Table Of ContentMarcoAbate
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
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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:
