noColor January6,2014 7x10 Frontiers in Complex Dynamics noColor January6,2014 7x10 PrincetonMathematicalSeries EDITORS:PHILLIPA.GRIFFITHS,JOHNN.MATHER,ANDELIASM.STEIN 1. TheClassicalGroupsbyHermannWeyl 8. TheoryofLieGroups:IbyC.Chevalley 9. MathematicalMethodsofStatisticsbyHaraldCrame´r 14. TheTopologyofFibreBundlesbyNormanSteenrod 17. IntroductiontoMathematicalLogic,Vol.IbyAlonzoChurch 19. HomologicalAlgebrabyH.CartanandS.Eilenberg 28. ConvexAnalysisbyR.T.Rockafellar 30. SingularIntegralsandDifferentiabilityPropertiesofFunctionsbyE.M.Stein 32. Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss 33. E´taleCohomologybyJ.S.Milne 35. Three-DimensionalGeometryandTopology,Volume1byWilliamP.Thurston. EditedbySilvioLevy 36. Representation Theory of Semisimple Groups: An Overview Based on Ex- amplesbyAnthonyW.Knapp 38. SpinGeometrybyH.BlaineLawson,Jr.,andMarie-LouiseMichelsohn 43. Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory IntegralsbyEliasM.Stein 44. TopicsinErgodicTheorybyYa.G.Sinai 45. CohomologicalInductionandUnitaryRepresentationsbyAnthonyW.Knapp andDavidA.Vogan,Jr. 46. Abelian Varieties with Complex Multiplication and Modular Functions by GoroShimura 47. RealSubmanifoldsinComplexSpaceandTheirMappingsbyM.SalahBaouendi, PeterEbenfelt,andLindaPreissRothschild 48. EllipticPartialDifferentialEquationsandQuasiconformalMappingsinthe PlanebyKariAstala,TadeuszIwaniec,andGavenMartin 49. APrimeronMappingClassGroupsbyBensonFarbandDanMargalit 50. Advances in Analysis: The Legacy of Elias M. Stein edited by Charles Feffer- man,AlexandruIonesco,D.H.Phong,andStephenWainger 51. FrontiersinComplexDynamics:InCelebrationofJohnMilnor’s80thBirth- dayeditedbyAraceliBonifant,MikhailLyubich,andScottSutherland noColor January6,2014 7x10 Frontiers in Complex Dynamics In Celebration of John Milnor’s 80th Birthday Edited by Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD noColor January6,2014 7x10 Copyright(cid:13)c 2014byPrincetonUniversityPress PublishedbyPrincetonUniversityPress 41WilliamStreet,Princeton,NewJersey08540 IntheUnitedKingdom:PrincetonUniversityPress 6OxfordStreet,Woodstock,Oxfordshire,OX201TW AllRightsReserved ISBN978-0-691-15929-4 LibraryofCongressControlNumber:2013944768 BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinLATEX The publisher would like to acknowledge the editors of this volume for providing the camera-readycopyfromwhichthisbookwasprinted. Printedonacid-freepaper.∞ press.princeton.edu PrintedintheUnitedStatesofAmerica 10987654321 noColor January6,2014 7x10 Contents Preface xi Introduction 1 1. Bibliography 8 PartI. OneComplexVariable 13 J.MILNOR: ArithmeticofUnicriticalPolynomialMaps 15 1. Introduction 15 2. Periodicorbits 15 3. Proofs 18 4. Postcriticallyfinitemaps 20 5. Bibliography 23 T. BOUSCH: Les racines des composantes hyperboliques de M sont desquartsd’entiersalge´briques 25 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORIN: Dynamical coresoftopologicalpolynomials 27 1. Introductionandthemainresult 27 2. Preliminaries 29 3. Dynamicalcore 35 4. Bibliography 47 X.BUFFANDTANLEI: Thequadraticdynatomiccurvesaresmooth andirreducible 49 1. Introduction 49 2. Dynatomicpolynomials 50 3. Smoothnessofthedynatomiccurves 53 4. Irreducibilityofthedynatomiccurves 61 5. Bibliography 71 J.H.HUBBARDANDD.SCHLEICHER: Multicornsarenotpathconnected 73 1. Introduction 73 2. Antiholomorphicandparabolicdynamics 77 3. Bifurcationalongarcsandthefixed-pointindex 80 4. Parabolicperturbations 85 5. Parabolictreesandcombinatorics 89 6. Non-pathwiseconnectivity 95 7. Furtherresults 98 8. Bibliography 101 noColor January6,2014 7x10 vi CONTENTS J.KIWI: Leadingmonomialsofescaperegions 103 1. Introduction 103 2. Statementoftheresults 104 3. Puiseuxseriesdynamicsstatement 106 4. One-parameterfamilies 108 5. Bibliography 118 R. L. DEVANEY: LimitingbehaviorofJuliasetsofsingularlyperturbed rationalmaps 121 1. Introduction 121 2. Elementarymappingproperties 123 3. Juliasetsconvergingtotheunitdisk 126 4. Thecasen >2 128 5. Otherc-values 131 6. Bibliography 133 A. DEZOTTI AND P. ROESCH: On (non-)local connectivity of some Juliasets 135 1. Localconnectivity 136 2. Rationalmaps 141 3. Douady-Sullivancriterion 143 4. Thecaseofinfinitelysatellite-renormalizablequadraticpolynomials: amodel 150 5. Bibliography 161 G.LEVIN: Perturbationsofweaklyexpandingcriticalorbits 163 1. Introduction 163 2. Polynomials 167 3. Rationalfunctions 176 4. Part(a)ofTheorem3.6 182 5. Part(b)ofTheorem3.6 189 6. Bibliography 195 D.MEYER: Unmatingofrationalmaps:Sufficientcriteriaandexamples 197 1. Introduction 197 2. Moore’stheorem 199 3. Matingofpolynomials 201 4. Equatorsandhyperbolicrationalmaps 204 5. Anexample 209 6. Asufficientcriterionformating 211 7. Connections 216 8. Criticalportraits 220 9. Unmatingthemap 221 10. Examplesofunmatings 224 11. Amatingnotarisingfromapseudo-equator 228 12. Openquestions 232 13. Bibliography 232 noColor January6,2014 7x10 CONTENTS vii Y. JIANG: A framework toward understanding the characterization of holomorphicdynamics 235 1. Characterization 235 2. Obstruction 237 3. Review 237 4. Geometry 242 5. Geometrization 248 6. Appendix on Transcendental Functions by Tao Chen, Yunping Jiang,andLindaKeen 252 7. Bibliography 254 PartII. OneRealVariable 259 C.G.MOREIRAANDD.SMANIA: Metricstabilityforrandomwalks (withapplicationsinrenormalizationtheory) 261 1. Introduction 261 2. ExpandingMarkovmaps,randomwalks,andtheirperturbations 265 3. Statementsofresults 269 4. Preliminaries 272 5. Stabilityoftransience 276 6. Stabilityofrecurrence 288 7. Stabilityofthemultifractalspectrum 296 8. Applicationstoone-dimensionalrenormalizationtheory 312 9. Bibliography 321 S. VAN STRIEN: Milnor’s conjecture on monotonicity of topological entropy:Resultsandquestions 323 1. Motivation 323 2. Milnor’smonotonicityofentropyconjecture 324 3. Ideaoftheproof 326 4. Openproblems 331 5. Bibliography 334 W.P.THURSTON: Entropyindimensionone 339 1. Introduction 339 2. Specialcase:Pisotnumbers 346 3. Constructingintervalmaps:Firststeps 351 4. Secondstep:ConstructingamapforλN 355 5. Powersandroots:CompletionofproofofTheorem1.3 355 6. Mapsofasterisks 358 7. Entropyinboundeddegree 359 8. Traintracks 365 9. Splittinghairs 367 10. Dynamicextensions 369 11. Bipositivematrices 370 12. Tracks, doubletracks, zipping and a sketch of the proof of Theorem1.11 374 13. Supplementarynotes(mostlybyJohnMilnor) 376 14. Bibliography 383 noColor January6,2014 7x10 viii CONTENTS PartIII.SeveralComplexVariables 385 M. ARIZZI AND J. RAISSY: OnE´calle-Hakim’stheoremsinholomorphic dynamics 387 1. Introduction 387 2. Notation 389 3. Preliminaries 389 4. Characteristicdirections 390 5. Changesofcoordinates 397 6. Existenceofparaboliccurves 407 7. Existenceofattractingdomains 421 8. Parabolicmanifolds 423 9. Fatoucoordinates 439 10. Fatou-Bieberbachdomains 444 11. Bibliography 447 M. ABATE: Indextheoremsformeromorphicself-mapsoftheprojective space 451 1. Introduction 451 2. Theproof 453 3. Bibliography 460 S.CANTAT: Dynamicsofautomorphismsofcompactcomplexsurfaces 463 1. Introduction 463 2. Hodgetheoryandautomorphisms 466 3. Groupsofautomorphisms 476 4. Periodiccurves,periodicpoints,andtopologicalentropy 478 5. Invariantcurrents 485 6. Entirecurves,stablemanifolds,andlaminarity 489 7. FatouandJuliasets 498 8. Themeasureofmaximalentropyandperiodicpoints 502 9. Complements 505 10. Appendix:Classificationofsurfaces 507 11. Bibliography 509 R. DUJARDIN: Bifurcation currents andequidistribution in parameter space 515 1. Prologue:normalfamilies,currentsandequidistribution 518 2. BifurcationcurrentsforfamiliesofrationalmappingsonP1 524 3. Higherbifurcationcurrentsandthebifurcationmeasure 534 4. BifurcationcurrentsforfamiliesofMo¨biussubgroups 547 5. Furthersettings,finalremarks 557 6. Bibliography 559 PartIV. LaminationsandFoliations 567 T.-C. DINH, V.-A. NGUYEˆN AND N. SIBONY: Entropy for hyperbolic RiemannsurfacelaminationsI 569 1. Introduction 569 noColor January6,2014 7x10 CONTENTS ix 2. Poincare´ metriconlaminations 571 3. Hyperbolicentropyforfoliations 578 4. Entropyofharmonicmeasures 585 5. Bibliography 591 T.-C. DINH, V.-A. NGUYEˆN AND N. SIBONY: Entropy for hyperbolic RiemannsurfacelaminationsII 593 1. Introduction 593 2. Localmodelsforsingularpoints 595 3. Poincare´ metriconleaves 604 4. Finitenessofentropy:thestrategy 611 5. Adaptedtransversalsandtheircoverings 614 6. Finitenessofentropy:endoftheproof 619 7. Bibliography 621 V. MUN˜OZ AND R. PE´REZ-MARCO: Intersection theory for ergodic solenoids 623 1. Introduction 623 2. Measuredsolenoidsandgeneralizedcurrents 624 3. Homotopyofsolenoids 627 4. Intersectiontheoryofsolenoids 630 5. Almosteverywheretransversality 638 6. Intersectionofanalyticsolenoids 641 7. Bibliography 644 H.GARC´IA-COMPEA´N,R.SANTOS-SILVAANDA.VERJOVSKY: Invariants offour-manifoldswithflowsviacohomologicalfieldtheory 645 1. Introduction 645 2. Asymptoticcyclesandcurrents 647 3. Overview of cohomological quantum field theory: Donaldson- Witteninvariants 650 4. Donaldson-Witteninvariantsforflows 653 5. Donaldson-WitteninvariantsforKa¨hlermanifoldswithflows 659 6. SurveyonSeiberg-Witteninvariants 667 7. Seiberg-Witteninvariantsforflows 669 8. Aphysicalinterpretation 672 9. Finalremarks 673 10. Bibliography 674 ColorPlates C-1 PartV. GeometryandAlgebra 677 W. GOLDMAN: Two papers which changed my life: Milnor’s seminal workonflatmanifoldsandbundles 679 1. Introduction 679 2. Gauss-Bonnetbeginnings 679 3. TheMilnor-Woodinequality 682 4. Maximalrepresentations 684