The Arithmetic of Dynamical Systems

518 Pages·2007·7.76 MB·English

by  Joseph H. Silverman (auth.)

GraduTaetxet s
inM athematics
JosephS iHl.v erman
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TheA rithmetic
DynamicSayls tems
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5 MAcL ANCEa.t egforort ihWeeos r king 38G RAU/EFRIRTTCZHSES.e veCroamlp lex
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6 HuGHEPsI/PPErRo.j ePcltainvees . 39A RVESAOnIN n.v ittaoCt *i-oAnl gebras.
7 SERRA ECo.u risnAe r ithmetic4.0 KEMYEINSLNLEIKNDAePnPu.m erable
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8 TAKE/uZnAIRNAGx.i omSaetTtih ce ory. MarkCohva i2nnsd. ed.
9 HuMPHRIEnYtSr.o dtuoc tALiliogene - 41 APOSTOL. FMuondcutalinaodrn s
braasnR de persentTahteioorny . DiricShelrieinNte u sm bTehre ory.
10C oHEAN C.o urisnSei mpHloem otopy 2nedd .
Theory. 42 J..S- EPRRLEin.e Raerrpe sentoaft ions
CoNAWY.F unctoifOo nneCs o mplex FinGirtoeu ps.
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VairabIl2.en edd . 43 GILLMAN/RJiEnRogIfsS ON.
12B EALASd.v anMcaetdh ematical ContinFuuonucst ions.
Analysis. KENDEIlGe.m enAtlagreyb raic
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13A NDERSON/RFiUnLagLnsEd R . Geometry.
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16W INTTEhRSe.t rucotfFu ireel ds. Mathematicians.
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IntrodtuoMc attihoenm aLtoigciacl. Logic.
23 GREULBi.n eAalrg e4btrehad .. 54 GRAVEARTIKWIC NoSm.b inawtiotrhi cs
24 HoLMGEeSo.m eFturnicct ional Emphaosnti hsTe h eoorfGy r aphs.
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25 HEWI/TTSTRBOEMRRGe.a l Aabnsdt ract OperaTthoeroI rE:yl emeonft s
Analysis. Functional Analysis.
26 MANEASl.g ebTrhaeiocr ies. 56 MASSEY. ATlopgoelboArgnay i:c
27 KELLGEeYn.e Tropaolol gy. Introduction.
28Z ARISKUIE/CLSo.Am Mmutative 57C ROWEL.L /IFnotxrotdou cKtniootn
AlgeVboria..I . Theory.
29Z ARISKUIEICLSo.Am Mmutative 58K oBZLp.I- TaNduimcb epr-sa,d ic
AlgreaVb.oi .II. AnalyasnZides t,a -Fun2cnetdid o.n s.
30J ACOBLSeOcNt.iu nrAe bss tArlagcetb ra5 9L ANCGy.c lotFoimeilcd s.
IB.a sCiocn cepts. 60 ARNOLaDt.hMemaMteitchaoildn s
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IIL.i neAalgre bra. 61W HITEHEElAeDm.eo nfHt osm otopy
32J ACOBLSeOcNt.iu nAr bess trAlagcetb ra Theory.
III. oThfFe ioerlyd Gsa laonids 62 KARPGOALOV/AMKEORVL.Z I
Theroy. Fundameonft aTtlhhsee o orfGy r oups.
33 HIRHS.DC iefrfenTtopiooalgly . 63 BoLLO.GB rAaSpThh eory.
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JoseHp.hS ilverman
TheA rithmoeft ic
DynamiScyaslt ems
Wit1h1I llustrations
�Springer
JoseHpS.hi lverman
DepartomfMe antth ematics
BroUwnn iversity
ProvidReI0n 2c9e1,2
USA
[email protected] ro.ewdnu
EditoBroiaarld
S.A xler K.AR.i bet
MathemDaetpiacrnstt m e MathemDaetpiacersnt tm
San FrancUinsicvoe rSstiattye UniveorfCs ailtiyf Boernrikae,l ey
SaFnr ancCiAs9 c4o1,3 2 BerkeClA9e 4y7,2 0-384
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[email protected] [email protected] eartkehedlue y.
MathemSautbijcCeslc ats sif(2i00c)0a1:t10 i-1o1,nG1 991,4 G9397,1- ,30 7lF0
ISBN-91738:- 0-387-69e9-0I3S-B5N9 -7183-:0 -387-69904-2
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Preface
Thbiosio dske sitpgorn oeavdp i adftroeth h ree aidnetarom aalng oatfmw aot ion
venearraoebmfala set hteiDmcyasn,a SmyiscatanelNdm u sm berM aTnohyfe ory.
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http://www.math.brown.edu/-jhs/ADSHome.html
Acknowledgments
Thaeu thhacosor n sagu rletaestdo muiarwnncr yei stt hibinosgoE kv.ea rtyt empt
habse meandt geoi pvreo aptietbrru ftoairlbo luntt hm eo sstta nrdeasMruudlc ths .
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vi Preface
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supapnopdra tt diuerntichmneeag nh yo uorcsc uiwpnri iettdhbi inosgo k.
JosHeS.pli hv erman
Jan1u,a ry
2007
Contents
Preface v
Introduction 1
Exercises 7
1 AnI ntrodtuoCc ltaisoDsnyi ncaamli cs 9
1.R1a tMioanpaanstld hP ere ocjtLiivn.ee . . . . 9. .
12. CriPtoiiacnnattdlhRs ei enmna-HuFrowrimtuzl a . 12
13. PerPiooidaninMtcdus l tipliers 18
14. ThJeu Sleaitna d tShee.t . Fa.t.ou.. . . . .2.2
15. ProeposePf re triPiooid.ni tc.s . . . . .2 7. . . . .
16. DynaSmyiscAtaselsm tosetc dAoi l ageGbrroauipcs 28
Exer.ci.s.es. . . . . . .. . . . .3.5 . . . . . . .
2 DynaomvieLcros cF aile lGdosoR:de duction 43
2.T1h Neo narcChhiomreddae.la nM. e tric 43
2.P2e rPiooidaninTtcdhs Pe riorep se. r t.i . 47
2.R3e duocPfto iiaonnnMtd as pM so du.l o 48
p
2.T4h Ree suolafRt aatniMtoa pn. a l. . 53
2.R5a tiMoanpasGl ow oRiedtd hu .c tion 58
2.P6e rPiooidaninGtcdos oR de du.c tion 62
27. P erPiooidaninDtcdys n ami.c al Units 69
Exceirs.e.s . . . ... . . . . . . .74
3 DynamoivceGsrl obFaiell ds 81
3.H1e iFguhntc .ti.o.ns. ... . . . . . 81
32. HeightG eFoumne.ctt..ri y.o n s and 89
3.T3h Uen oirfBmo undCeojdcnenteu.sr se 95
34. CanonicaDly nHaeSmiyigschtatelsm sa nd 97
3.L5o cCaaln onic.al. Heights 120
3.D6i ophAapnpitrmioanxte.i .on.. . . . 140
3.I7n tPeogirinOantrl sb . i.ts. ... . . . . 108
3.I8n teEgsrtailfmroiPa totyiie nOnstr sb. i ts 121
3.P9e rPiooidaninGtcdas lG oriosu ps 122
vii
viii Contents
3.E1q0u idiasnPtdrr eipbPeurotiiionodtnis c 126
31.R1a mifiUcnaitDtiyson nai tnao nmdi c Field1s2 9
Exceirs.e.s . ... . . . 135
4 FamiolfDi yensa mical Systems 147
41. DynmaiPtcoo lynomials . . 184
4.Q2u adProaltyinacon Dmdyi naalMtsoo dmuCilucar .rv es 155
4.T3h e SpoaRfca et iFounnaclt. i o.n s. . .1 86.
Ratd .
4.T4h Meo dSuplaic oeDf y naSmyisct.ae lm .s . .1 47. .
Md
4.P5e rPiooidMniutcpls lt,iia enMrdus l,t Sippelciterra .1 97. .
4.T6h Meo dSuplaMic2 oe Df y naSmyiscotaDfele m gs2r ee 188
4.A7u tpohmoiarsnTmdwis s .t s 195
4.G8e nTehreaoolTfr wi ys .t s. .. .. . . . 199.
4.T9wi sotRfsa tional Maps . . . 20.3 . . . . . .
41.0Fi eolDfde sfi nainttdhiF eoi noeM flo dd uli 206
4.1M 1iniRmeasulal ntMdai nntiMsmo adle ls 218
Exceirs.e s. . . . . . . . 2.2 4 . . . . . . .
5 DynamoivceLsro cFaile lBdaRsde: d uction 239
5.A1b soVllauuatenCesdo mpl.et.i.on.s 240
52. A Proinm eNro naArncyahslii.mse. d.ea..n . . .2.42
5.N3e wPtoolny agntodhMn eas x iMmoudmuP lruisn. c ipl2e48
54. The NonaJrucalhniFidaam tSeoedute sa n 254
2
5.T5h Dey mniaoc(fsz- z)jp. . . . . 257
5.A6N onarcMhoinmTtehedeleo. ar ne m 263
5.P7e rPiooidaninttcdhJs eu Slei.ta . . 268
5.N8o narcWhaidnmeerDdioenmagan i ns 276
5.G9r eFeunn catnLidoo cHnaesli ghts 287
5.D1y0mn iaocnBs e vrkiSocpha ce 294
Exceirs.e.s . . . ... . . . . . . 321
6 Dynamics AAslsgoecbiGraratoieucdp st o 325
61. PowMearp st hMaeun ldta itGpirvloeiu cp 325
6.C2h ebPyoslhyenvo mials . . . .3 2.8 .
6.3P riAom nEe lprlt Ciiucr ves . . . 336
6.G4e nPerroaeplose Lf ra ttMitaepss 350
6.F5l eLxaitbMtlaeeps s . . . . . . 3.5 5
6.R6i LgaitdMt aep.ss . . . . . . . 3.6 4
67. U noirfBmo ufnoLdras t Mtaep.ss 368
6.A8ffi nep hMioasrnCmdos m muFtaimni.gl ies 375
Exer.ci.s.es. . . . ... . . . . .38.0. . .
Conentts ix
7 DynamiinDc ism enGsrieoanTt hearn One 387
7.D1y mniaocRfsa tMioanpoasnPl r e ocjtSipva.ec e 388
7.P2r iomAnel rg ebraic Geometry .4 0.2 . . . . . . . .
7.T3h Weei Hle iMgahcth. i.ne. . . .. . . . .4.0.7
7.D4y naomnSiu acrcswfe isNt ohn comImnluvutoti.in ogn s 410
Exceirs.e..s. . . . . . . . .. . . .42.7. . . . . . . .
NotoenEs x ercises 441
LiosfNto tation 445
Reefrences 451
Index 473