Cross-sections of multibrot sets

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by  Line Baribeau

MindaConferenceProceedingsmanuscriptNo.
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Cross-sections of multibrot sets
LineBaribeau · ThomasRansford
DedicatedtoDavidMindaontheoccasionofhisretirement
Received:date/Accepted:date
7
1
0 Abstract We identify the intersection of the multibrot set of zd+c with the rays
2
R+ω,whereωd−1=±1.

n
a Keywords Mandelbrotset·Multibrotset
J
MathematicsSubjectClassification(2010) 37F45
9
1

] 1 Introduction
S
D
Letd beanintegerwithd≥2.Givenc∈C,wedefine
.
h
at pc(z):=zd+c and p[cn]:=pc◦···◦pc (ntimes).
m
ThecorrespondinggeneralizedMandelbrotset,ormultibrotset,isdefinedby
[
(cid:110) (cid:111)
1 Md := c∈C:sup|p[cn](0)|<∞ .
v n≥0
5
OfcourseM isjusttheclassicalMandelbrotset.Computer-generatedimagesofM
3 2 3
andM arepicturedinFigure1.Multibrotsetshavebeenextensivelystudiedinthe
5 4
5 literature.Schleicher’sarticle[5]containsawealthofbackgroundmaterialonthem.
0
. TRsupportedbygrantsfromNSERCandtheCanadaresearchchairsprogram
1
0 LineBaribeau
7 De´partementdemathe´matiquesetdestatistique,Universite´Laval,
1 1045avenuedelaMe´decine,Que´bec(QC),CanadaG1V0A6
: E-mail:[email protected]
v
i ThomasRansford
X De´partementdemathe´matiquesetdestatistique,Universite´Laval,
r 1045avenuedelaMe´decine,Que´bec(QC),CanadaG1V0A6
a Tel.:[phone]ext2738
Fax:[phone]
E-mail:[email protected]
2 LineBaribeau,ThomasRansford
Fig.1 ThemultibrotsetsM3andM4
Wementionheresomeelementarypropertiesofmultibrotsets.Firstofall,they
exhibit(d−1)-foldrotationalinvariance,namely
M =ωM (ω ∈C, ωd−1=1). (1)
d d
Indeed, for these ω, writing φ(z):=ωz, we have φ−1◦p ◦φ = p , so p[n](0)
c c/ω c
[n]
remains bounded if and only if p (0) does. (In fact, the rotations in (1) are the
c/ω
only rotational symmetries of M . The paper of Lau and Schleicher [1] contains an
d
elementaryproofofthisfact.)
Also,writingD(0,r)forthecloseddiskwithcenter0andradiusr,wehavethe
inclusions
D(0,α(d))⊂M ⊂D(0,β(d)),
d
where
α(d):=(d−1)d−d/(d−1) and β(d):=21/(d−1).
The first inclusion follows from the fact that, if |c| ≤ α(d), then the closed disk
D(0,d−1/(d−1))ismappedintoitselfby p ,andconsequentlythesequence p[n](0)is
c c
bounded.Forthesecondinclusion,weobservethat,if|c|>β(d),thenbyinduction
|p[n+2](0)|≥(2d)n(|c|d−2|c|)foralln≥0,andtheright-handsideofthisinequality
c
tendstoinfinitywithn.
Whend isodd,wehave
M ∩R=[−α(d),α(d)]. (2)
d
This equality was conjectured by Parise´ and Rochon in [3], and proved by them in
[4].Also,whend iseven,wehave
M ∩R=[−β(d),α(d)]. (3)
d
This equality was also conjectured in [3], and subsequently proved in [2]. When
d=2,itreducestothewell-knownequalityM ∩R=[−2,1].
2 4
Cross-sectionsofmultibrotsets 3
Byvirtueoftherotation-invarianceproperty(1),theequalities(2)and(3)yield
informationabouttheintersectionofM withcertainraysemanatingfromzero.In-
d
deed,ifωd−1=1,then
M ∩R+ω ={tω :0≤t≤α(d)},
d
andifωd−1=−1andd iseven,then
M ∩R+ω ={tω :0≤t≤β(d)}.
d
Thisleavesopenthecasewhenωd−1=−1andd isodd.Thepurposeofthisnoteis
tofillthegap.Thefollowingtheoremisourmainresult.
Theorem1.1. Ifωd−1=−1andd isodd,then
M ∩R+ω ={tω :0≤t≤γ(d)},
d
where
γ(d):=d−d/(d−1)(cid:0)sinh(dξ )+dsinh(ξ )(cid:1), (4)
d d
andξ istheuniquepositiverootoftheequationcosh(dξ )=dcosh(ξ ).
d d d
Whend=3,onecanusetherelationcosh(3x)=4cosh3x−3coshxtoderivethe
(cid:112)
exactformulaγ(3)= 32/27,whichyields
Corollary1.2. M ∩iR={iy:|y|≤(cid:112)32/27}.
3
√
Incomparison,notethat(2)givesM ∩R={x:|x|≤2/ 27}.SeeFigure1.
3
Thefirstfewvaluesofα(d),β(d),γ(d)aretabulatedinTable1forcomparison.
Table1 Valuesofα(d),β(d),γ(d)for2≤d≤12
d α(d) β(d) γ(d)
2 0.250000000 2.000000000 1.100917369
3 0.384900179 1.414213562 1.088662108
4 0.472470394 1.259921050 1.078336651
5 0.534992244 1.189207115 1.069984489
6 0.582355932 1.148698355 1.063192242
7 0.619731451 1.122462048 1.057591279
8 0.650122502 1.104089514 1.052904317
9 0.675409498 1.090507733 1.048928539
10 0.696837314 1.080059739 1.045514971
11 0.715266766 1.071773463 1.042552690
12 0.731314279 1.065041089 1.039957793
Itcanbeshownthatγ(d)>1foralld,andthat
γ(d)=21/d+O((logd)2/d2)) asd→∞.
Thesestatementswillbejustifiedlater.
4 LineBaribeau,ThomasRansford
2 ProofofTheorem1.1
Inthissectionwesupposethatd isanoddintegerwithd≥3.Ifωd−1=−1,then,
writingφ(z):=ωz,wehaveφ−1◦p ◦φ =q ,where
c c/ω
q (z):=−zd+c.
c
ThusM ∩R+ω =ω(N ∩R+),where
d d
(cid:110) (cid:111)
N := c∈C:sup|q[n](0)|<∞ .
d c
n≥0
WenowseektoidentifyN ∩R+.Weshalldothisintwostages.
d
Lemma2.1. Letd beanoddintegerwithd≥3.Then
N ∩R+=[0, µ(d)],
d
where
µ(d):=max(cid:8)a−bd :a,b≥0, ad+bd =a+b(cid:9).
Proof. Considerfirstthecasec∈[0,1].Inthiscasewehaveq (0)=candq (c)=
c c
−cd+c≥0.Sinceq isadecreasingfunction,itfollowsthatq ([0,c])⊂[0,c],and
c c
[n]
inparticularthatq (0)isbounded.Hencec∈N forallc∈[0,1].
c d
Consider now the case c∈[1,∞). Then q (0)=c and q[2](0)=−cd+c≤0.
c c
[2n]
As q is a decreasing function, it follows that q (0) is a decreasing sequence and
c c
[2n+1] [n]
q (0)isanincreasingsequence.If,further,c∈N ,thenq (0)isbounded,and
c d c
[2n+1] [2n]
bothofthesesubsequencesconverge,sayq (0)→aandq (0)→−b,where
c c
a,b≥0. We then have q (−b)=a and q (a)=−b, in other words bd+c=a and
c c
ad−c=b. Adding these equations gives ad+bd =a+b. Summarizing what we
haveproved:ifc∈N ∩[1,∞),thenc=a−bd,wherea,b≥0andad+bd =a+b.
d
Conversely, if c is of this form, then q (−b)=a and q (a)=−b, so [−b,a] is a
c c
q -invariant interval containing 0, which implies that q[n](0) remains bounded, and
c
hencec∈N .Combiningtheseremarks,wehaveshownthat
d
N ∩[1,∞)={a−bd :a,b≥0, ad+bd =a+b}∩[1,∞). (5)
d
Theconditionthatad+bd =a+bcanbere-writtenash(a)=−h(b),whereh(x):=
xd−x. Viewed this way, it is more or less clear that the right-hand side of (5) is a
closed interval containing 1, so N ∩[1,∞)=[1,µ(d)], where µ(d) is as defined in
d
thestatementofthelemma.
Finally,puttingallofthistogether,wehaveshownthatN ∩R+=[0,µ(d)].
d
Nextweidentifyµ(d)moreexplicitly.
Lemma2.2. µ(d)=γ(d).
Cross-sectionsofmultibrotsets 5
Proof. Wereformulatethemaximizationproblemdefiningµ(d).Set
S:={(a,b)∈R2:a,b≥0},
f(a,b):=a−bd,
g(a,b):=ad+bd−a−b.
Weareseekingtomaximize f overS∩{g=0}.ThesetS∩{g=0}iscompactand
f iscontinuous,sothemaximumiscertainlyattained,sayat(a ,b ).Noticealsothat
0 0
∇g(cid:54)=0ateverypointofS∩{g=0}.Therearetwocasestoconsider.
Case1:(a ,b )∈∂S.Theconditionthatg(a ,b )=0thenimpliesthat
0 0 0 0
(a ,b )=(0,0),(0,1)or(1,0).
0 0
Thecorrespondingvaluesof f(a ,b )are0,−1,1respectively.Clearlywecanelim-
0 0
inate the first two points from consideration. As for the third, we remark that the
directionalderivativeof f at(1,0)along{g=0}inthedirectionpointingintoS is
(cid:112)
equalto1/ 1+(d−1)2,whichisstrictlypositive.So(1,0)cannotbeamaximum
of f either.
Case 2: (a ,b )∈int(S). In this case, by the standard Lagrange multiplier ar-
0 0
gument, we must have ∇f(a ,b )=λ∇g(a ,b ) for some λ ∈R. Writing this out
0 0 0 0
explicitly,weget
1=λ(dad−1−1),
0
−dbd−1=λ(dbd−1−1).
0 0
Dividingthesecondequationbythefirstandthensimplifying,weobtain
a b =d−2/(d−1).
0 0
Thus a =d−1/(d−1)eξ and b =d−1/(d−1)e−ξ for some ξ ∈R. With this notation,
0 0
theconstraintg(a ,b )=0translatestocosh(dξ)=dcosh(ξ),andthevalueof f at
0 0
(a ,b )is
0 0
f(a ,b )=a −bd = a0−b0 +ad0−bd0 =d−d/(d−1)(cid:0)dsinh(ξ)+sinh(dξ)(cid:1).
0 0 0 0 2 2
Therearepreciselytworootsofcosh(dξ)=dcosh(ξ),onepositiveandonenegative.
Necessarilythepositiverootgivesrisetothemaximumvalueof f,therebyshowing
thatµ(d)=γ(d).
Remark. Clearly f(1,0)=1.ThetreatmentofCase1aboveshowsthat f doesnot
attain its maximum over S∩{g=0} at (1,0), and so µ(d)>1. This shows that
γ(d)>1,therebyjustifyingastatementmadeintheintroduction.
ProofofTheorem1.1. Combiningthevariousresultsalreadyobtainedinthissection,
wehave
M ∩R+ω =ω(N ∩R+)=ω[0,µ(d)]=ω[0,γ(d)].
d d
ThisconcludestheproofofTheorem1.1.
6 LineBaribeau,ThomasRansford
3 Anasymptoticformulaforγ(d).
Ouraimistojustifythefollowingstatementmadeintheintroduction.
Proposition3.1. Ifγ isdefinedasin(4),then
γ(d)=21/d+O((logd)2/d2) asd→∞. (6)
Thereisnoneedtosupposethatd isanintegerhere.
Proof. We begin by deriving an asymptotic formula for ξ as d →∞. On the one
d
hand,since
edξd ≥cosh(dξ )=dcosh(ξ )≥d,
d d
we certainly have ξ ≥(logd)/d. On the other hand, since the unimodal function
d
(coshx)/xtakesthesamevaluesatξ anddξ ,wemusthaveξ ≤η ≤dξ ,where
d d d d
η isthepointatwhich(coshx)/xassumesitsminimum.Thus
edξd
≤cosh(dξ )=dcoshξ ≤dcoshη,
d d
2
whence
logd (cid:16)1(cid:17)
ξ = +O .
d
d d
Thisisnotyetpreciseenough.Substitutingintotheequationcosh(dξ )=dcosh(ξ ),
d d
weobtain
edξd (cid:16)1(cid:17) (cid:16)(logd)2(cid:17)
+O =d+O ,
2 d d
whence
log(2d) (cid:16)(logd)2(cid:17)
ξ = +O .
d d d3
Thisisgoodenoughforourneeds.
Wenowestimateγ(d)asd→∞.Firstofall,wehave
(cid:16)(logd)3(cid:17)
dsinh(ξ )=dξ +O(dξ3)=log(2d)+O .
d d d d2
Also
(cid:16) (cid:16)(logd)2(cid:17)(cid:17) (cid:16)(logd)2(cid:17)
sinh(dξ )=sinh log(2d)+O =d+O .
d d2 d
Hence
(cid:16) (cid:17) d
logγ(d)=log dsinh(dξ )+sinh(dξ ) − logd
d d
d−1
(cid:16) (cid:16)(logd)2(cid:17)(cid:17) (cid:16) 1 (cid:16) 1 (cid:17)(cid:17)
=log d+log(2d)+O − 1+ +O logd
d d d2
log(2d) (cid:16)(logd)2(cid:17) (cid:16) logd (cid:16)logd(cid:17)(cid:17)
=logd+ +O − logd+ +O
d d2 d d2
log2 (cid:16)(logd)2(cid:17)
= +O .
d d2
Finally,takingexponentialsofbothsides,weget(6).
Cross-sectionsofmultibrotsets 7
Acknowledgements ThefirstauthorthankstheorganizersoftheConferenceonModernAspectsofCom-
plexGeometry,heldattheUniversityofCincinnatiinhonorofTaftProfessorDavidMinda,fortheirkind
hospitalityandfinancialsupport.
References
1. Lau,E.,Schleicher,D.:Symmetriesoffractalsrevisited.Math.Intelligencer18(1),45–51(1996)
2. Parise´,P.O.,Ransford,T.,Rochon,D.:Tricomplexdynamicalsystemsgeneratedbypolynomialsof
odddegree.Preprint(2016)
3. Parise´,P.O.,Rochon,D.:Astudyofdynamicsofthetricomplexpolynomialηp+c.NonlinearDynam.
82(1-2),157–171(2015)
4. Parise´, P.O., Rochon, D.: Tricomplex dynamical systems generated by polynomials of odd degree.
Preprint(2015)
5. Schleicher,D.:OnfibersandlocalconnectivityofMandelbrotandMultibrotsets.In:Fractalgeometry
andapplications:ajubileeofBenoˆıtMandelbrot.Part1,Proc.Sympos.PureMath.,vol.72,pp.477–
517.Amer.Math.Soc.,Providence,RI(2004)