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1、August21,201019:11WSPC/S1793-0421203-IJNTS1793042110003356InternationalJournalofNumberTheoryVol.6,No.5(2010)10111025cWorldScientificPublishingCompanyDOI:10.1142/S1793042110003356AFINITENESSPROPERTYFORPREPERIODICPOINTSOFCHEBYSHEVPOLYNOMIALSSU-IONIH∗,‡andTHOMASJ.TUCKER†,§∗De
2、partmentofMathematics,UniversityofColoradoatBoulderCampusBox395,BoulderCO80309-0395,USA†DepartmentofMathematics,UniversityofRochesterRochesterNY14627,USA‡ih@math.colorado.edu§ttucker@math.rochester.eduReceived3March2008Accepted15June2009LetKbeanumberfieldwithalgebraicclosur
3、eK,letSbeafinitesetofplacesofKcontainingtheArchimedeanplaces,andletϕbeaChebyshevpolynomial.Weprovethatifα∈Kisnotpreperiodic,thenthereareonlyfinitelymanypreperiodicpointsβ∈KwhichareS-integralwithrespecttoα.Keywords:Chebyshevpolynomials;equidistribution;integralpoints;preperio
4、dicpoints.MathematicsSubjectClassification2010:11G05,11G35,14G05,37F10,11J86,11J71,11G501.IntroductionLetKbeanumberfieldwithalgebraicclosureK,letSbeafinitesetofplacesofKcontainingtheArchimedeanplaces,andletα,β∈K.WesaythatβisS-integralrelativetoαifnoconjugateofβmeetsanyconjuga
5、teofαatprimeslyingoutsideofS.Moreprecisely,thismeansthatforanyprimev/∈SandanyK-embeddingsσ:K(α)→Kvandτ:K(α)→Kv,wehave
6、σ(β)−τ(α)
7、v≥1if
8、τ(α)
9、v≤1;and
10、σ(β)
11、v≤1if
12、τ(α)
13、v>1.Notethatthisdefinitionextendsnaturallytothecasewhereαisthepointatinfinity.WesaythatβisS-integralrelativetot
14、hepointatinfinityif
15、σ(β)
16、v≤1forallv/∈SandallK-embeddingsσ:K(β)→Kv.Thus,ourS-integralpointscoincidewiththeusualS-integerswhenαisthepointatinfinity.In[4],thefollowingconjectureismade.Conjecture1.0.1.LetKbeanumberfield,andletSbeafinitesetofplacesofKthatcontainsalltheArchimedeanpl
17、aces.Ifϕ:P1→P1isanonconstantKK1011August21,201019:11WSPC/S1793-0421203-IJNTS17930421100033561012S.-I.Ih&T.J.Tuckerrationalfunctionofdegreed>1andα∈P1(K)isnon-preperiodicforϕ,thenthereareatmostfinitelymanypreperiodicpointsβ∈P1(K)thatareS-integralwithrespecttoα.In[4],itisprove
18、dthatthisconjectureholdswhenϕisamultiplication-by-n(forn≥2)maponanGmoronanellipticcurve.R
