ComparisonsofQuadraticMatingMethodsSlices,MedusasandClusters,ThurstonEquivalenceandSharedMatings

ChrisKingAquadraticmatingconsistsofaquadraticrationalfunction,whoseJuliasetscontaintwocomplementarycomponents,eachofwhichistheJuliasetofaquadraticpolynomiallocatedoncomplimentaryhemispheresoftheRiemannsphere.Twoapproachescanbeused.Inthefirst,oneoftherationalfunction’scriticalpointsisgivenafixedperiodandtheotherisallowedtovary,formingaparameterplaneofmatings.Inthesecond,includingMedusa,asequenceofcoefficientsiscombinatoriallygeneratedfromexternalangles,usingtheoryofThurstonandothersdevelopedbyJohnHubbard.IncludedareobservationsofJuliasetmatingsutilizingMedusa(Boyd&Henriksen2012)andPerk(0)modulispaceslices(Devaneyetal.2013)inDarkHeart(King2016)andPer3(0)inMandel(Jung2014),alsoincludingChéritat’s(2015)andSharland’s(2012)matingexamples.

Fig1:Left:PeriodicmodulispaceslicesandJuliasetmatingsusingrationalfunctions(Devaneyetal.2013),whicharealso

generalformatingaglobalarrayofJuliasetswithlowperiodcases.Right:Medusamatings,complementthosecreatedbytheslicemethod,matingJuliasetsdefinedbyanytwoexternalangles,usingcoefficientsproducedbytheMedusaalgorithm.

ThePerk(0)generatingfunctionsare: f1(z) = z2 + c,  f2 (z) = c / (z

2 + 2z),  f3(z) = (z−1)(z− c / (2− c)) / z2 and

f4 (z) = (z− 4c / (10c+1))(z− (1+ 2c) / (1+ 6c)) / z2 .Medusamatingsareoftheform f (z) = (az2 +1− a) / (bz2 +1− b) .

TheMedusamethodfavoursJuliasetsidentifiedbyexternalanglesasshowninfig3,buttheslicemethodcanreadilyfindmatingswithirrationalflows,providedtheyarematedwithlowperiodattractorsasinfig2right.

Fig2:Left:Medusa[1/7,1/3][9/31,1/3]and[1/1,1/15].Right:Per4(0)matingswithaSiegeldiscandapd5parabolicset.

AlthoughMedusacandiverge,orremainunstableforsomevalues,itdoesgivecomparableresultsformanyexamplesofperiodicdomainswithoddexternalangles,wherethemethodsappeartobehomologous.Forexample(3,2)(5,2)and(1,4)abovehavehomologousMedusaJuliamatings[1/7,1/3][9/31,1/3]and[1/1,1/15]showninfig2,implyingthefunctions,whiledifferent,haveconjugatedynamics.However,itismorechallenging

tofindcorrespondencesinsomeothercases,althoughbothapproachesappeartogivevalidmatings.Compareforexample[1/71/15]infig3with(3,4)aboveandtheexamplefromthesmallerperiod3bulbfig3right,withlocationshownat(a).Allthreearetopologicallydistinctmatings.

Fig3:[1/71/15]matingperiods3and4andthekeyPer4(0)period3matingat(a)indicatesdifferences,whichmaybedue

tothegeneratingfunctionoftheparameterplanePer4(0)sincetheright-handimageisasymmetric,asinfigs7,8.

Neitherdoesthe[1/71/7]self-matinginfig4appeartobehomologoustoanyofthoseofPer3(0),possiblyduetoitssuppressionofpd3,althoughitdoesappearhomologoustothatofArnaudChéritat’sThurstonalgorithm.

Fig4:[1/71/7]whichdoesn’tappearinPer3(0)andanequivalentmatingbyArnaudChéritat(2015).

Medusacorrectlyportraysboth[1/7,1/5]and[1/5,1/7]matingtheperiod3bulbtoitsperiod4dendriticMandelbrot,asshowninfig1,anditcanportraytwodendriticMandelbrotsatelliteJuliasonthesamesideofthex-axisasshownbelowfor[5/311/5]and[1/55/31],shownbelowleftandcentre.Butthe[1/5,1/5]and[5/31,5/31]self-matings,shownatrighthavedistinctappearances.Significantlythecoefficientsof[1/5,1/5]arecomplexconjugates.Thesecondshowsitsstructuretobeacomplementaryfractalinthedetailright.

Fig5:Medusamatings[5/311/5]and[1/55/31],and[1/5,1/5]and[5/31,5/31]mainbodybetweenperiodbulb

JuliasanddendriticMandelbrotJuliasets.AsituationwheresomethingprovocativehappensistheMedusamatingbetweentheperiod3bulbJuliaset(Rabbit)withtheJuliasetoftheperiod3Mandelbrotonthenegativexdendrite(Airplane).TheMedusaalgorithmfor[1/7,3/7]and[3/7,1/7]don’tlookatfacevaluelikeamatingbetweenabulbandadendriticMandelbrotJulia,aswesawin[1/5,1/7]andtheyareapparentlyhomologoustooneanotherasshowninfig6.

Fig6:[1/7,3/7]and[3/7,1/7]comparedwithTomSharland’sandArnaudChéritat’sversions.

Thesamesituationasinfig6applissalsoto[1/511,255/511](fig1),whichisalsoamatingbetweenaperiod9bulbandaperiod9dendriticMandelbrotonthenegativex-axis.WolfJunghaspointedoutthattheseaspectscanbeexplainedbysharedmatings-‘differentpairsofpolynomialsmaygivethesamerationalmap.OneofthesimplestexamplesisthatthematingofRabbitandAirplaneisthesameasthematingofAirplaneandRabbit,uptoarescaling.Infactthemapcanberescaledsuchthatitisinvariantunderinversion1/z,althoughitisnotaself-mating.Moreover,thefactthatsixFatoucomponentsmeetatasinglepoint,canbeexplainedintermsofrayconnections’.ThisexampleisconfirmedagaintheimagerightfromTomSharland’s(2012)Harvardlecture.ThurstonequivalencemeansthatJuliasetsofmatingsareuniqueuptoconjugacyclassesviaMobiustransformations.Infig7weexplorethismatingusingtwoversionsofPer3(0).Theperiod3dendriticMandelbrot(a)hasamatinglookingaswewouldexpect,inbothDarkHeart(upperrow)andMandel(lowerrow)–veryobviouslytheAirplaneandRabbit.Theotherperiod3regionsintheparameterplanesare(b)whichisnothomologoustofig6and(c),whichdiffersinDarkHeart,butisidenticaltofig6inMandel,raisingaquestionabouttherelationshipbetweenthemandwhethertheJuliasetsformhomologousmatingsunderaMobiustransformation.

Fig7:DarkHeartandMandelversionsof(3,3)Per3(0)matingsshowsubtledifferencesoftopology.

Thetwoparameterplanesillustratedinfig8appearatfirstsighttobeidenticalbuthavesubtledifferencesintheirtopologytotherightofthecentralbasinwhichramifiesintotheJuliasets.TherationalfunctioninMandelis f3

M (z) = (z2 + c3 − c−1) / (z2 − c2 ) withperiod3criticalorbit∞→1→−c andcriticalpoint0,whiletheoneinDarkHeart(Devaneyetal.2013)is f3

D (z) = (z−1)(z− c / (2− c)) / z2with∞→1→ 0 andcriticalpointc.Bothappeartogivevalidmatingsdespitetheasymmetry,sopresumablymustdifferbyaMobiustransformation.

Fig8:RunninginDarkHeart,thetwoPer3(0)parameterplanesandtheirJuliasetshavesubtledifferences.

Toseekaresolutionfortheperiod4caseweneedtogenerateasymmetricJuliaspectrumbyconfiningthezeroandinfinitecriticalpointstozeroandinfinityasisthecasefortheperiod3versioninMandel:

f (z) = z2 + pz2 + q

, ∞→1→−c,  ⇒1+ p = −c(1+ q),  g = −c2, p = −c(1− c2 )−1= c3 − c−1,   f (z) = (z2 + c3 − c−1) / (z2 − c2 )

Wenowneedtoassignanarbitrarypointatoretainthecorrectdegreesoffreedomasshownbelow.

f (z) = z2 + pz2 + q

, ∞→1→ a→−c,  1→ a⇒1+ p = a(1+ q),  a→−c⇒ a2 + p = −c(a2 + q),  −c→∞⇒ g = −c2,  

p = a(1− c2 )−1,  a2 + a(1− c2 )−1= −ca2 + c3, a2 + a((1− c)− (c2 − c+1) = 0, a = (c−1)± (c−1± 5c2 − 6c+ 5) / 2,  

f (z) = (z2 + (c−1± 5c2 − 6c+ 5)(1− c2 ) / 2−1) / (z2 − c2 )

Becausethisgeneratingfunctionnowinvolvesafractionalpowerofc,thecomplexparameterplanebecomessplit,resultingintwo“fermionic”parameterplanesconnectedbytheellipticsplitillustrated(right)infig9below.ComparisonofthesewiththeMedusamatingsforthe6periodfourlocationsinthequadraticMandelbrotset(left)oftheseshowsthatthetwoplanesprovideafullrepresentationofthematings,withallthesecasesandconfirmstheconsistencyofthetwomatingmethods.

Fig9:GlobalcorrespondencebetweenMedusamatingsforalltheperiod4typesandthe“fermionic”Per4(0).

WenowexploresharedorequivalentmatingsfurtherinMedusa.ThetwodendriticMandelbrotmatings[3/7,1/5]and[1/5,3/7]toprowfig10appeartobeequivalentto[1/7,6/15]and[6/15,1/7]ontheperiod3and2x2bulbs,againsuggestingsharedmatings.

Fig10:[1/53/7]givesthesameMedusamatingas[1/76/15]

TomSharland(2012)notesthatthetwomatingsinfig11areknowntobeequivalent.IndeedMedusanotonlygivesidenticalcoefficientsforboth,buttheinversemating[7/151/5]ishomologoustotheoriginal,eventhoughtheFatoubasinsofzero(black)andinfinity(shadedorange)havebeenexchanged.

Fig11:Equivalentmatings[1/57/15]and[4/56/15]withtheirJuliasetandthatoftheinversematings.

ClusteringistheconditionwherethecriticalorbitFatoucomponentsgrouptogethertoformaperiodiccycle.Tomcommentsthatthematingsrightinfig11allhaveperiod3clustercycleswiththesameintrinsicdata.Buttheycertainlydon’talllookthesame!Insimplecases,(periods1&2)thecombinatorialdataofaclustercompletelydefinesarationalmap,butinperiod3theexperimentalpicturessuggestnot.

Fig12:Equivalentperiod3clustermatingscorrespondtoTomSharlnd’simages,providedyoupickthe

appropriatepairofratiosintheleft-handfigures.Someappeartobetopologicallydistinct.Nowlet’sturntoevendenominatorswherewehaveraystoMisiurewiczpointsonthedendrites.Thereisnoproblemwiththefirstdenominatorbeingevenas[1/4,1/7]showsusbelowleft,and[1/4,1/511]atright,butifwechoose[1/7,1/4],wegettheinfiniteJuliasetshowncentre.NotingthatMedusahasplacedtheJuliasetoverinfinityinsteadofzero-thecorrectthingtodoas[1/4]hasnointeriorbasinsoitshouldsitoninfinity,wecan

maketheMobiustransformation az2 +1− abz2 +1− b

→(1− b)z2 + b(1− a)z2 + a

andwehaveaniceJuliasetwhosecoefficientsare

distinctfromthoseof[1/4,1/7]whichisotherwisehomologousto[1/4,1/7].

Fig13:Left:[1/6,1/7]withdendritictreedetail.Centre:[1/7,1/6]anditsMobiusinversion.Right:[1/4,1/511]

[1/4,1/6]and[1/4,1/2]alsolooktobeplausiblebecausetheyarematingachaoticdendriticJuliasettoanotherone,sothewholeplaneisclosetoJulia…butisthisthecaseifoneshouldhavecomplimentaryshading?

Fig14:Starryskywithsymmetries.MedusamatingoftwodendriticJuliasets[1/4,1/2]

ArnaudChéritat’sThurstonalgorithmdoesgiveclearevolutionaryportraitsofmatingsofdendriticJuliasetsincluding[1/65/14],infig15.HenotesthataccordingtoShishikuraandMilnor,thisgivesaLattèsmap.

Fig15:ThreestagesinArnaudChéritat’s(2015)movieofdendriticJuliasetmating[1/65/14]appearstosolvethis.

Medusaiterationsremainedunstableforthesevalues.ManymoreavailableatArnaud’slinkbelow.

References

1. BoydSuzanne,HenriksenChristian(2012)TheMedusaAlgorithmForPolynomialMatingsConformalGeometryandDynamics16,161-183arXiv:1102.5047.Download:http://www.math.cornell.edu/~dynamics/Matings/

2. ChéritatArnaud(2015)PolynomialmatingsontheRiemannspherehttps://www.math.univ-toulouse.fr/~cheritat/MatMovies/

3. DevaneyR,FagellaN,GarijoA,JarqueX(2013)SierpinskicurveJuliasetsforquadraticrationalmapsarXiv:1109.0368.4. JungWolf(2014)Mandelhttp://mndynamics.com/indexp.html5. KingChris(2016)DarkHeartPackage2.0http://dhushara.com/DarkHeart/6. SharlandThomas(2012)PolynomialMatingsandRationalMapswithClusterCycles

www.math.uri.edu/~tsharland/Harvard.pdf