Table Of ContentAlgorithms 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
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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
