Getting acquainted with fractals

189 Pages·2007·1.68 MB·English

by  Gilbert Helmberg

de Gruyter Textbook
Helmberg · Getting Acquainted with Fractals

Gilbert Helmberg
Getting Acquainted
with Fractals
≥
Walter de Gruyter
Berlin · New York
Gilbert Helmberg
Kalkofenweg 5
6020 Innsbruck
Austria
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Helmberg,Gilbert.
Gettingacquaintedwithfractals/byGilbertHelmberg.
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ISBN978-3-11-019092-2(hardcover:alk.paper)
1.Fractals. I.Title.
QA614.86.H45 2007
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Preface
Tosomeone,havingheardaboutfractalsbutnotyetacquaintedwiththem,theymight
seemtoberegardedwithsuspicion: Howcould“real”objects–accessiblebysightand
not onlybythought –bereplicasofarbitrarilysmall partsofthemselves? Howcould
a continuous path which runsalmost everywhere parallel tosealevel climbup toany
height? Howcouldacontinuouscurvepassthrougheverypointofasquare?
Gettingacquaintedwithfractalsopensaglimpseintoaworldofwonders,butthese
wonders are strongly supported by a frame of serious mathematics in which various
of its branches play together: geometry, analysis, linear algebra, topology, measure
theory,functionsofcomplexvariables,algebra,... .
Ihavetriedtodojusticetobothaspects: thefascinationofgeometricobjectsaswell
as the serious mathematical background – as far as an advanced undergraduate level.
At some points, where the technicalities would transgress this level, I have at least
indicatedwhereaninterestedreadercouldfindthewholestory. Ihopethepresentation
adds something worthwhile to the many remarkable books on this topic which also
leadmuchfartherintotheworldoffractals.
Thesebooksalsocontainsomethingwhichareadermightmissinthepresentone: I
havechosentoavoidthepossibilityoffrustratingthereaderbyexpectinghimtodoex-
ercises;hewillfindtheminabundanceinthementionedbooks(e.g.[Barnsley,1988],
[Falconer,1990]) if he wants to. However, it is at least my intention to make acces-
sible – via the internet address http://techmath.uibk.ac.at/helmberg–
the programs producing the illustrations, thus enabling the reader to create and play
withfractalsaccordingtohisowntaste.
MythanksareduetothedeGruyterPublishingCompany,inparticulartoDr.Plato,
fortheirinterestinandsupportofthisbookproject. Myfirstbookhasbeendedicated
to my parents, my wife, and my two eldest children, but there are more people who
meanverymuchtome. Thereforethisbookisdedicated
toChri,Moni,andMui.
Innsbruck,Cavalese,August2006 GilbertHelmberg

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Fractalsanddimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Thegameofdeletingandreplacing . . . . . . . . . . . . . . . . . . . . 1
1.2 Thebox-countingdimension . . . . . . . . . . . . . . . . . . . . . . . 50
1.3 TheHAUSDORFF dimension . . . . . . . . . . . . . . . . . . . . . . . . 55
2 Iterativefunctionsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.1 Thespaceofcompactsubsetsofacompletemetricspace . . . . . . . . 63
2.2 Contractionsinacompletemetricspace . . . . . . . . . . . . . . . . . 70
2.3 AffineiterativefunctionsystemsinR2 . . . . . . . . . . . . . . . . . . 74
3 Iterationofcomplexpolynomials . . . . . . . . . . . . . . . . . . . . . . . 109
3.1 GeneraltheoryofJULIA sets . . . . . . . . . . . . . . . . . . . . . . . 111
3.2 JULIAsetsforquadraticpolynomials . . . . . . . . . . . . . . . . . . . 121
3.3 TheMANDELBROT set. . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.4 GenerationofJULIA sets. . . . . . . . . . . . . . . . . . . . . . . . . . 150
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Listofsymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Contents(detailed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

1 Fractals and dimension
1.1 The game of deleting and replacing
The word “fractal” comes fromthe Latinword “frangere” (with past participle “frac-
tus”) which means “to break”, “to destroy”. Let us begin with exploring how such a
destructionprocessmaystillgeneratesomenewmathematicalobjectdisplayinginter-
estingfeatures.
1.1.1 The CANTOR set
Letusdefineanoperationf (suchanoperationiscommonlycalledanoperator)work-
ing on any closed segment [a,b] ⊂ R (= the real line) by deleting the open middle
third ]a+ b−a,b− b−a[,andletusdenotetheinterval[0,1]⊂RbyA . Application
3 3 (0)
off toA deletestheinterval ]1,2[ andproducesaclosedset
(0) 3 3
(cid:2) (cid:3) (cid:2) (cid:3)
A = 0,1 ∪ 2,1 ,
(1) 3 3
the union of the two disjoint closed intervals A = [0,1] and A = [2,1], each of
0 3 1 3
whichhaslength 1. Ifweapplyf nowtoA wegetaclosedset
3 (1)
A = f(A ) = f(f(A )) ⊂ A
(2) (1) (0) (1)
consisting of four disjoint intervals A , A , A , A of length 1 = 1 each.
0,0 0,1 1,0 1,1 9 32
Since we want to continue the application of f, in order toavoid theclumsy notation
f(f(...))letususethenotation
f(0)(A) := A,
f(1)(A) := f(A),
f(k+1)(A) := f(f(k)(A)).
(Weshall call the indexk the levelof the construction.) Applied to our intervalA
(0)
thisallowsustodefineasequenceofclosedsetsA (1≤k <∞)by
(k)
A := f(k)(A )
(k) (0)
satisfying
A ⊃ A ⊃ ··· ⊃ A ⊃ A ⊃ ··· . (1.1)
(0) (1) (k) (k+1)
The set A is the union of 2k closed intervals A (j ∈ {0,1}, 1 ≤ i ≤ k) of
(k) j1,...,jk i
length 1 each. Asequence{A }∞ aswellbehavedasindicatedby(1.1)raisesthe
3k (k) k=1
question whether there exists, in some sense, a limit set A. Indeed, by a well known