资源描述：
《frontiers in complex dynamics》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。
1、FrontiersincomplexdynamicsCurtisT.McMullen∗MathematicsDepartmentUniversityofCaliforniaBerkeleyCA94720December10,20061IntroductionRationalmapsontheRiemannsphereoccupyadistinguishednicheinthegeneraltheoryofsmoothdynamicalsystems.First,rationalmapsarecomplex-analytic,soabroadspect
2、rumoftechniquescancontributetotheirstudy(quasiconformalmappings,potentialtheory,algebraicgeometry,etc.)Therationalmapsofagivendegreeformafinitedimensionalmanifold,soex-plorationofthisparameterspaceisespeciallytractable.Finally,someoftheconjecturesonceproposedforsmoothdynamicalsy
3、stems(andnowknowntobefalse),seemtohaveadefinitechanceofholdinginthearenaofrationalmaps.Inthisarticlewesurveyasmallconstellationofsuchconjectures,cen-teringaroundthedensityofhyperbolicrationalmaps—thosewhicharedynamicallythebestbehaved.Wediscusssomeoftheevidenceandlogicunderlying
4、theseconjectures,andsketchrecentprogresstowardstheirreso-lution.∗BasedonalecturepresentedtotheAMS-CMS-MAAjointmeeting,VancouverBC,August16,1993.SupportedinpartbytheNSF.1991MathematicsSubjectionClassifica-tion.Primary30D05,58F23.1Ourpresentationentailsonlyabriefaccountofthebasics
5、ofcomplexdynamics;amoresystematicexpositioncanbefoundinthesurveyarticles[Dou1],[Bl],and[EL],therecentbooks[Bea]and[CG],andMilnor’slecturenotes[Mil4].2HyperbolicrationalmapsArationalmapf:Cb→CbisaholomorphicdynamicalsystemontheRie-mannsphereCb=C∪{∞}.Anysuchmapcanbewrittenasaquoti
6、entP(z)azd+...+a0df(z)==Q(z)b0zd+...+bdoftworelativeprimepolynomialsPandQ.Thedegreeoffcanbedefinedtopologicallyoralgebraically;itisthenumberofpreimagesofatypicalpointz,aswellasthemaximumofthedegreesofPandQ.Thefundamentalprobleminthedynamicsofrationalmapsistounderstandthebehavior
7、ofhighiteratesnf(z)=(f◦f◦...◦f)(z).
8、{z}ntimesAnyrationalmapofdegreed>1hasbothexpandingandcontractingfeatures.Forexample,fmustbeexpandingonaverage,becauseitmapstheRiemannsphereoveritselfdtimes.Indeed,withrespecttothesphericalmetric(normalizedtohavetotalareaone),Zn02nk(f)k=d→∞,Cb
9、sothederivativeoffnisverylargeonaverage.Ontheotherhand
