Comparisons of Quadratic Mating Methods Slices, Medusas and Clusters, Thurston Equivalence and Shared Matings Chris King A quadratic mating consists of a quadratic rational function, whose Julia sets contain two complementary components, each of which is the Julia set of a quadratic polynomial located on complimentary hemispheres of the Riemann sphere. Two approaches can be used. In the first, one of the rational function’s critical points is given a fixed period and the other is allowed to vary, forming a parameter plane of matings. In the second, including Medusa, a sequence of coefficients is combinatorially generated from external angles, using theory of Thurston and others developed by John Hubbard. Included are observations of Julia set matings utilizing Medusa (Boyd & Henriksen 2012) and Perk(0) moduli space slices (Devaney et al. 2013) in Dark Heart (King 2016) and Per3(0) in Mandel (Jung 2014), also including Chéritat’s (2015) and Sharland’s (2012) mating examples. Fig 1: Left: Periodic moduli space slices and Julia set matings using rational functions (Devaney et al. 2013), which are also general for mating a global array of Julia sets with low period cases. Right: Medusa matings, complement those created by the slice method, mating Julia sets defined by any two external angles, using coefficients produced by the Medusa algorithm. The Perk(0) generating functions are: f 1 ( z ) = z 2 + c , f 2 ( z ) = c /( z 2 + 2 z ), f 3 ( z ) = ( z − 1)( z − c / (2 − c)) / z 2 and f 4 ( z ) = ( z − 4c / (10c + 1))( z − (1 + 2c) / (1 + 6c)) / z 2 . Medusa matings are of the form f ( z ) = (az 2 + 1 − a)/(bz 2 + 1 − b) . The Medusa method favours Julia sets identified by external angles as shown in fig 3, but the slice method can readily find matings with irrational flows, provided they are mated with low period attractors as in fig 2 right. Fig 2: Left: Medusa [1/7,1/3] [9/31,1/3] and [1/1,1/15]. Right: Per4(0) matings with a Siegel disc and a pd 5 parabolic set. Although Medusa can diverge, or remain unstable for some values, it does give comparable results for many examples of periodic domains with odd external angles, where the methods appear to be homologous. For example (3,2) (5,2) and (1,4) above have homologous Medusa Julia matings [1/7,1/3] [9/31,1/3] and [1/1,1/15] shown in fig 2, implying the functions, while different, have conjugate dynamics. However, it is more challenging
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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
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