Table Of ContentFractal Geometry
Mathematical Foundations and Applications
Third Edition
Kenneth Falconer
UniversityofStAndrews,UK
Thiseditionfirstpublished2014
©2014JohnWiley&Sons,Ltd
LibraryofCongressCataloging-in-PublicationDataappliedfor.
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-1-119-94239-9
Setin10/12ptTimesbyLaserwordsPrivateLimited,Chennai,India
12014
Contents
Prefacetothefirstedition ix
Prefacetothesecondedition xiii
Prefacetothethirdedition xv
Coursesuggestions xvii
Introduction xix
PARTI FOUNDATIONS 1
1 Mathematicalbackground 3
1.1 Basicsettheory 3
1.2 Functionsandlimits 7
1.3 Measuresandmassdistributions 11
1.4 Notesonprobabilitytheory 17
1.5 Notesandreferences 24
Exercises 24
2 Box-countingdimension 27
2.1 Box-countingdimensions 27
2.2 Propertiesandproblemsofbox-countingdimension 34
*2.3 Modifiedbox-countingdimensions 38
2.4 Someotherdefinitionsofdimension 40
2.5 Notesandreferences 41
Exercises 42
3 Hausdorffandpackingmeasuresanddimensions 44
3.1 Hausdorffmeasure 44
3.2 Hausdorffdimension 47
3.3 CalculationofHausdorffdimension–simpleexamples 51
3.4 EquivalentdefinitionsofHausdorffdimension 53
*3.5 Packingmeasureanddimensions 54
*3.6 Finerdefinitionsofdimension 57
*3.7 Dimensionprints 58
*3.8 Porosity 60
3.9 Notesandreferences 63
Exercises 64
4 Techniquesforcalculatingdimensions 66
4.1 Basicmethods 66
4.2 Subsetsoffinitemeasure 75
4.3 Potentialtheoreticmethods 77
*4.4 Fouriertransformmethods 80
4.5 Notesandreferences 81
Exercises 81
5 Localstructureoffractals 83
5.1 Densities 84
5.2 Structureof1-sets 87
5.3 Tangentstos-sets 92
5.4 Notesandreferences 96
Exercises 96
6 Projectionsoffractals 98
6.1 Projectionsofarbitrarysets 98
6.2 Projectionsofs-setsofintegraldimension 101
6.3 Projectionsofarbitrarysetsofintegraldimension 103
6.4 Notesandreferences 105
Exercises 106
7 Productsoffractals 108
7.1 Productformulae 108
7.2 Notesandreferences 116
Exercises 116
8 Intersectionsoffractals 118
8.1 Intersectionformulaeforfractals 119
*8.2 Setswithlargeintersection 122
8.3 Notesandreferences 128
Exercises 128
PARTII APPLICATIONSANDEXAMPLES 131
9 Iteratedfunctionsystems–self-similarandself-affinesets 133
9.1 Iteratedfunctionsystems 133
9.2 Dimensionsofself-similarsets 139
9.3 Somevariations 143
9.4 Self-affinesets 149
9.5 Applicationstoencodingimages 155
*9.6 Zetafunctionsandcomplexdimensions 158
9.7 Notesandreferences 167
Exercises 167
10 Examplesfromnumbertheory 169
10.1 Distributionofdigitsofnumbers 169
10.2 Continuedfractions 171
10.3 Diophantineapproximation 172
10.4 Notesandreferences 176
Exercises 176
11 Graphsoffunctions 178
11.1 Dimensionsofgraphs 178
*11.2 Autocorrelationoffractalfunctions 188
11.3 Notesandreferences 192
Exercises 192
12 Examplesfrompuremathematics 195
12.1 DualityandtheKakeyaproblem 195
12.2 Vitushkin’sconjecture 198
12.3 Convexfunctions 200
12.4 Fractalgroupsandrings 201
12.5 Notesandreferences 204
Exercises 204
13 Dynamicalsystems 206
13.1 Repellersanditeratedfunctionsystems 208
13.2 Thelogisticmap 209
13.3 Stretchingandfoldingtransformations 213
13.4 Thesolenoid 217
13.5 Continuousdynamicalsystems 220
*13.6 Smalldivisortheory 225
*13.7 Lyapunovexponentsandentropies 228
13.8 Notesandreferences 231
Exercises 232
14 Iterationofcomplexfunctions–JuliasetsandtheMandelbrotset 235
14.1 GeneraltheoryofJuliasets 235
14.2 Quadraticfunctions–theMandelbrotset 243
14.3 Juliasetsofquadraticfunctions 248
14.4 Characterisationofquasi-circlesbydimension 256
14.5 Newton’smethodforsolvingpolynomialequations 258
14.6 Notesandreferences 262
Exercises 262
15 Randomfractals 265
15.1 ArandomCantorset 266
15.2 Fractalpercolation 272
15.3 Notesandreferences 277
Exercises 277
16 BrownianmotionandBrowniansurfaces 279
16.1 Brownianmotioninℝ 279
16.2 Brownianmotioninℝn 285
16.3 FractionalBrownianmotion 289
16.4 FractionalBrowniansurfaces 294
16.5 Lévystableprocesses 296
16.6 Notesandreferences 299
Exercises 299
17 Multifractalmeasures 301
17.1 Coarsemultifractalanalysis 302
17.2 Finemultifractalanalysis 307
17.3 Self-similarmultifractals 310
17.4 Notesandreferences 320
Exercises 320
18 Physicalapplications 323
18.1 Fractalfingering 325
18.2 Singularitiesofelectrostaticandgravitationalpotentials 330
18.3 Fluiddynamicsandturbulence 332
18.4 Fractalantennas 334
18.5 Fractalsinfinance 336
18.6 Notesandreferences 340
Exercises 341
References 342
Index 357
Preface to the first edition
I am frequently asked questions such as ‘What are fractals?’, ‘What is fractal
dimension?’,‘Howcanonefindthedimensionofafractalandwhatdoesittellus
anyway?’or‘Howcanmathematicsbeappliedtofractals?’Thisbookendeavours
toanswersomeofthesequestions.
Themainaimofthebookistoprovideatreatmentofthemathematicsassoci-
atedwithfractalsanddimensionsatalevelwhichisreasonablyaccessibletothose
whoencounterfractalsinmathematicsorscience.Althoughbasicallyamathemat-
icsbook,itattemptstoprovideanintuitiveaswellasamathematicalinsightinto
thesubject.
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
methodsfortheircalculationareintroduced.Thengeometricalpropertiesoffrac-
tals are investigated in much the same way as one might study the geometry of
classical figures suchas circlesorellipses: locally, acirclemaybeapproximated
byalinesegment,theprojectionor‘shadow’ofacircleisgenerallyanellipse,a
circletypicallyintersectsastraightlinesegmentintwopoints(ifatall)andsoon.
There are fractal analogues of such properties, usually with dimension playing a
keyrole.Thus,weconsider,forexample,thelocalformoffractalsandprojections
andintersectionsoffractals.
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,randomfractalsandsomephysicalapplications.
Therearemanydiagramsinthetextandfrequentillustrativeexamples.Com-
puter 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
producefurtherdrawingsforthemselves.
Itishopedthatthebookwillbeausefulreferenceforresearchers,byprovid-
inganaccessibledevelopmentofthemathematicsunderlyingfractalsandshowing
howitmaybeappliedinparticularcases.Thebookcoversawidevarietyofmath-
ematicalideasthatmayberelatedtofractalsand,particularlyinPartII,providesa
flavourofwhatisavailableratherthanexploringanyonesubjectintoomuchdetail.
Theselectionoftopicsistosomeextentattheauthor’swhim–therearecertainly
someimportantapplicationsthatarenotincluded.Someofthematerialdatesback
toearlytwentiethcentury,whilstsomeisveryrecent.
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
coveredwouldfillalargevolume.However,itishopedthatenoughinformationis
includedtoenablethosewhowishtodosotopursueanytopicfurther.
It would be possible to use the book as a basis for a course on the mathe-
maticsoffractals,atpostgraduateor,perhaps,final-yearundergraduatelevel,and
exercisesareincludedattheendofeachchaptertofacilitatethis.Hardersections
and proofs are marked with an asterisk and may be omitted without interrupting
thedevelopment.
Anefforthasbeenmadetokeepthemathematicstoalevelthatcanbeunder-
stood by a mathematics or physics graduate and, for the most part, by a diligent
final-yearundergraduate.Inparticular,measuretheoreticideashavebeenkepttoa
minimum,andthereaderisencouragedtothinkofmeasuresas‘massdistributions’
onsets.Providedthatitisacceptedthatmeasureswithcertain(intuitivelyalmost
obvious) properties exist, there is little need for technical measure theory in our
development.
Resultsarealwaysstatedpreciselytoavoidtheconfusionwhichwouldother-
wise 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
harderproofsthatarenotavailableinthatformelsewherehavebeenincluded,in
particularthoseonsetswithlargeintersectionandonrandomfractals.)Suitabledia-
gramscanbeahelpinunderstandingtheproofs,manyofwhichareofageometric
nature.Somediagramsareincludedinthebook;thereadermayfindithelpfulto
drawothers.
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
andmassdistributionswhich,itishoped,willbefoundadequate.Thesectionon
probabilitytheorymaybehelpfulforthechaptersonrandomfractalsandBrownian
motion.
Withthewidevarietyoftopicscovered,itisimpossibletobeentirelyconsistent
in the use of notation, and inevitably, sometimes, there has to be a compromise
betweenconsistencywithinthebookandstandardusage.
Inthepastfewyears,fractalshavebecomeenormouslypopularasanartform,
withtheadventofcomputergraphics,andasamodelofawidevarietyofphysical
phenomena.Whilstitispossibleinsomewaystoappreciatefractalswithlittleor
noknowledgeoftheirmathematics,anunderstandingofthemathematicsthatcan
beappliedtosuchadiversityofobjectscertainlyenhancesone’sappreciation.The
phrase‘thebeautyoffractals’isoftenheard–itistheauthor’sbeliefthatmuchof
theirbeautyistobefoundintheirmathematics.
Itisapleasuretoacknowledgethosewhohaveassistedinthepreparationofthis
book.PhilipDrazinandGeoffreyGrimmettprovidedhelpfulcommentsonpartsof
themanuscript.PeterShiarlygavevaluablehelpwiththecomputerdrawings,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
thebook.
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
alsotypedtheentiremanuscriptinamostexpertway.
Finally,IwouldliketothankmywifeIsobelforhersupportandencouragement,
whichextendedtoreadingvariousdraftsofthebook.
April1989 KennethJ.Falconer
Bristol
Preface to the second edition
It is 13 years since Fractal Geometry–Mathematical Foundations and Applica-
tions 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
fewyears,withchanginginterestsandapproachestomathematicsteaching,many
universities have introduced undergraduate and postgraduate courses on fractal
geometryandaconsiderablenumberhavebeenbasedonpartsofthisbook.
Thus, this edition has two main aims. Firstly, it indicates some recent devel-
opments 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
furtherexercises.
Parts of the book have been rewritten. In particular, multifractal theory has
advancedconsiderablysincethefirsteditionwaspublished,sothechapteron‘Mul-
tifractalMeasures’hasbeencompletelyrewritten.Thenotesandreferenceshave
beenupdated.Numerousminorchanges,correctionsandadditionshavebeenincor-
porated, and some of the notation and terminology has been changed to conform
withwhathasbecomestandardusage.Manyofthediagramshavebeenreplaced
to take advantage of the more sophisticated computer technology now available.
Wherepossible,thenumberingofsections,equationsandfigureshasbeenleftas
inthefirstedition,sothatearlierreferencestothebookremainvalid.
Furtherexerciseshavebeenaddedattheendofthechapters.Solutionstothese
exercisesandadditionalsupplementarymaterialmaybefoundontheWorldWide
Webathttp://www.wileyeurope.com/fractal.
In1997,asequel,TechniquesinFractalGeometry,waspublished,presentinga
varietyoftechniquesandideascurrentinfractalresearch.Readerswishingtostudy
fractalmathematicsbeyondtheboundsofthisbookmayfindthesequelhelpful.
Iammostgratefultoallwhohavemadeconstructivesuggestionsonthetext.
Inparticular,IamindebtedtoCarmenFernández,GwynethStallardandAlexCain
