texts in Mathematics
Abbott: Understanding Analysis. Daepp/Gorkin: Reading, Writing,
Anglin: Mathematics: A Concise History and and Proving: A Closer Look at
Philosophy. Mathematics.
Readings in Mathematics. Devlin: The Joy of Sets: Fundamentals
Anglin/Lambek: The Heritage of Thales. of-Contemporary Set Theory. Second edition.
Readings in Mathematics. Dixmier: General Topology.
Apostol: Introduction to Analytic Number Driver: Why Math?
Theory. Second edition. Ebbinghaus/Flum/Thomas: Mathematical
Armstrong: Basic Topology. Logic. Second edition.
Armstrong: Groups and Symmetry. Edgar: Measure, Topology, and Fractal
Axler: Linear Algebra Done Right. Second Geometry. Second edition.
edition. Elaydi: An Introduction to Difference
Beardon: Limits: A New Approach to Real Equations. Third edition.
Analysis. Erdõs/Surányi: Topics in the Theory of
Bak/Newman: Complex Analysis. Second Numbers.
edition. Estep: Practical Analysis on One Variable.
Banchoff/Wermer: Linear Algebra Through Exner: An Accompaniment to Higher
Geometry. Second edition. Mathematics.
Beck/Robins: Computing the Continuous Exner: Inside Calculus.
Discretely Fine/Rosenberger: The Fundamental Theory
Berberian: A First Course in Real Analysis. of Algebra.
Bix: Conics and Cubics: A Concrete Fischer: Intermediate Real Analysis.
Introduction to Algebraic Curves. Second Flanigan/Kazdan: Calculus Two: Linear and
edition. Nonlinear Functions. Second edition.
Brèmaud: An Introduction to Probabilistic Fleming: Functions of Several Variables.
Modeling. Second edition.
Bressoud: Factorization and Primality Foulds: Combinatorial Optimization for
Testing. Undergraduates.
Bressoud: Second Year Calculus. Foulds: Optimization Techniques: An
Readings in Mathematics. Introduction.
Brickman: Mathematical Introduction to Franklin: Methods of Mathematical
Linear Programming and Game Theory. Economics.
Browder: Mathematical Analysis: An Frazier: An Introduction to Wavelets Through
Introduction. Linear Algebra.
Buchmann: Introduction to Cryptography. Gamelin: Complex Analysis.
Second Edition. Ghorpade/Limaye: A Course in Calculus and
Buskes/van Rooij: Topological Spaces: From Real Analysis
Distance to Neighborhood. Gordon: Discrete Probability.
Callahan: The Geometry of Spacetime: An Hairer/Wanner: Analysis by Its History.
Introduction to Special and General Relavitity. Readings in Mathematics.
Carter/van Brunt: The Lebesgue– Stieltjes Halmos: Finite-Dimensional Vector Spaces.
Integral: A Practical Introduction. Second edition.
Cederberg: A Course in Modern Geometries. Halmos: Naive Set Theory.
Second edition. Hämmerlin/Hoffmann: Numerical
Chambert-Loir: A Field Guide to Algebra Mathematics.
Childs: A Concrete Introduction to Higher Readings in Mathematics.
Algebra. Second edition. Harris/Hirst/Mossinghoff: Combinatorics and
Chung/AitSahlia: Elementary Probability Graph Theory.
Theory: With Stochastic Processes and an Hartshorne: Geometry: Euclid and Beyond.
Introduction to Mathematical Finance. Fourth Hijab: Introduction to Calculus and Classical
edition. Analysis. Second edition.
Cox/Little/O’Shea: Ideals, Varieties, and Hilton/Holton/Pedersen: Mathematical
Algorithms. Second edition. Reflections: In a Room with
Croom: Basic Concepts of Algebraic Topology. Many Mirrors.
Cull/Flahive/Robson: Difference Equations. Hilton/Holton/Pedersen: Mathematical Vistas:
From Rabbits to Chaos From a Room with Many Windows.
Curtis: Linear Algebra: An Introductory Iooss/Joseph: Elementary Stability and
Approach. Fourth edition. Bifurcation Theory. Second Edition.
(continued after index)
Gerald Edgar
Measure, Topology,
and Fractal Geometry
Second Edition
Gerald Edgar
Department of Mathematics
The Ohio State University
Columbus, OH 43210-1174, USA
[email protected]
Editorial Board
S. Axler K.A. Ribet
Mathematics Department Department of
San Francisco State Mathematics
University University of California
San Francisco, CA 94132 at Berkeley
USA Berkeley, CA 94720-3840
[email protected] USA
[email protected]
ISBN: 978-0-387-74748-4 e-ISBN: 978-0-387-74749-1
Library of Congress Control Number: 2007934922
Mathematics Subject Classification (2000): 28A80, 28A78, 28-01, 54F45
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Dedication
Arnold Ross
(1906–2002)
[37]
Preface
What is a fractal? Benoit Mandelbrot coined the term in 1975. There is (or
should be) a mathematical definition, specifying a basic idea in geometry.
Thereisalsoafigurative useofthetermtodescribephenomenathatapprox-
imate this mathematical ideal. Roughly speaking, a fractal set is a set that
is more “irregular” than the sets considered in classical geometry. No matter
how muchthesetismagnified, smaller andsmaller irregularities becomevisi-
ble.Mandelbrotarguesthatsuchgeometricabstractionsoftenfitthephysical
world better than regular arrangements or smooth curves and surfaces. On
page 1 of his book, The Fractal Geometry of Nature, he writes, “Clouds are
not spheres, mountains are not cones, coastlines are not circles, and bark is
not smooth, nor does lightning travel in a straight line”. [44, p. 1]∗
To define fractal, Mandelbrot writes: “A fractal is by definition a set for
which the Hausdorff–Besicovitch dimension strictly exceeds the topological
dimension”. [44,p.15]Itmightbesaidthatthisbookisameditationonthat
verse. Study of the Hausdorff dimension requires measure theory (Chap. 5);
studyoftopologicaldimensionrequiresmetrictopology(Chap.2).Note,how-
ever,thatMandelbrotlaterexpressedsomereservationsaboutthisdefinition:
“In science its [the definition’s] generality was to prove excessive; not only
awkward, but genuinely inappropriate ... This definition left out many ‘bor-
derlinefractals’,yetittookcare,moreorless,ofthefrontier‘against’ Euclid.
Butthefrontier‘against’truegeometricchaoswasleftwideopen.”[44,p.159]
We will discuss in Chap. 6 a way (proposed by James Taylor) to repair the
definition.Mandelbrothimself,inthesecondprintingof[44,p.459],proposes
“... to use ‘fractal’ without a pedantic definition, to use ‘fractal dimension’
as a generic term applicable to all the variants in Chapter 39, and to use in
each specific case whichever definition is the most appropriate”. I have not
adopted the first of these suggestions in this book, since a term without a
“pedantic definition” cannot be discussed mathematically. I have, however,
used the term “fractal dimension” as suggested.
∗ Bracketed numbers like this refer to the references collected on p. 257.
VIII Preface
This is a mathematics book. It is not about how fractals come up in
nature; that is the topic of Mandelbrot’s book [44]. It is not about how to
draw fractals on your computer. (I did have a lot of fun using a Macintosh
to draw the pictures for the book, however. There will be occasional use of
the Logo programming language for illustrative purposes.) Complete proofs
of the main results will be presented, whenever that can reasonably be done.
For some of the more difficult results, only the easiest non-trivial case of the
proof (such as the case of two dimensions) is included here, with a reference
to the complete proof in a more advanced text.
The main examples that will be considered are subsets of Euclidean space
(in fact, usually two-dimensional Euclidean space); but as we will see, it is
helpful to deal with the more abstract setting of “metric spaces”.
This book deals with fractal geometry. It does not cover, for example,
chaotic dynamical systems. That is a separate topic, although it is related.
Anotherbookofthesamesizecouldbewritten∗onit.Thisbookdoesnotdeal
with the Mandelbrot set. Some writing on the subject has left the impression
that “fractal” is synonymous with “Mandelbrot set”; that is far from the
truth. This book does not deal with stochastic (or random) fractals; their
rigorous study would require more background in probability theory than I
have assumed for this book.
Prerequisites
(1) Experience in reading (and, preferably, writing) mathematical proofs is
essential, since proofs are included here. I will say “necessary and suffi-
cient” or “contrapositive” or “proof by induction” without explanation.
Readers without such experience will only be able to read the book at
a more superficial level by skipping many of the proofs. (Of course, no
mathematics student will want to do that!)
(2) Basicabstractsettheorywillbeneeded.Forexample,theabstractnotion
of a function; finite vs. infinite sets; countable vs. uncountable sets.
(3) The main prerequisite is calculus. For example: What is a continuous
function, and why do we care? The sum of an infinite series. The limit
of a sequence. The least upper bound axiom (or property) for the real
numbersystem.Proofsoftheimportantfactsareincludedinmanyofthe
modern calculus texts.
Advice for the reader
Here is some advice for those trying to read the book without guidance from
an experienced instructor. The most difficult (and tedious) parts of the book
are probably Chaps. 2 and 5. But these chapters lead directly to the most
important parts of the book, Chaps. 3 and 6. Most of Chap. 2 is independent
∗ For example, [16].
Preface IX
of Chap. 1, so to ease the reading of Chap. 2, it might be reasonable to
interspersepartsofChap.2withpartsofChap.1.Inasimilarway,Chaps.3,
4,and5aremostlyindependentofeachother,sopartsofthesethreechapters
could be interspersed with each other.
There are many exercises scattered throughout the text. Some of them
deal with examples and supplementary material, but many of them deal with
the main subject matter at hand. Even though the reader already knows it, I
mustrepeat:Understandingwillbegreatlyenhancedbyworkontheexercises
(evenwhenasolutionisnotfound).Answerstosomeoftheexercisesaregiven
elsewhere in the book, but in order to encourage the reader to devote more
work to the exercises,I have not attempted to make them easyto find. When
an exercise is simply a declarative statement, it is to be understood that it is
to be proved. (Professor Ross points out that if it turns out to be false, then
you should try to salvage it.) Some exercises are easy and some are hard. I
have even included some that I do not know how to solve. (No, I won’t tell
you which ones they are.)
TakealookattheAppendix.Itisintendedtohelpthereaderofthebook.
Thereisanindexofthemaintermsdefinedinthebook;anindexofnotation;
and a list of the fractal examples discussed in the text.
Some illustrations are not referred to in the main text. An instructor who
knows what they are may use them for class assignments at the appropriate
times.
Someofthesectionsthataremoredifficult,ordealwithlesscentralideas,
aremarkedwithanasterisk(*).Theyshouldbeconsideredoptional.Asection
of “Remarks” is at the end of each chapter. It contains many miscellaneous
items, such as: references for the material in the chapter; more sophisticated
proofs that were omitted from the main text; suggestions for course instruc-
tors.
Notation
Mostnotationusedhereiseitherexplainedinthetext,orelsetakenfromcal-
culusandelementarysettheory.Afewremindersandadditionalexplanations
are collected here.
Integers: Z={··· ,−2,−1,0,1,2,···}.
Natural numbers or positive integers: N={1,2,3,···}.
Real numbers: R=(−∞,∞).
Intervals of real numbers:
(a,b) ={x: a<x<b},
(a,b] ={x: a<x≤b}, etc.
The notation (a,b) also represents an ordered pair, so the context must be
used to distinguish.
Set difference or relative complement: X \A={x∈X : x(cid:5)∈A}.
X Preface
If f: X → Y is a function, and x ∈ X, I will use parentheses f(x) for
the value of the function at the point x; if C ⊆ X is a set, I will use square
brackets for the image set f[C]={f(x): x∈C}.
The union of a family (Ai)i∈I of sets, written
(cid:2)
A ,
i
i∈I
consistsofallpointsthatbelongtoatleastoneofthesetsA .Theintersection
i
(cid:3)
A
i
i∈I
consists of the points that belong to all of the sets Ai. The family (Ai)i∈I is
said to be disjoint iff A ∩A =∅ for any i(cid:5)=j in the index set I.
i j
The supremum (or least upper bound) of a set A ⊆ R is written supA.
By definition u = supA satisfies (1) u ≥ a for all a ∈ A, and (2) if y ≥ a for
all a∈A, then y ≥u. Thus, if A is not bounded above, we write supA=∞,
andifA=∅,wewritesupA=−∞.Theinfimum (orgreatestlowerbound)
is infA. The upper limit of a sequence (x )∞ is
n n=1
limsupx = lim supx .
n k
n→∞ n→∞ k≥n
And, if α(r) is defined for real r >0,
limsupα(r)= lim sup α(r).
r→0 s→00<r<s
Similar notation is used for the lower limit or liminf.
The sign (cid:10)(cid:11) signals the end of a proof.
The origin of the book
I offered a course in 1987 at The Ohio State University on fractal geometry.
It was intended for graduate students in mathematics, and it was based on
the books of Hurewicz and Wallman [35] and Falconer [23]. I tried to keep
the prerequisites at a low enough level that, for example, a graduate stu-
dent in physics could take the course. The prerequisites listed were: metric
topology and Lebesgue measure.∗ When the course was announced, I began
getting inquiries from many other types of students, who were interested in
studyingfractalgeometrymorerigorously,butdidnothaveeventhisminimal
background. For example, a student in computer science with a strong back-
ground in calculus would still have required two more years of mathematics
∗ Then I found, to my surprise, that Lebesgue integration is not considered nec-
essary for physics students. I suppose the fact that I find this incredible is an
illustration of my ignorance of how mathematics is applied in practice.
Preface XI
study (“Advanced Calculus” and “Introductory Real Analysis”) before being
preparedforthecourse.Thisbookisintendedtofitthissortofstudent.Only
a small part of those two courses is actually required for the study of fractal
geometry,atleastatthemostelementarylevel.Therequiredtopicsfrommet-
rictopologyandmeasuretheoryarecoveredinChaps.2and5.(Mathematics
students may be able to skip much of these two chapters.)
Thisbookisdirectlyderivedfromnotespreparedforuseinacourseoffered
in1988inconnectionwiththeprogramfortalentedhighschoolstudentsthat
is run here at The Ohio State University by Professor Arnold Ross for eight
weeks every summer. The influence of these young students can be seen in
many small ways in the book. (In particular, 1.5.7 and 1.6.1.) Past practice
in the Ross program suggested the fruitful use of ultrametric spaces.
Parts of the manuscript were read by Manav Das, Don Leggett, William
McWorter, Lorraine Rellick, and Karl Schmidt. Their comments led to many
improvements in the manuscript. I would like to thank the many peo-
ple at Springer-Verlag New York, especially mathematics editor Ru¨diger
Gebauer, mathematics assistant editor Susan Gordon, mathematics editor
Ulrike Schmickler-Hirzebruch (editor of the “Undergraduate Texts in Mathe-
matics” series), production editor Susan Giniger, and Kenneth Dreyhaupt.
...
Columbus, Ohio Gerald A. Edgar
March 1990
Second Edition
Arnold Ross retired from his summer program in 2000 at age 94. He passed
away on September 25, 2002. But the summer program has continued, with
Professor Daniel Shapiro as director.
Although this Second Edition is substantially the same as the First, there
were many changes. Most important was an increased emphasis on the pack-
ing measure (Sect. 6.2), so that now it is often treated on a par with the
Hausdorffmeasure.Thetopologicaldimensionswererearranged(Chap.3),so
that the covering dimension is the major one, and the inductive dimensions
are the variants. A “reduced cover class” notion was introduced to help in
proofs for Method I or Method II measures (p. 149). Research results since
1990 that affect these elementary topics have been taken into account. Some
examples have been removed (golden rectangle, Barnsley wreath). Other ex-
amples have been added (Barnsley leaf, Li lace, Julia set). Terminology has
beenchanged(addressingfunction,self-referentialequation,Lipschitzα,con-
stituent). Notation has been changed (spectral radius). Most of the figures
have been re-drawn.