MarcoAbate HolomorphicDynamicsonHyperbolicRiemannSurfaces De Gruyter Studies in Mathematics | Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob,Swansea, United Kingdom Zenghu Li, Beijing, China Guozhen Lu,Storrs, USA Karl-Hermann Neeb, Erlangen, Germany René L.Schilling, Dresden, Germany Volkmar Welker, Marburg, Germany Volume 89 Marco Abate Holomorphic Dynamics on Hyperbolic Riemann Surfaces | MathematicsSubjectClassification2020 Primary:37F99;Secondary:30C80;30F45;30J99;37F44 Author Prof.Dr.MarcoAbate UniversitàdiPisa DipartimentodiMatematica LargoPontecorvo5 56127Pisa Italy [email protected] ISBN978-3-11-060105-3 e-ISBN(PDF)978-3-11-060197-8 e-ISBN(EPUB)978-3-11-059874-2 ISSN0179-0986 LibraryofCongressControlNumber:2022944255 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2023WalterdeGruyterGmbH,Berlin/Boston Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksGmbH,Leck www.degruyter.com Contents Introduction|VII 1 TheSchwarzlemmaandRiemannsurfaces|1 1.1 TheSchwarz–Picklemma|2 1.2 ThePoincarédistance|8 1.3 Theupperhalf-plane|14 1.4 Fixedpointsofautomorphisms|19 1.5 MultipointSchwarz–Picklemmas|28 1.6 Riemannsurfaces|36 1.7 HyperbolicRiemannsurfacesandtheMonteltheorem|54 1.8 Boundarybehavioroftheuniversalcoveringmap|67 1.9 ThePoincarémetric|73 1.10 TheAhlforslemma|84 1.11 Blochdomains|88 2 BoundarySchwarzlemmas|96 2.1 TheJulialemma|97 2.2 Stolzregionsandnontangentiallimits|113 2.3 TheJulia–Wolff–Carathéodorytheorem|125 2.4 TheLindelöftheorem|136 2.5 TheWolfflemma|143 2.6 TheautomorphismgroupofhyperbolicRiemannsurfaces|147 2.7 TheBurns–Krantztheorem|153 3 DiscretedynamicsonRiemannsurfaces|158 3.1 Thefixed-pointcase|159 3.2 TheWolff–Denjoytheorem|166 3.3 TheHeinstheorem|169 3.4 StabilityoftheWolffpoint|182 3.5 ModelsonRiemannsurfaces|184 3.6 RandomiterationonBlochdomains|192 3.7 Randomiterationofsmallperturbations|201 4 Discretedynamicsontheunitdisk|212 4.1 Ellipticdynamics|213 4.2 Superattractingdynamics|217 4.3 Hyperbolicdynamics|221 4.4 Parabolicdynamics|226 4.5 Modelsontheunitdisk|234 4.6 Thehyperbolicstep|240 VI | Contents 4.7 Parabolictypeandboundarysmoothness|251 4.8 Boundaryfixedpoints|260 4.9 Backwarddynamics|266 4.10 Commutingfunctions|286 5 ContinuousdynamicsonRiemannsurfaces|294 5.1 Algebraicsemigrouphomomorphisms|295 5.2 One-parametersemigroups|297 5.3 One-parametersemigroupsonRiemannsurfaces|300 5.4 Theinfinitesimalgenerator|304 5.5 ThecontinuousWolff–Denjoytheorem|313 5.6 TheBerkson–Portaformula|315 5.7 One-parametersemigroupsontheunitdisk|320 A Appendix|325 A.1 TheHurwitztheorems|325 A.2 TheFatouuniquenesstheorem|326 A.3 Holomorphicfunctionswithnonnegativerealpart|328 A.4 Sequences|331 A.5 Topologicalgroups|334 Bibliography|337 Index|353 Introduction Inthelast50years,dynamicalsystemshavebecomeoneofthemainobjectsofstudy in mathematics, with many applications outside mathematics. It is a huge subject that can be considered from many points of view. There are discrete and continu- ousdynamicalsystems;therearelocalandglobaldynamicalsystems;thereareone- dimensionaldynamicalsystemsandinfinitely-dimensionaldynamicalsystems;there aremeasurabledynamicalsystems,topologicaldynamicalsystems,smoothdynami- calsystems—andthereareholomorphicdynamicalsystems. Thisbookisdevotedtoa(relativelysmall)portionofthe(quitevast)areaofholo- morphic dynamical systems: one-dimensional dynamical systems on Riemann sur- faces; more specifically, on hyperbolic Riemann surfaces. The investigation of one- dimensionalholomorphicdynamicalsystemsstartedinthesecondhalfofthenine- teenthcentury,moreorlessinthesameyearswhenPoincarébegantounderstandthe importanceofdynamicalsystemsandstartedtoinvestigatetheminearnest.About 150yearslater,thefieldofone-dimensionalholomorphicdynamicalsystemsisstill averyactiveareaofresearch,bothonnonhyperbolicRiemannsurfaces(mainlythe ℂ̂ ℂ Riemann sphere and the complexplane ) and on hyperbolic Riemann surfaces 𝔻 (the unit disk and all Riemann surfaces whose universal cover is the disk), with severalnewpapersappearingeveryyear.Therearemanybooksdescribingthebasics ofholomorphicdynamicsontheRiemannsphere(see,e.g.,[287]);ontheotherhand, theonlybookdevotedtoholomorphicdynamicsonhyperbolicRiemannsurfacesas farasIknowis[3],thathasbeenoutofprintsinceatleast20yearsago(moreabout this later). By the way, I do not know of any introductory book on the dynamics of holomorphicfunctionsontheplane,acuriousholeintheliterature. Letmenowdescribeabitmorepreciselywhatthisbookisabout.Adiscreteholo- morphicdynamicalsystemisgivenbyaholomorphicself-mapf ofacomplexman- ifold M. (A continuous holomorphic dynamical system is instead usually given by a holomorphicvectorfieldonacomplexmanifold;inthisbook,however,weshalltake aslightlydifferentpointofview,asIshallexplainbelow.)Asoftenhappenswithdy- namicalsystems,theobjectofstudyisclassical,inthiscaseholomorphicmaps;itis thekindofquestionsthatoneasksontheseobjectsthatcharacterizethefield.Namely, weassociatetof thesequence{fν}ofiteratesoff,wherefνisthecompositionoff with itselfν ∈ℕtimes;wealsoassociatetoeachpointz ∈Xitsorbit{fν(z)}.Indynamical systemswearetheninterestedintheasymptoticbehaviorofthesequenceofiterates, thatis,whathappensasνgoestoinfinity:isthesequenceconvergent?Ifitisnotcon- vergent,canweanywaydescribethesetofaccumulationpoints?Whataboutsingle orbits,dotheyallhavethesamebehaviorordifferentpointsthatcanbehavediffer- ently?Whataboutstability,thatis,whathappensifweperturbthestartingpointof theorbitoreventheoriginalfunctions?Dochaoticbehaviorsappear?Andsoon.This bookshalltryandgivesomeanswersfordynamicalsystemsdefinedonhyperbolic Riemannsurfaces.Inthiscase,theMonteltheorempreventstheappearanceofchaos; https://doi.org/10.1515/9783110601978-201 VIII | Introduction so, as we shall see, the flavor of the theory and the kind of results we shall obtain isquitedifferentfromthecaseofholomorphicdynamicalsystemsonnonhyperbolic Riemannsurfaces—eventhoughrecentlyithasbeendiscoveredthatthetheorydevel- opedforhyperbolicRiemannsurfacescanbeusefulforunderstandingthebehavior ofdynamicalsystemsinthecomplexplane(see,e.g.,[38]). As anticipated above, the investigation of this subject began with the works of Schröder[368,369]in1870andKœnigs[244]in1883.Theyweremainlyinterestedin thelocalsituationforholomorphicfunctionsofonevariable.Letz beapointofthe 0 ℂ complexplane andf aholomorphicfunctiondefinedinaneighborhoodofz such 0 ( )= thatf z z .Thenthebehaviorofthesequenceofiteratesoff nearz dependson 0 0 0 | ′( )|< thevalueofthederivativeoff atz .Morespecifically,if f z 1everypointzsuffi- 0 0 cientlyclosetoz isattractedbyz (i.e.,fν(z)→z asν→+∞)whileif|f′(z )|>1the 0 0 0 0 pointsarerepelledawayfromz —or,ifyouprefer,theyareattractedbyz underthe 0 0 actionoff−1,whichisdefinedinaneighborhoodofz .Finally,if|f′(z )|=1(andthere 0 0 isaboundedneighbourhoodofz sentintoitselfbyf;otherwisemorecomplicated 0 thingscanhappen),thebehaviorof{fν}iscyclic,withafiniteperiodiff′(z )isaroot 0 ofunity.AsweshallseeinChapter4,thislocalbehaviorhasglobalrepercussions;in <| ′( )|< averyprecisesense,if0 f z 1thenthelinearmapgivenbythemultiplication 0 ′( ) byf z isagoodmodelforthedynamicsoff.In1904,Böttcher[71]wasabletogive 0 ′( ) = amodelalsowhenf z 0.Furthermore,forglobalmapsinhyperbolicRiemann 0 | ′( )|≤ | ′( )|= surfacesnecessarily f z 1andwhen f z 1thenf isanautomorphismwith 0 0 simpledynamics;soinourcontext,themoreinterestingcaseiswhenf hasnofixed points. The first really deep work on global holomorphic dynamical systems has been donebyJulia[215]in1918.Heinvestigatedthedynamicsofrationalfunctionsdefined ℂ̂ ontheRiemannsphere anddiscoveredthattheglobalbehaviorofthesequenceof iteratesisbothcomplicatedandfascinating.Nearfixedpointsitispossibletoadapt andclarifythelocaldescription,butnewphenomenaarise,linkedforinstancetothe distributionofperiodicpoints(i.e.,fixedpointsoffνwithν>1).Amainproblemwas ∈ ℂ̂ thedescriptionoftheJuliaset off,thatis,ofthesetofpointsz suchthatthe 0 sequenceofiterates{fν}isnotequicontinuousinanyneighborhoodofz .Theidea 0 isthatif{fν}isequicontinuousinaneighborhoodofz thenthereisasubsequence 0 {fνk}converginguniformlynearz andthenthebehaviorofthesequenceofiteratesis 0 somehowundercontrol.Inotherwords,theJuliasetisinsomesensethesingularset fortheasymptoticbehaviorof{fν};itisthesetwherechaoticbehaviorappears. Slightlylater,inaseriesofpapersFatou[146–148]extendedanddeepenedJulia’s work,alsoinvestigatingholomorphicdynamicalsystemsonthecomplexplanegener- atedbytranscendentalentirefunctions[149].Again,amainroleisplayedbytheJulia set,definedreplacingthenotionofequicontinuitybythenotionofnormality:theFa- touset,thecomplementoftheJuliaset,isthelargestopensubsetwherethesequence ofiteratesisnormalinthesenseofMontel. Introduction | IX AfterFatou,thestudyofdynamicalsystemsgeneratedbyrationalandentirefunc- tionsmomentarilylostitsimpetus.BesidestheworksofCremer[132,133],Siegel[378], Töpfer[391],andBaker[29–32],mainlydevotedtothestudyofperiodicpoints,both locallyandgloballybyusingNevanlinna’sdistributionvaluetheory,andBrolin[86], devotedtoadeepinvestigationoftheiterationofpolynomialsoflowdegree,anda fewothers,nothingreallynewappeared. Thesituationchangedcompletelyinthe1970sand1980swhentheworkofBr- juno,Hermann,Sullivan,Douady,Hubbard,andmanyothersshedacompletelynew lightonthetopic,showingitsdeeprelationshipwiththetheoryofquasi-conformal mappingsandopeningthegatesforafloodofexcitingnewanddeepresultsthatis stillgoingonnowadays,thankstosomanymathematicians(includingsomeFields medalists)thatitisimpossibletolisttheirnameshere. However,thisisnotthesubjectofthisbook.Ashintedabove,amainsourceof ℂ̂ ℂ complexityinthestudyofholomorphicdynamicalsystemson and isthatthese- quenceofiteratesisnotnormaleverywhere,andthuschaosappears.Ontheother hand,theMonteltheoremimpliesthatonhyperbolicRiemannsurfacesthewholese- quence of iterates is normal everywhere. This completely changes the situation. In fact,bynormality,thesequenceofiteratesisrelativelycompactinasuitablefunc- tionspaceandthecompactnesshasstrongconsequencesonthedynamicsoff.For instance,asmentionedbefore,iff isaholomorphicself-mapofahyperbolicRiemann | ′( )|≤ | ′( )|= surfacewithafixed-pointz ,then f z 1;moreover, f z 1ifandonlyiff is 0 0 0 ′( ) = anautomorphismandf z 1ifandonlyiff istheidentity.Thiscanbeobtained 0 bynoticingthatthesequenceofiteratesshouldhaveaconvergingsubsequenceand, therefore,thecoefficientsoftheTaylorexpansionoffνatz cannottendtoinfinityas 0 ν→+∞;since(fν)′(z )=f′(z )ν,weget|f′(z )|≤1andfromthisitisnottoodifficult 0 0 0 toprovetherestoftheassertion(seeTheorem3.1.10).Itshouldberemarkedthatthe strengthofthisapproachwascompletelyunderstoodonlyafteritsapplication(due toH.Cartan[104,105]andtoCarathéodory[98]inthe1930s)tothetheoryofholo- morphicmapsofseveralcomplexvariables,probablybecauseinonevariableitwas initiallysomehowconcealedbytheSchwarz–Picklemma. Thuswehavethehopetobeabletounderstandtheholomorphicdynamicson hyperbolic Riemann surfaces by using the Montel theorem and the Schwarz–Pick 𝔻 lemma.AsalreadyremarkedbyJulia[215],iff isaholomorphicfunctionof into itselfwithafixed-pointz ∈ 𝔻,thenthebehaviorof{fν}canbeeasilyderivedbythe 0 | ′( )|< Schwarz–Picklemma:if f z 1,thenz isgloballyattractive(andnotjustlocally 0 0 | ′( )| = attractiveasalreadyprovedbyKœnigs)andif f 0 1thenf isanon-Euclidean rotationaboutz . 0 Thenewideasneededtostudywhathappenswhenf hasnofixedpointswere ∈𝜕𝔻 ∈𝔻 providedbyWolff[414–416]andDenjoy[135]in1926.Letτ ;thenasz tends toτ,thePoincarédisksofcenterzandfixedEuclideanradiustendtoahorocycleatτ, 𝜕𝔻 thatistoanEuclideandiskinternallytangentto atτ.ThenWolffprovedasortof Schwarzlemmaforholomorphicfunctionswithoutfixedpoints,usingthehorocycles: