The Intrigues and Delights of Kleinian and Quasi … › DarkHeart › quasif ›...
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The Intrigues and Delights of Kleinian and Quasi-Fuchsian Limit Sets CHRIS KING Preprint published 2019 Math Intelligencer https://doi.org/10.1007/s00283-018-09856-6 Introduction: My research in dynamical systems has inexorably led me from the better-known fractal dynamics of Julia and Mandelbrot sets, including those of the Riemann Zeta function 1 , to the more elusive forms of Kleinian limit sets. Julia sets and their universal atlases, Mandelbrot sets, have become ubiquitous features of the mathematical imagination, demonstrating the power of computer algorithms to generate, in scintillating detail, visualizations of complex dynamical systems, complementing theory with vivid counter-examples. However Gaston Julia’s original explorations took place abstractly, long before the explosion of computing technology, so it was impossible to appreciate their fractal forms and it was only after Beniot Mandelbrot popularized the set that now bears his name that the significance of Julia’s work again became recognized. It is a historical irony, very pertinent to this article, that the Mandelbrot set was actually discovered by Robert Brooks and Peter Matelski 2 as part of a study of Kleinian groups. It was only because Mandelbrot was a fellow at IBM that he had the computing resources to depict the set in high resolution, as a universal multi-fractal atlas. Julia sets and are generated by iterating a complex polynomial, rational or other analytic function, such as exponential and harmonic functions. For a non- linear function, exemplified by the quadratic iteration z → f c ( z ) = z 2 + c , successive iterations either follow an ordered pattern, tending to a point, or periodic attractor, or in a complementary set of cases, behave chaotically, displaying the butterfly effect and other features of classical chaos. The Julia set is the set of complex values on which the iteration is chaotic and the complementary set, where it is ordered, is named after the Julia set’s co- discoverer Pierre Fatou. The Mandelbrot set M, fig 1(h), becomes an atlas of all the Julia sets , by starting from the critical point: z 0 : f '( z 0 ) = 0 (the last point to escape to infinity) and applying the iteration for every complex c-value. In a fascinating demonstration of algorithmic topology, if c ∈ M the fractal Julia set J c of c is topologically connected, otherwise it forms a fractal dust, or Cantor set, with the most ornate and challenging examples lying very close to, or on, the boundary of M. Alongside images on the internet of complex fractals, you will also find a more esoteric class of fractal sets that often look like medieval arboreal tapestries, and go variously by the names of Kleinian and quasi-Fuchsian limit sets, but the programs that generate them are much more difficult to find – to such an extent that I resolved to generate accessible multi-platform versions in the public domain to enable anyone to explore them. To make these fascinating systems freely accessible and to aid further research, I have thus developed an interactive website: Kleinian and Quasi-Fuchsian Limit Sets: An Open Source Toolbox, 3 with freely available cross-platform software, including a Matlab toolbox and a generic C script, a hundred times faster, that will compile and generate images on any GCC-compatible operating system, as well as a MacOS viewing app, enabling full dynamical exploration of the limit sets at will. These limit sets also have an intriguing history, which I will sketch only briefly, as it is elucidated in definitive and engaging detail in Indra’s Pearls: The Vision of Felix Klein 4 . Interspersed below are short references (e.g. IP1 ) so one can refer to an expanded discussion of each topic by browsing the book. Fig 1: Julia sets and Kleinian limit sets (a-c) are two classes of complex fractals, the former being the set on which a non-linear function, is chaotic, and the latter the limit set of two interacting Möbius transformations. Just as Julia sets (g) have the Mandelbrot set (h), as atlas, limit sets with Tb=2, as in fig 3 have the Maskit slice (f) as atlas, with the critical states on the upper boundary. In this field, we again witness an evolution, where research and discovery has at first been driven by J c
Transcript of The Intrigues and Delights of Kleinian and Quasi … › DarkHeart › quasif ›...
The Intrigues and Delights of Kleinian and Quasi-Fuchsian Limit Sets
CHRIS KING Preprint published 2019 Math Intelligencer https://doi.org/10.1007/s00283-018-09856-6
Introduction:Myresearchindynamicalsystemshasinexorablyledmefromthebetter-knownfractaldynamicsofJuliaandMandelbrotsets,includingthoseoftheRiemannZetafunction1,tothemoreelusiveformsofKleinianlimitsets.Juliasetsandtheiruniversalatlases,Mandelbrotsets,havebecomeubiquitousfeaturesofthemathematicalimagination,demonstratingthepowerofcomputeralgorithmstogenerate,inscintillatingdetail,visualizationsofcomplexdynamicalsystems,complementingtheorywithvividcounter-examples.HoweverGastonJulia’soriginalexplorationstookplaceabstractly,longbeforetheexplosionofcomputingtechnology,soitwasimpossibletoappreciatetheirfractalformsanditwasonlyafterBeniotMandelbrotpopularizedthesetthatnowbearshisnamethatthesignificanceofJulia’sworkagainbecamerecognized.Itisahistoricalirony,verypertinenttothisarticle,thattheMandelbrotsetwasactuallydiscoveredbyRobertBrooksandPeterMatelski2aspartofastudyofKleiniangroups.ItwasonlybecauseMandelbrotwasafellowatIBMthathehadthecomputingresourcestodepictthesetinhighresolution,asauniversalmulti-fractalatlas.Juliasetsandaregeneratedbyiteratingacomplexpolynomial,rationalorotheranalyticfunction,suchasexponentialandharmonicfunctions.Foranon-linearfunction,exemplifiedbythequadraticiteration z→ fc (z) = z
2 + c ,successiveiterationseitherfollowanorderedpattern,tendingtoapoint,orperiodicattractor,orinacomplementarysetofcases,behavechaotically,displayingthebutterflyeffectandotherfeaturesofclassicalchaos.TheJuliasetisthesetofcomplexvaluesonwhichtheiterationischaoticandthecomplementaryset,whereitisordered,isnamedaftertheJuliaset’sco-discovererPierreFatou.TheMandelbrotsetM,fig1(h),becomesanatlasofalltheJuliasets ,bystartingfromthecriticalpoint: z0 : f '(z0 ) = 0 (thelastpointtoescapetoinfinity)andapplyingtheiterationforeverycomplexc-value.Inafascinatingdemonstrationofalgorithmictopology,if c ∈M thefractalJuliaset Jc ofcistopologicallyconnected,otherwiseitformsafractaldust,orCantorset,withthemostornateandchallengingexampleslyingverycloseto,oron,theboundaryofM.Alongsideimagesontheinternetofcomplexfractals,youwillalsofindamoreesotericclassof
fractalsetsthatoftenlooklikemedievalarborealtapestries,andgovariouslybythenamesofKleinianandquasi-Fuchsianlimitsets,buttheprogramsthatgeneratethemaremuchmoredifficulttofind–tosuchanextentthatIresolvedtogenerateaccessiblemulti-platformversionsinthepublicdomaintoenableanyonetoexplorethem.Tomakethesefascinatingsystemsfreelyaccessibleandtoaidfurtherresearch,Ihavethusdevelopedaninteractivewebsite:KleinianandQuasi-FuchsianLimitSets:AnOpenSourceToolbox,3withfreelyavailablecross-platformsoftware,includingaMatlabtoolboxandagenericCscript,ahundredtimesfaster,thatwillcompileandgenerateimagesonanyGCC-compatibleoperatingsystem,aswellasaMacOSviewingapp,enablingfulldynamicalexplorationofthelimitsetsatwill.Theselimitsetsalsohaveanintriguinghistory,whichIwillsketchonlybriefly,asitiselucidatedindefinitiveandengagingdetailinIndra’sPearls:TheVisionofFelixKlein4.Interspersedbelowareshortreferences(e.g.IP1)soonecanrefertoanexpandeddiscussionofeachtopicbybrowsingthebook.
Fig1:JuliasetsandKleinianlimitsets(a-c)aretwo
classesofcomplexfractals,theformerbeingthesetonwhichanon-linearfunction,ischaotic,andthelatterthelimitsetoftwointeractingMöbiustransformations.JustasJuliasets(g)havetheMandelbrotset(h),asatlas,limitsetswithTb=2,asinfig3havetheMaskitslice(f)asatlas,
withthecriticalstatesontheupperboundary.Inthisfield,weagainwitnessanevolution,whereresearchanddiscoveryhasatfirstbeendrivenby
Jc
abstracttheoreticaladvances,whicharethenfollowedbytheemergenceofinnovativecomputationalapproachesthatyieldcriticalexamplesdisplayingtherichnessandvarietythatgivesthetheoryitsfullvalidation.ThefieldofKleiniangroupswasfoundedbyFelixKleinandHenriPoincaré,thespecialcaseofSchottkygroupshavingbeenelucidatedafewyearsearlierbyFriedrichSchottky.Theensuingstory,asdepictedinIndra’sPearlsbeginswithavisittoHarvardbyBenoitMandelbrotthatleadstoDavidMumfordsettingupacomputationallaboratorytoexplore“somesuggestive19thcenturyfiguresofreflectedcircleswhichhadfascinatedFelixKlein”.CarolineSeriesalongwithDavidWright,assistedbyCurtMullenwhohadheldsummerpositionsatIBM,andothercollaboratorssuchasLindaKeen,thenbegantoinvestigatecomputationalexplorationoftheselimitsets.InDavidWright'swords:"Taketwoverysimpletransformationsoftheplaneandapplyallpossiblecombinationsofthesetransformationstoapointintheplane.Whatdoestheresultingcollectionofpointslooklike?"Indra’sPearlscontinuesthishumanandmathematicaljourneyinelegantdetail,notingthecontributionsofmanyresearchers,includingBernardMaskitandTroelsJørgensentotheunfoldingexplorationofthearea.AMathematicalNexus:BycontrastwithJuliasets,Kleinianlimitsetsarealgebraic‘attractors’generatedbytheinteractionoftwoMöbiusmapsa,bandtheirinversesA=a-1,B=b-1.MöbiusmapsIP62arefractionallineartransformationsoperatingontheRiemannsphereR,fig2(a),representedbycomplexmatrices.Compositionofmapsisthusequivalenttomatrixmultiplication:
z→ a(z) = pz+ qrz+ s
≈p qr s
⎡
⎣⎢⎢
⎤
⎦⎥⎥
z1
⎡
⎣⎢
⎤
⎦⎥ :ps–qr=1(1).
Möbiustransformationscanbeelliptic,parabolic,hyperbolic,orloxodromica,intermsoftheirtraceTr(a)=p+s,ascomplexgeneralizationsofthetranslations,rotationsandscalingsofaffinemapsoftheEuclideanplane.TheymapcirclestocirclesonR,countinglinesinthecomplexplaneasgreatcirclesonR.Möbiusmapsarethussimplerinnaturethananalyticfunctionsandarelinearandinvertible,soasingleMobiusmapwillnotgenerateaninterestingfractal.However,whentwoormoreMöbiusmapsinteract,theiterationsofpointsinR,aParabolicmapshaveTr2(a)=4andactliketranslations, withonefixedpointonR,ellipticmapshave0<Tr2(a)<4andbehavelikerotationsonR,hyperbolicandloxodromicmapshaverealandcomplexvalueslyingoutsidethesevaluesonCandbehavelike(spirally)expandingandcontractingversionsofscalingsonR.
underthetwotransformationsandtheirinverses,tendasymptoticallytowardsasetwhichisconservedunderthetransformations,oftenacomplexfractal,calledthelimitsetofthegroup.Whenthetwotransformationsactingtogetherformcertainclassesofgroup,includingKleinian,quasi-FuchsianandSchottkygroups,theirlimitsetsadoptformswithintriguingmathematicalproperties.Becausetheyformaninterfacebetweenwidelydifferentmathematicalareas,theirdynamicsdrawtogetherdiversefields,fromgrouptheory,throughtopology,Riemannsurfacesandhyperbolicgeometry,tochaosandfractalanalysis,perhapsmoresothananyrelatedarea.Dependingontheparticulargroup,itsactionbmaytransformthetopologyoftheRiemannsphereintothatofanotherkindoftopologicalsurface,suchasatwo-handledtorus,sometimeswithpinchedhandles,fig2(e-g).AKleiniangroupisadiscretecsubgroupof2x2complexmatricesofdeterminant1moduloitscenter-thesetofelementsthatcommutewitheveryelementofthegroup.ASchottkygroupIP96isaspecialformofKleiniangroup,consistingoftwo(ormore)ofMöbiusmapsaandbonR,eachofwhichmapstheinsideofonecircleContotheoutsideoftheotherD.Ifthetwopairsofcirclesareentirelydisjointasinfig2(b),thegroupactioninducesatopologicaltransformationofR,generatingamulti-handledtorusasinfig2(e).Aquasi-FuchsiangroupIP161isaKleiniangroupwhoselimitsetiscontainedinaninvariantJordancurve,ortopologicalcircle,sotheirlimitsetsarefractalcircles.Theunderlyingspacesofquasi-FuchsianandotherKleiniangroups,suchasthatoftheApolloniangasketfig2(d),canbedefinedbypinchingthetopologyasinfig2(f,g),sothatthelimitsetbecomesafractaltilingbycirclesasinfig3(a).Thegroupinteractions,whendiscrete,inducetilingsinthecomplexplane,whichalsoillustratetransformationsofhyperbolicgeometryd.bThisnewtopologyarisesfromthequotientofthecomplementofthelimitsetundertheactionofthegroup.cAdiscretegroupisagroupoftransformationsofanunderlyingspacepossessingthediscretetopology,wherethereisacollectionofopensetsinwhicheachopensetcorrespondstojustoneelementofthegroup.Discretegroupsthusappearnaturallyasthesymmetriesofdiscretestructures,suchastilingsofaspace.dThegroupofallMöbiustransformationsisisomorphictotheorientation-preservingisometriesofhyperbolic3-space,andhencegroupactionsofsuchtransformationscangeneratehyperbolictilings,suchasthoseofMauriceEscher’s“AngelsandDemons”,representingthemodulargroup,aswellasthelimitsetsdepictedinfig3.
Astheparametersofthesemapsvary,thelimitsetsapproachaboundaryinparameterspace,wherethelimitsetcanbecomearationalcirclepacking,asinfig3(a-c)orirrationallydegenerate(f,g),formingaspace-filingfractal,beyondwhichtheprocessdescendsintochaoticdynamicsfig6(d),asthegroupsbecomenon-discreteandtheirorbitsbecomeentangled.LiketheMandelbrotset,forcertainclassesoflimitset,onecangenerateanatlas,calledtheMaskitslice,fig1(f),whoseupperboundarycontainstheselimitingcases.AninsightfulwaytoportraytheiractionsistodeterminewhattheMöbiustransformationsdotokeycirclesinvolvedinthedefinitionsofaandb.AgoodstartingpointiswithpairsofSchottkymaps.Ifallthecirclesaredisjoint,whenthecirclesaretransformedbybothmaps,theirimagesformacascadeofcircles,whichtendinlimittothetotally-disconnectedfractaldustofaCantorset,fig2(b).Astheparametersp,q,r,andsofeacharevaried,sothatcirclepairscometogetherandtouch,asin(c),theCantorsettransformsintoacircularlimitset.
Fig2:(a)TheRiemannsphere–thecomplexplaneclosedbyasinglepointatinfinity.(b)SchottkymapsformingCantordust.(c)Touchingcirclesformingacircularlimitset.(d,g)Fourcircles,alltouching,forminganApolloniangasket.(e)SchottkymapsperformasurgeryofRtoformadoublehandledtorus.(f)Ifthecommutatorisparabolic,
asinquasi-Fuchsianlimitsets,whichformafractaltopologicalcircle,orJordancurve,whosecomplements,quotientbythegroupintoapairofRiemannsurfaces,thedoubletorusbecomespinched.(g)Whenoneorbothofaandb,ortheirwords,arealsoparabolic,asinthegasket,(d)andthoseinfig3,one,ortwofurtherpinchesoccur.Thereisanalternativeway,independentlyofSchottkygroups,touniquelydefinethematricesviatheirtracesTa,Tb,whereTa=p+s,asin(1),ifwefocusonmapswherethecommutatoraba–1b–1isparabolic,withatraceof–2.Astracesarepreservedunderconjugacyofmappingsb=cac-1,wecansimplifythediscussiontoconjugacyclasses,reducingthenumberoffreevariablesinthetwomatricestoauniquesolution.This,process,affectionatelyknowninIndra’sPearlsas‘Grandma’srecipe’IP227,enablesustoexplorenewclassesoflimitset,asillustratedinfig2(d)forthe
Apolloniangasket,whichcanalsoberepresentedvia4circles,allofwhichtouch,asshowninblack.TheasymmetricnatureofthetwomatrixmappingsgeneratedbyGrandma’srecipecanbeseeninfig1(b,d),wherebotharecolouredbythefirstiterate.Althoughthecommutatorisparabolic(resultinginthesinglepinchweseeinfig2(f),thegeneratorsa,andbcanbeloxodromic,andcantendtoparabolicaswell,withthetransitionfromthequasi-Fuchsiansetoffig1(b)whereTa=Tb=2.2totheApolloniangasketfig3(a),whereTa=Tb=2,resultinginthetwofurtherpinchesinfig2(g)andineachoftheexamplesinfig3,wherebisparabolicwithTb=2.Becausethelimitsetistheplacewhereallpointsareasymptoticallymappedunderinteractionofthetransformationsandtheirinverses,onecansimplypickeachpointontheplaneandrepeatedlymapitbyarandomsequenceofthetwotransformationsandtheirinversesandthesepointswillbecomeasymptoticallydrawntothelimitset.Thetroublewiththisprocessisthat,althoughitcancrudelyportrayanylimitset,includingchaotic,non-discreteexamples,theasymptoticiteratesaredistributedexponentiallyunevenly,sothatsomepartsofthelimitsetarevirtuallynevervisitedandthelimitsetisincompletelyandonlyveryapproximatelyportrayed,asinfig6(d).Bycontrastwiththenon-linearfunctionsofJuliasets,wheredepictionalgorithms,suchasmodifiedinverseiterationfig1(g)anddistanceestimation,dependonanalyticfeatures,suchasderivativesandpotentialfunctions,anaccuratedescriptionofKleiniangrouplimitsetsrequirestakingfullstrategicadvantageoftheunderlyingalgebraicpropertiesofthegrouptransformations.DescendingtheSpiralLabyrinth:Thedepth-firstsearchalgorithmIP141todepictthelimitsetsisingeniousandextremelyelegant.Theaimistotraversethealgebraicspaceofallwordcombinationsofthegeneratorsa,b,a-1,b-1inawaywhichdrawsacontinuouspiecewiselinearapproximationtothefractalofanydesiredresolution.Todothis,wegenerateacorkscrewmaze,wherewetraversedeeperanddeeperlayersofthesearchtree,turningrightateachdescentandthenentereachtunnelsuccessively,movinganticlockwisearoundthegenerators,andretreatingwhenwereachtheinverseofthemapweenteredby,afterthreeleftturns,sincethemapanditsinversecancel.Ateachnodeweretainarecordofourjourneydown,ourAriadne’sthread,bymultiplyingthesuccessivematricesaswedescend,andmakeacriticaltest:Theparaboliccommutatorhasasinglefixedpoint,whichrepresentsaninfinitelimitofcyclicrepeatsofthegenerators.Ifweapplythecomposedorbitmatrixtothefixedpointofthe
clockwisecommutatoranddothesamefortheanti-clockwisecommutator’sfixedpointandthesetwoarewithinanepsilonthreshold,thismeansgoingthe‘oppositeway’aroundthelocalfractalleavesuswithinresolution,sowedrawalinebetweenthetwoandterminatethedescent,retreatingandturningintovacanttunnelsandexploringthem,untilwefindourselvesbackattherootofthetree.Usingthefixedpointshastheeffectofproducingatheoreticallyinfiniteorbitoftransformationsthatcancarryustoanypartofthelimitset,andbecausewearetraversinggeneratorspacesystematically,wewilltraversepartsofthelimitsetthatarevisitedexponentiallyrarelyinarandomprocess.NavigatingtheMaskitSlice:IfwefocusonlimitsetswherebisparabolicwithTb=2,andonlyTavaries,thisgivesusacomplexparameterplanecalledtheMaskitsliceIP287,fig1(f),forminganatlasoflimitsets,justastheMandelbrotsetdoesforJuliasets,sowecanexploreandclassifytheirproperties.Todefinelocationsontheslice,weneedtobeabletodeterminethetracesTathatcorrespondtorationalcasesinvolvingfractionalmotion.Keytothisarethegeneratorwordsformedbythemaps.Forexample3/10infig3(c)hasaparabolicgeneratorworda3Ba4Ba3Bsymbolicallyexpressingthe3/10ratio.Forgeneralfractionalvaluesp/rwecandeterminethetracesrecursivelyfromtwokeyrelations–extendedMarkovandGrandfatherIP192:(a)Taba-1b-1=T2a+T2B+T2aB–Ta.TB.TaB–2and(b)Tmn=Tm.Tn–Tm-1n.
Fig3:(a)Apolloniangasketfraction0/1Ta=2,shadedtohighlightseparatecomponentsinthemapping,(b)the1/15limitsetwiththe15stepsoftheparabolicword
a15Binred.Intriguingly,as1/n–>0thelimitsetsdonottendinlimittothegasket,althoughtheirtracesandmatricesdo,astheycontainanadditionalfourfold
symmetry.(c)The3/10setcolouredbythelastiterate.(d)Afreediscretegroup.(e)Singly-degenerateLRsetof
theGoldenmean.(f)L10Rspiraldegeneracy.
WecanusetheserelationstosuccessivelymovedowntheFareytreeoffractionsIP291,seefig1(e),becauseneighboursonthetreehavegeneratorwordscombiningviaFareymediantswp+q/r+s=wp/rwq/s,andsothegrandfatheridentityimpliesTwp+q/r+s=Twp/rTwq/s–Twp–q/r–sbecause,in
thelastterm,thecombinedwordm-1nhasneatlycancellinggeneratorsinthemiddle.SinceTaba-1b-1=–2in(a),forp/r,thisreducestosolvinganr-thdegreepolynomial.Ifweplotallthepolynomialsolutionstothep/rlimitsets,forr≤nwehavetheMaskitsliceIP287,fig1(f),whoseupperboundarycontainseachofthep/rtracevaluesasfractalcusps.Aboveandontheboundaryarediscretegroups,whichgeneratefractaltilingsofthecomplementinR,whilevalueswithinthesliceproducenon-freegroupsorchaoticnon-discretesets,whichcangenrallybeportrayedstochastically,fig6(d),butnotbydepth-firstsearch.TherationalcuspsIP273provideintriguingexamplesofcirclepackingswithdeeplinkstohyperbolicgeometry.However,irrationallydegeneratespace-fillinglimitsetsremainedenigmatic,untilakeyexampletracevaluewasfoundIP314,namedafterTroelsJørgensen,whopositedtheexistenceofsuchlimitsets5.Thesolutioniscompletelynatural–theGoldenmeanlimitoftheFibonaccifractionsarisingfromarepeatedLRmoveontheFareytreeresultinginthelimitsetoffig3(e).WecanapproachsuchvaluesbyuseofNewton’smethodtosolvetheequationsoftheascendingfractions.Thisleadsontofurthercases,suchasspiraldegeneracyIP320,wherewerepeatatentoonesetofmoves,L10R,upthelefthandslopeoftheMaskitsliceupperboundary,andintoaspiralingsequenceofeversmallercuspsintheslice,givingrisetothelimitsetoffig3(f).Sinceirrationalnumbersareuncountable,suchlimitsetsaretheoverwhelmingmajorityofcases,althoughhardertoaccess.DoubledegeneracyattheEdgeofChaos:Theaboveexamplesaresinglydegenerate,becauseonlyhalftheregionisengulfed,andthequestionarises,isitpossibletofindapairoftracesgeneratingalimitsetwhichisentirelyspace-filling,permeatingthecomplexplanewithafractaldimensionof2?ThisquestbecameamathematicalepiphanyIP331.ItispossibletogeneratealimitsetconjugatetoTroels’exampleinthefollowingway.TomakeaLRjourneydowntheFareytreetoahigherFibonaccifraction,suchas987/1597,aboutthemiddle,wefind21/34and13/21.ItturnsoutthatthewordsintermsofaandBforthesestepsalsoformgeneratorswhichcanbeusedinreversetodefineaandB.IfweapplyGrandma’salgorithmtotheirtracescalculatedbymatrixmultiplicationoftheaandBwords,wegetanotherlimitsetconjugatetotheoriginal,asinfig4(a).ThereasoningthatthesetwomidpointtraceswereverysimilarledtotheideathatadoublydegeneratelimitsetcorrespondingtotheGoldenmeanmightarisefromsimplyapplyingendlessLRFareymoves
(a,B)→L(a,aB)→
R(a2B,aB) ,whichleavethe
generatorsunchanged:(a,B)=(a2B,aB),givingtwoequationswhich,combinedwithGrandfatherandMarkovleadstotheconjugatetracesolutions(3±31/2i)/2,asshowninfig4(b).
Fig4:(a)LimitsetconjugatetotheGoldenmeanlimitset.(b)Thedoublydegeneratespace-fillinglimitsetoffractaldimension2,colouredbythethirditeratetohighlightthefractallineconnectingcentres.(c)Figure8knot,whose3Dcomplementisgluedfromhyperbolic3-spaceby(b).
Anintriguingfeatureisthat,inadditiontoa,b,thereisathirdinducedparabolicsymmetryc,becausethe(LR)∞movemustarisefromaMöbiusmapconjugatingthelimitsettoitself,whichhasthesamefixedpointastheparaboliccommutator,andconjugateswithit,althoughperformingadistinct'translation'flippingthesetsboundingthejaggedlineconnectingthecentres.RobertRiley6hasshownthatthe3transformationsresultinagluingof3Dhyperbolicspace,viathegroup’sdiscrete3D‘tiling’,tobecomethe3-spherewiththefigure8knot(fig4inset)removedIP388.
Fig5:The‘HolyGrail’–doublespiraldegeneracy,with
thetracesreversedinthesecondimage.Theconvergencetospace-fillingisveryslowandthefullresolutionimagetook29hoursingenericCandwouldhavetaken112daysininterpretedMatlabcode.Althoughthetracesarenolongerconjugates,thesetsareclearlydegenerateonbothcomplementcomponentsinthesamewayasinfig4.
Tounravelthespirallydegeneratecase,Iexaminedtherecursiverelationsfromtheendless(L10R)∞moveandfoundtheycouldbeexploitedtoproduceapairoftracesbysolvingasystemof11equations:Ta=1.936872603726+0.021074052017i,andTb=1.878060670271–1.955723310188i.WhentheseareinputintotheDFSalgorithm,wehavethedoublespiraldegeneracyshowninfig5.TheBondageofRelationships:ThereisafurtherenchantingcollectionoflimitsetsIP353,wherethegroupisnon–freeandhasacommutatorthat
squarestotheidentity–(aba–1b–1)2=I.Thiscausesthetracetobe0,butanextendedgrandma’salgorithmIP261applies,therecursiverelations(a)and(b)canbesolvedtogiveapolynomial,andthelimitsetscanbedepictedbyDFS,aslongasgeneratorwordsthatshort-circuittotheidentityaretreatedasdead-ends.Thisrequiresanautomaticgroup–involvingalook-uptable(automaton)whereallthegrowingwordstringsthatcouldendinashort-circuit,aswedescendthesearchtree,areaccounted,byiterativelycomposingtheirstatesandexitingwhenadeathstateisreached.Infig6(a-c),arethreesuchlimitsets,withthe“inner”regionshighlightedinyellowtodistinguishthemfromthecomplementarywhiteregions,fromwhichtheyareisolatedbythelimitset.
Fig6:Non-freelimitsets,(a)Ta(1/60)Tb=2,(b)aquasi-Fuchsian“dragon”(c)L10Rspirallydegenerate.(d):A
successionofnon-freeandnon-discretechaoticlimitsetswhereTa=Tb=2underavaryingcommutatortrace,
usingthestochasticalgorithm.FallingintotheChaoticAbyss:Onecanalsofreelyexplorethewilderlimitsets,usingthestochasticalgorithm,whichrecursivelymapschosenpoints,suchasMöbiustransformationfixedpoints,usingarandomsequenceofthefourgenerators.Thiscanvisualizelimitsetsinlowerfidelitywithoutanyrestriction,includingnon-free,andnon-discretechaoticsystems,asillustratedinfig6(d).References:1ChrisKing2011FractalGeographyoftheRiemannZetaFunctionarXiv:1103.5274.2RobertBrooksandPeterMatelski,Thedynamicsof2-generatorsubgroupsofPSL(2,C),inIrwinKra,ed.1981RiemannSurfacesandRelatedTopics:Proceedingsofthe1978StonyBrookConference,PrincetonUniversity.3ChrisKing2018KleinianandQuasi-FuchsianLimitSets:AnOpenSourceToolboxhttp://dhushara.com/quasi/4DavidMumford,CarolineSeriesandDavidWright2002Indra’sPearls:TheVisionofFelixKleinCambridgeISBN978-0-521-35253-6.5TroelsJorgensen1977Compact3-ManifoldsofConstantNegativeCurvatureFiberingOvertheCircleAnn.Math.Ser2,106(1)61-72.6RobertRiley2013ApersonalaccountofthediscoveryofhyperbolicstructuresonsomeknotcomplementsExpo.Math.31104-115doi:10.1016/j.exmath.2013.01.003.
