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Algorithms and Computation in Mathematics Volume 23 • Editors ArjehM.Cohen HenriCohen DavidEisenbud MichaelF.Singer BerndSturmfels Mark Braverman Michael Yampolsky (cid:127) Computability of Julia Sets y With31Figures ABC Authors Mark Braverman Michael Yampolsky Department of Computer Science Department of Mathematics University of Toronto University of Toronto Sandford Fleming Building 40 St George Street 10 King’s College Road Toronto, ON Toronto, ON Canada M5S 2E4 Canada M5S 3G4 [email protected] [email protected] LibraryofCongressControlNumber:2008935319 MathematicsSubjectClassification(2000):03D15, 03D80, 37F10, 37F50, 65Y20, 68Q05, 68Q17 ISSN1431-1550 ISBN 978-3-540-68546-3 e-ISBN 978-3-540-68547-0 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c 2009 Springer-VerlagBerlinHeidelberg Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorandSPi usingaSpringerLATEXmacropackage Coverdesign:the cover is designed by deblik, Berlin Printedonacid-freepaper SPIN:12228874 46/SPi 543210 To ourfamilies Preface Among all computer-generated mathematical images, Julia sets of rational maps occupy,perhaps,themostprominentposition.Theirbeautyandcomplexitycanbe fascinating.Theyalsoholdadeepmathematicalcontent,andnumericalexperiments havebecomeadefiningfeatureofthesubjectofComplexDynamics. ComputationalhardnessofJuliasetsisthemainsubjectofthisbook.Bydefini- tion,acomputablesetintheplanecanbevisualizedonacomputerscreenwithan arbitrarilyhighmagnification.Inthisdefinitiontherunningtimeofthevisualization algorithmisnotlimited. CountlessprogramstovisualizeJuliasetshavebeenwritten.Yet,aswewillsee, itispossibletoconstructivelyproduceexamplesofquadraticpolynomials,whose Juliasetsarenotcomputable. In a way, this resultis striking – it says thatwhile a dynamicalsystem can be de- scribednumericallywithanarbitraryprecision,thepictureofthedynamicscannot bevisualized. Asoneindicationofhowunusualthisis,considerthefollowing.Anotherinter- estingobjectforaquadraticpolynomialisitsfilledJuliaset.Itisobtainedby“filling in”alltheholesintheJuliasetitself.Indoingthis,thecomputablepropertiesofthe picturecanchangedramatically: afilledJuliasetisalwayscomputable. Thenon-computabilityphenomenonisverysubtle,andindescribingitwewillre- quireaverypreciseanalyticmachinery.Manyofthetechniquesweuse haveonly become available in the last few years. Perversely, we are able to construct non- computableexamplesofJuliasetsbecauseweunderstandJuliasetssowell. Non-computability turns out to be rare. Most Julia sets are computable. Their computationalhardness,however,mayvary.Therunningtimerequiredtoproduce ahigh-resolutionimageofacomputableJuliasetmaybeprohibitivelyhigh.Already we have seen some furthersurprises– a class of Julia sets (Julia sets of quadratic polynomialswithparabolicorbits)empiricallythoughtofashardtocomputeturns outtobeeasy(andwithapracticalalgorithm). vii viii Preface Our understanding of the computational complexity of Julia sets is in its first stages. Examplesof a truly pathologicalkind (Julia sets of quadratic polynomials withCremerperiodicorbits)turnouttoalwaysbecomputable.Noinformativepic- turesofthistypehaveeverbeenproduced,astherunningtimeofallpresentlyexist- ingalgorithmsrendersthemimpractical.However,itisnotknowniftheyareever computationallyhard.Thisisprobablythecaseatleastsometimes,butitmayalso be possible thatsome of them can be visualized effectivelyby a cleveralgorithm. Manyinterestingproblemsawaitfurtherstudyhere. The goalofthe presentbookis to summarizeourpresentknowledgeaboutthe computationalpropertiesofJuliasetsinafashionthatisasself-containedaspossi- ble.WhilewehavefoundtheinterplaybetweentheoreticalComputerScienceand DynamicalSystemsextremelyfruitful,itmakesthepresentationmorechallenging. Wehavestriventomakethebookaccessibleandinterestingtoexpertsinbothfields. Thebookassumesnopriorknowledgeofcomputabilitytheory,andonlyabasicfa- miliaritywithcomplexanalysis. Westartthebookwithanintroductiontocomputabilitytheory(Chapter1)anda surveyofthebasicprinciplesofdynamicsofrationalmaps(Chapter2).InChapter 3webeginthestudyofthecomputabilityandcomplexityofJuliasetsbylookingat sometypicalexamples.We discussthegeneralpositiveresultsinChapter4.Non- computabilityappearsinChapter5.Chapter6servestounderstandthetopological structureofnon-computableexamplesinmoredetail. Thematerialweviewas“optionalreading”is typesetlikethis. It is either notdirectly related to the main storyline,or is too technical,and is di- rectedtowardsexpertsinoneofthetwofields. Acknowledgments ItisourpleasuretothankourfriendandcolleagueIliaBinderforthemanyuseful discussions on the computability of Julia sets. Some of the material presented in thisbookisbasedonourjointworks.WearegratefultoJohnMilnorforhisinterest in our work, and many insightful comments and questions, which have inspired muchofourstudy.WethankourcolleagueMichaelShubforusefuldiscussionsof our results. Alan Baker’s comments were very helpful to us in understanding the connectionbetweennumber-theoreticalpropertiesofparametersandcomputability ofJuliasets.M.B.wouldliketothankStephenCookforhisguidanceandforsharing hisinsightsonComplexityTheory.OurcolleagueBorisKhesinwasinstrumentalin gettingthisbooktothepublisher. WethanktheMathematicsandComputerScienceDepartmentsoftheUniversity ofTorontoforthewonderfulworkingenvironment.M.Y.wishestothankIHE´S,and FIM at ETHZ, andboth M.Y. and M.B. thankIPAM at UCLA, for the hospitality extendedtousduringourvisitstotheseinstituteswhileworkingonthisbook.Our researchwaspartiallysupportedbyNSERC,throughitsCanadaGraduateScholar- ship(M.B.)andaDiscoveryGrant(M.Y.). As thisworkwas takingshape,the questions,enthusiasm,and evenskepticism ofourcolleagueshavebeenaninvaluablemotivationforus.Wethankthemall. ix Contents 1 IntroductiontoComputability................................... 1 1.1 Discretecomputabilityandcomplexity......................... 1 1.2 Computabilityandcomplexityofrealnumbersandfunctions...... 5 1.3 ComputabilityandcomplexityofsubsetsofRk.................. 11 1.4 Weaklycomputablesets ..................................... 14 1.5 Set-valuedfunctionsanduniformity........................... 17 2 DynamicsofRationalMappings ................................. 21 2.1 GeneralfactsaboutRiemannsurfacesandthehyperbolicmetric ... 21 2.2 Juliasetsofrationalmappings................................ 27 3 FirstExamples................................................. 37 3.1 Acasestudy:hyperbolicJuliasets ............................ 37 3.2 Mapswithparabolicorbits................................... 49 3.3 ComputingJuliasetswithparabolicorbitsefficiently............. 53 3.4 LackofuniformcomputabilityofJuliasets..................... 60 4 PositiveResults ................................................ 65 4.1 ComputabilityoffilledJuliasets.............................. 65 4.2 Juliasetswithoutrotationdomains ............................ 69 4.3 ComputableJuliasetsofSiegelquadratics...................... 70 4.4 Robustcomputability ....................................... 75 5 NegativeResults ............................................... 81 5.1 SiegeldisksandCremerpoints ............................... 81 5.2 Non-computableJuliasets ................................... 90 5.3 ThecomplexityofJuliasets..................................103 5.4 Proofsofthemaintechnicallemmas...........................107 5.5 Numbertheoryandcomputability.............................111 5.6 Quadraticswithnon-computableJuliasetsarerare...............113 xi xii Contents 6 ComputabilityvsTopology......................................119 6.1 Howcantheboundaryofacomputablesetbenon-computable?....119 6.2 LocallyconnectedquadraticJuliasets .........................120 6.3 LocalconnectednessversuscomputabilityofJθ .................127 6.4 Non-computablelocallyconnectedJuliasets....................136 References.....................................................146 Index .............................................................149 List of Notation B(y,r) theballwithcentery∈Rn andradiusr; B(Y,r) ther-neighborhoodofthesetY inRn; U theunitdiskinC; D thesetofdyadicrationals; Cˆ theRiemannsphere; T thecircleR/Z; M theMandelbrotset; Crit(R) thesetofcriticalpointsofarationalmapR; Postcrit(R) thepostcriticalsetofR; K∗ thesetofallcompactsubsetsofRn; n RC thesetofallcomputablerealnumbers; φ M anoracleTuringMachine; T thecircleR/Z; S1 theunitcircle{|z|=1}⊂C; C thesetoffiniteunionsofcloseddyadicballsinRk; RC thefieldofcomputablerealnumbers; CC thefieldofcomputablecomplexnumbers; fn unlessotherwisespecified,then-thiterate f◦f◦···◦f; (cid:2) (cid:3)(cid:4) (cid:5) n J(R) theJuliasetofthefunctionR; K(p) thefilledJuliasetofthepolynomialp; J theJuliasetJ(z2+c); c K thefilledJuliasetK(z2+c); c B thesetofBrjunonumbers; a (cid:6)b ,or a =O(b )andb =O(a ); n n n n n n a =Θ(b ) n n a (cid:2)b a =O(b ). n n n n xiii

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