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1、FrontiersincomplexdynamicsCurtisT.McMullen∗MathematicsDepartmentUniversityofCaliforniaBerkeleyCA94720December10,20061IntroductionRationalmapsontheRiemannsphereoccupyadistinguishednicheinthegeneraltheoryofsmoothdynamicalsystems.First,rationalmapsarecomplex-analytic,soabroadspectrumo
2、ftechniquescancontributetotheirstudy(quasiconformalmappings,potentialtheory,algebraicgeometry,etc.)Therationalmapsofagivendegreeformafinitedimensionalmanifold,soex-plorationofthisparameterspaceisespeciallytractable.Finally,someoftheconjecturesonceproposedforsmoothdynamicalsystems(an
3、dnowknowntobefalse),seemtohaveadefinitechanceofholdinginthearenaofrationalmaps.Inthisarticlewesurveyasmallconstellationofsuchconjectures,cen-teringaroundthedensityofhyperbolicrationalmaps—thosewhicharedynamicallythebestbehaved.Wediscusssomeoftheevidenceandlogicunderlyingtheseconject
4、ures,andsketchrecentprogresstowardstheirreso-lution.∗BasedonalecturepresentedtotheAMS-CMS-MAAjointmeeting,VancouverBC,August16,1993.SupportedinpartbytheNSF.1991MathematicsSubjectionClassifica-tion.Primary30D05,58F23.1Ourpresentationentailsonlyabriefaccountofthebasicsofcomplexdynamic
5、s;amoresystematicexpositioncanbefoundinthesurveyarticles[Dou1],[Bl],and[EL],therecentbooks[Bea]and[CG],andMilnor’slecturenotes[Mil4].2HyperbolicrationalmapsArationalmapf:Cb→CbisaholomorphicdynamicalsystemontheRie-mannsphereCb=C∪{∞}.AnysuchmapcanbewrittenasaquotientP(z)azd+...+a0df(
6、z)==Q(z)b0zd+...+bdoftworelativeprimepolynomialsPandQ.Thedegreeoffcanbedefinedtopologicallyoralgebraically;itisthenumberofpreimagesofatypicalpointz,aswellasthemaximumofthedegreesofPandQ.Thefundamentalprobleminthedynamicsofrationalmapsistounderstandthebehaviorofhighiteratesnf(z)=(f◦f
7、◦...◦f)(z).
8、{z}ntimesAnyrationalmapofdegreed>1hasbothexpandingandcontractingfeatures.Forexample,fmustbeexpandingonaverage,becauseitmapstheRiemannsphereoveritselfdtimes.Indeed,withrespecttothesphericalmetric(normalizedtohavetotalareaone),Zn02nk(f)k=d→∞,Cbsothederivativeoffnisverylar
9、geonaverage.Ontheotherhand
