3.3Maintypesoftwin-begettingrelatorsW(A;B)........223.4Basicexamples.. - PDF document

3.3Maintypesoftwin-begettingrelatorsW(A;B)........223.4Basicexamples...........................223.5Exampleswithbuilt-insymmetries...............343.6Furtherexamples.........................434SomeLietheory.Active/passivesubalgebras.Divisorsanduniversalkernels.464.1Theactive/passiveltrationofLie[a;b].............464.2Dimensions.............................524.3Universalconstraints........................534.4Lagrangianconstraints.......................544.5Powerconstraints..........................564.6Proofs................................585Generic,low-complexityidentity-tangenttwins.605.1Finiteco-dimension.Thegeneralpicture.............605.2Fixedratiop=qandnocontinuousparameter..........615.3Fixedratiop=qandonecontinuousparameter..........625.4Freeratiop=qandonecontinuousparameter..........636Nonidentity-tangenttwins.656.1Twinsoftangency(0;q)......................656.2Simpleexamples..........................666.3Twinsoftangency(0;0)......................686.4Simpleexamples..........................686.5Pre-identity-tangent`twins':type(0;0)............686.6Simpleexamples..........................696.7Siblings...............................696.8Simpleexamples..........................717Analyticnatureoftwins.727.1Multiplierjaj6=1:convergence..................727.2Multiplierjaj=1:divergence...................747.3Pre-identity-tangent`twins':convergence.............757.4Identity-tangenttwins:thegeneralpicture...........757.5Identity-tangenttwins:autonomousdierential-dierenceequa-tionsanddoubleresurgence....................767.6Identity-tangenttwins:canonicalaccelero-summability.....802 Inordertogenerateaboundgroupintheabovesense,fandgmustofcourseverifysomecompositionrelation:W(f;g)=idwith(1.1)W(f;g)=fm1gn1


fmrgnr(mi;ni2Z;r2N)Thepoint,however,isthatmostrelationsWforcefandgtocommuteor\nearly-commute",i.e.tobesimultaneouslyconjugatetohomographies.Infact,asweshallsee,onlynelyhonedandfairlyintricaterelationsWcanleadtosuitablesolutions(f;g)andgiverisetoboundgroupsGff;gg.1.2Motivations.(i)Originally,thequestionastotheexistenceofboundgroups(specically,foranalytic,identity-tangentlocalself-mappingsofC)wasraisedintheearly90sbyD.Cerveau,R.MoussuandothersinconnectionwiththeclassicationoflocalanalyticfoliationsonC2andtheholonomyoflocaldierentialequations(see[Ce]).Theexpectation,itseems,wasthatsuchgroupsdidn'texist.In1995,however,oneofuscameupwitharstseriesofexamples,namelythoselistedinthemid-partofx3(mainlyx3.3).Butthenthesubjectwasprovisionallylaidtorest,andonlyrecentlytakenupanewforsystematicinvestigation(ii)Anotherwayoflookingatthesamequestion([Ce])isbytryingtoextendtheso-calledTitsalternative.Initsclassicalformulation,theTitsalternativeappliestosubgroupsGGl(n;C)ofthelineargroup,anditstatesthat{eitherGisvirtuallysolvable,ieitcontainsanitesolvablesubgroupofniteindex{oritcontainsafree2-generatorsubgroup,whichisZariskidense.Soanaturalandimportantquestionis:doesasimilaralternativeholdforDiff(C;0)anditsformalcounterpart?Forthelatter,weshallshowthattheanswerisno.Fortheformeralso,theanswerisno,buttheidentity-tangentsub-caseisstillopen.(iii)Actually,boundgroupsofmappingsarebasicandnaturalobjectsintheirownrightandtheirstudyrequiresnoelaborateapology.Indeed,intheone-operation-onlycontextofgrouptheory,\beingintertwined"isforapairofmappings(f;g)theclosestequivalentonecouldthink4 Dieos(1.2)witha\multiplier"a=1aresaidtobeidentity-tangent.Whenais6=1butaunitroot,fissaidtobepre-identitytangent(sinceasuitableiterateisidentitytangent).Whenneitheristhecase,fisdeclarednon-identitytangent.Iffisidentity-tangent,thesubstitutionoperatorFistriangularwithrespecttothenaturalbasisfxngofC[[x]].Moreover,F�1istriangularwithazero-diagonal,andsoisF:=logF,butwhereasFisanautomorphismofthealgebraC[[x]]:F(': )F('):F( )(8'; 2C[[x]])(1.4)itslogarithmFisaderivationandthusoftheform:F=f(x)@withf(x)=Xn1anxn+1and@=@x:(1.5)Thetangencyorderp(2)ofanidentity-tangentdieofistheindexoftherstnonzerocoecientanin(1.2)oranin(1.5),whichamountstothesame,sincetheleadingcoecientscoincideinbothseries(ap=ap).Iffhastangencyorderp,thenundersome(formal,unramied)changeofcoordinateitcanbebroughttothenormalform:fnor(x)xf1�1 pxp+(1 21+p p2� p)x2p+


g(1.6)Fnorexp(Fnor)postcompositionbyfnor(1.7)Fnor�1 pxp+1(1+xp)�1@(1.8)withawell-denediterationresidue2C.Asaconsequence,afurther(ramied)changeofcoordinate:x7�!z:=x�p+log(x�p)(1.9)willturnfintotheplainunitshiftz7�!z+1withthexedpointmovingfromx=0toz=1.Iffispre-identity-tangent,itsmultiplierhasaprimitiveunitroota=exp(2p p)ofsomeorderp.Sotheiteratefpisidentity-tangent,andfitselfcanbenormalisedto:Fnor=R:exp(�1 pxp+1(1+xp)�1)@)(1.10)6 andlog(Gr(A;B))asasubsetof Lie(a;b),where Aland LiedenotethenaturalcompletionofAlandLieobtainedbyallowinginnitesums.ApolynomialfunctiononGr(A;B)isa(scalar-valued)functionPoftheform:P(Am1Bn1:::AmrBnr)P(rkm1;n1;:::;mr;nr)(1.12)withr2N;mi;ni2Z)withpolynomialdependenceonthevariablesmiandniandthenaturalconnectioncondition:P(rk:::;mj;0;mj+1;:::)P(r�1k:::;mj+mj+1;:::)(1.13)P(rk:::;nj;0;nj+1;:::)P(r�1k:::;nj+nj+1;:::)(1.14)AsubsetHofGr(A;B)issaidtobeof(polynomial)codimensiondifitisthezerolocusofdindependentpolynomialfunctionsonGr(A;B):H=fW2Gr(A;B);P1(W)=P2(W)=


Pd(W)=0g(1.15)If,foreachmonomialc=a1b1


asbsinAl(a;b)andeachelementW(A;B)=Am1Bn1


AmsBnsinGr(A;B),wedenePc(W)asthe(rational)coecientofcinthenaturalexpansionofW(ea;eb)thentheaggregateofallthesePcclearlyconstitutesacompletebutnon-freesetofpolynomialfunctionsonGr(A;B).Butifwelet runthroughsomeba-sis(egLyndonbasis)ofLie(a;b)anddeneP (W)asthecoecientoflog(W(ea;eb)alongthebasisvector ,thentheseP yieldasetofpoly-nomialfunctionsonGr(A;B)thatisstillcompletebutfreeaswell.1.5Campbell-Hausdortypeformulae.Weshallmakeconstantuseofthenotations: ba:=[b;a]:=ba�ab(1.16) BA:=fB;Ag:=B�1A�1BA(1.17)WeshallalsorequiretheCampbell-Hausdorformula:log(ebea)=b+a+1 2[b;a]+1 12[b;[b;a]]+1 12[a;[a;b]]+:::(1.18)8 i.e.thosewithlowest(polynomial)codimensionorwithotherlow\complex-ityindices".Wethensketchanaturalgeneralisation:boundgroupswithmorethantwogenerators(\siblings".)1Inthenextsection(x7)weinvestigatetheanalyticnatureofthetwinsconstructedthusfar.Sincethesolutions(F;G)ofanyequationW(F;G)=1aredeneduptoasimultaneousconjugacy:(F;G)7�!(H�1FH;H�1GH)(1.21)thepertinentquestionisnot,ofcourse,whetherthegeneralsolutioncon-verges,butrather:howsimplecanthepair(F;G)bemadeinasuitablechart?Oragain:howsimpledoesF(resp.G)becomeafteritstwinG(resp.F)hasbeennormalised?Fornon-identitytangenttwins,itisrathereasy(byapplyingthetheoryoftheso-called\sandwichequation")toshowthegenericanalycityofwell-chosenrepresentatives(F;G).Foridentity-tangenttwins,thepositionisexactlythereverse:weestablishtheirgenericdiver-genceandresurgence.Itwouldevenseemthatidentity-tangenttwinsarealwaysdivergentandalwaysresurgent,butaregularproof(especiallyoftheformer)appearstobealongwayo.Thelast,verysketchysection(x8)discussestwinsinthelargersettingoftransseriesandanalysablefunctions(asopposedtopowerseriesandanalyticorresurgentgerms)andbroachesanumberofside-issues,suchastheorderingoffreegroups.22Somegrouptheory.Alternatorsandwordfactorisation.Periodicautomorphisms.Throughoutthissection,GrandLiewillstandforGr(A;B)andLie(a;b),i.e.willdenotethetwo-generatorfreegroupresp.Liealgebra. 1Asectiondevotedtoyetanothergeneralisation{boundgroupsconsistingofhigher-dimensionaldieos{wasscrappedforlackofspace.2anothersuchside-issue,originallymeantforinclusioninthispaper,hasbeenremovedforlackofspaceandredirectedto[EV2].Itasks:whatisthe\proper"andmost\com-prehensive"notionofanalyticityonfreealgebras(Lieorassociative)?Werephrasethequestionsoastogiveitaclear-cutmeaning.Thenwesolveitandcomeupwithastartling,quitecounter-intuitiveanswer.10 becausetheCampbell-Hausdorformula(1.19)forbracketsinvolves,onitsright-handside,onlytermsofdegree1inaandb.Thenwehaveaseconddenition(provisionnallydistinguishedbydoublebraces)accordingtowhichGrffdgg(resp.Grffd1;d2gg)isthesubgroupgen-eratedbyallmulticommutatorsofalternanced(resp.(d1;d2)).Wespeakhereof\alternance"ratherthan\degree"toprecludeanyconfusionwiththe\degree"ofawordW(A;B).Amulticommutatorofalternancedisofcourseonewithexactlydarguments(whichwemaytaketobeA1orB1)andamulticommutatorofalternance(d1;d2)isonewithexactlyd1argumentsAmi(orA1)andd2argumentsBni(orB1).Thereasonwhywemayassumeallexponentsmiandnitobe1isthatbyrepeateduseoftheWitt-Hallidentities:fA;BgfB;Ag=1(2.8)fA;BCg=fA;CgfA;BgffA;Bg;Cg(2.9)fAB;Cg=fA;CgffA;Cg;BgfB;Cg(2.10)wemaybreakupanymulticommutatorintoaproductofmulticommutatorswithequal(orgreater)alternance,butwithargumentsoftheformA1orB1.Clearly,theGrffdggandGrffd1;d2ggdeneanewltrationonGr,i.e.theinclusions(2.7)extendtothedouble-bracedGrff


gg.Butinfact:Proposition2.1ThetwonaturalltrationsonGrcoincide:Grfdg=Grffdgg(2.11)Grfd1;d2g=Grffd1;d2gg(2.12)Corollary2.1Thequotients:Gr[d]:=Grfdg=Grfd+1g(2.13)Gr[d1;d2]:=Grffd1;d2gg=(Grffd1+1;d2gg:Grffd1;d2+1gg)(2.14)makesense(sincethegroupsrightoftheslasharedistinguishedsubgroupsofthoseleftoftheslash)anddeneabeliangroupsGr[d]andGr[d1;d2]thatareisomorphictoLie[d]andLie[d1;d2](orrathertotheiradditivesubmodulesspannedbymulticommutators).12 2.2Finitecriteriaforalternance(d1;d2).EachsubgroupGrfdghasnitecodimensioninGr(seex1.4).Infact,itscodimensionisexactlylie(2)+


+lie(d�1).But,savefor(d1;d2)=(1;1),eachsubgroupGrfd1;d2ghasinnitecodimension.Indeed,ifwegobythe(rst)denitionofGrfd1;d2g,checkingthatagivenwordW(A;B)isinGrfd1;d2gimpliescheckinginnitelymany\polynomial"identitiesP(W)=0oftype(1.12).Fortunately,thereexistcriteriainvolvinganitenumberofsteps.First,observethatanyW(A;B)inGrf1;1gmaybewritteninauniquewayasniteproductsoffactorsAn:=BnAB�norBm:=AmBA�m:W(A;B)=W1(An1;An2;:::;Ans1)(ni2IZ)(2.17)W(A;B)=W2(Bm1;Bm2;:::;Bms1)(mi2JZ)(2.18)Lemma2.2W(A;B)isinGrfd1;d2g�(d1;d2)�(1;1)
ifandonlyif:logW1(ean1;:::;eans1)=O(ad1)(2.19)logW2(ebm1;:::;ebms2)=O(bd2)(2.20)ThesymbolOmeansthattheright-handsidesareoftotaldegreed1(ord2)intheani(orbmj)regardedasfreeindependentvariables.Thelemmadirectlyfollowsfromthefactthatwithinthealgebra Liefa;bg(i.e.inthe`closure'derivedfromLiefa;bgbyallowinginnitesums),thesub-algebrasLiefan1;:::;ans1gorLiefbm1;:::;bms2gnitelygeneratedbyanitenumberofdistinctelementsanorbm:an:=enbae�nb=(exp(n b))a(n2Z)(2.21)bm:=emabe�ma=(exp(m a))b(m2Z)(2.22)arethemselvesfree.So,checkingthatW(A;B)hasalternance(d1;d2)involvesonlyanitenumberofstepswhich,however,steeplyincreaseswiths1ands2.14 (iii)thevariancevar(W)(resp.varA(W)orvarB(W)),denedasthenumberofdistinctpairs(mi;ni)(resp.asthenumberofdistinctvaluesassumedseparatelybymiorni)wherebymj:=m1+


mjandnj:=n1+


nj(iii)thealternancealt(W)(resp.(altA(W);altB(W)))denedasthesmall-estd(resp.(d1;d2))suchthatWbeinGrfdg(resp.Grfd1;d2g).Intheusualgraphicalrepresentationofwords,therstthreecomplexityindicesareimmediatetodetect,butnotsothealternance,whichdoesn't`meettheeye',atleastwhend1+d23.2.4PeriodicautomorphismsofGrLetAut(Gr)denotethegroupofallautomorphismsofGr.Itisknown(Nielsen'stheorem)tobegeneratedbythreeelementaryautomorphisms:(A;B)7�!(B;A);(A;B)7�!(A�1;B);(A;B)7�!(AB;A)(2.27)Forthesequel,weneedtoknow,uptoconjugacy,allthenitesubgroupsofAut(Gr)andinparticularalltheperiodic(unipotent)elementsofAut(Gr).Lemma2.3PeriodicautomorphismsofGr:Theynecessarilyhaveor-der1,2,3,4,thenumberofdistinctconjugacyclassesbeingrespectively1,4,1,1.Eachinvolution(order2)isconjugateeitherto(A;B)7�!(B;A)or(A;B)7�!(A�1;B)or(A;B)7�!(A�1;B�1)or(A;B)7�!(B�1;A�1).Eachautomorphismoforder3isconjugateto(A;B)7�!(B�1;AB�1).Eachautomorphismoforder4isconjugateto(A;B)7�!(B�1;A).Foraproof,see[L.S.],prop.4.6,p25,andthereferencesthereafter.Nowthatwehavethelistofallperiodicautomorphisms,itisaneasymattertoconstructallnon-cyclicnitesubgroupsofAut(Gr).Therearefourofthem(uptoconjugacy),namelythetwoabeliangroupsoforder4:Aut1fI;I1;I2;I3g(twogenerators)(2.28)Aut2fI;S1;S2;I3g(twogenerators)(2.29)16 correspondingwordsW(A;B)neverrangingoversubsetsofnitecodimen-sioninGrfA;Bg.Fortrulygenericexamplesandnitecodimensions,weshallhavetowaitforx5.Asinmostofthispaper,weshallhavetoworksimultaneouslyinthefourstructures:8:W(A;B)2GrfA;Bg�!~G03F##"w(a;b)2 Liefa;bg�!~L03F(3.1)~G0isthegroupofformal,identity-tangentdieosf:x7�!xf1+Panxng,usuallydenotedbythecorrespondingsubstitutionoperatorsF.~L0istheLiealgebraof~G0,withitsnaturalbasiselements:ln:=xn+1@:=xn+1@=@x(n=1;2;:::)(3.2)ThemapF7�!F:=logFfrom~G0into~L0isone-to-one,buttheinjec-tivemapW(A;B)7�!w(a;b)=logW(ea;eb)fromGrfA;Bginto Liefa;bgisofcoursefarfromsurjective.Amongthemapsfrom Liefa;bginto~L0,deservingofspecialattentionarethegradedmorphisms(iewhichrespectthenaturalgraduationofbothalgebras)andarenecessarilyoftheform:(a;b)7�!(lp;lq)(p;q2N)(3.3)Perhapsthequickestwaytoproducetwinsin~G0istoconsidermulticom-mutatorsW(A;B)(withalternance(d1;d2)�(1;1))whosecornercompo-nentw0(a;b)(denedasthehomogeneouscomponentofw(a;b)thatliesinLie[d1;d2])hasthefollowingthreeproperties:(i)w0(a;b)6=0.(ii)Itisacollapsor,i.e.itbelongstothekernelofsome(orall)graduedmorphismsof Lieinto~L0:w0(lp;lq)=0forsome(resp:all)p;qinNN(3.4)(iii)ItsdivisorDis6=0foralltinN.ThedivisorD(t)D(p;q;t)isapolynomialintcharacterised(forallp;qsuchthatw0(lp;lq)=0)bytheidentity:w0(lp+1lp+t;lq+2lq+t)(1(q�t)�2(p�t))D(t)lpd1+qd2+t+o(1;2)(3.5)18 exibilitytheyaord.Andsincethesepairsaredeneduptoacommoncon-jugacy,weshallpriviledge\normal-conormalforms",whichnormaliseoneofthetwinstoFnor(resp.Gnor)whiletheotherassumesarigidlyxedconor-malformGconor(resp.Fconor).Actually,forcompleterigiditywehavetodemandthatFconorandGconorshouldcontainnotermoftheformlp+qifFstartswithlpandGwithlq.Inthemosttypicalexamples,thesenamelywithexactlyoneparameter(otherthantheratiop=q,whichisdiscrete),weshallndthat:Fnor=fnor(x)@=(1�xp)�1( px1+p@)(3.9)Gconor=gconor(x)@=(1+xq+Xm;n0;m+n2(m;n)mnxmp+nq)( qx1+q@)(3.10)Fconor=fconor(x)@=(1+xp+Xm;n0;m+n2(m;n)mnxmp+nq)( px1+p@)(3.11)Gnor=gnor(x)@=(1�xq)�1( qx1+q@)(3.12)withacountableinnityofinvariants:(;);(;);(p;q);f(m;n);m;n2Ngorf(m;n);m;n2Ng(3.13)Anotherobjectofcentralimportance,onaccountbothofitsinvarianceand(anti)symmetryin(F;G),istheconnectorHnor=expHnorwhichconjugatesthenormaltotheconormalforms:Fconor=HnorFnor(Hnor)�1(3.14)Gconor=(Hnor)�1GnorHnor(3.15)butwhichmayalsobeconstructeddirectlyfromanysolution(F;G)bysetting:F=H1FnorH�11;G=H2GnorH�12(3.16)Hnor=H�12H1(3.17)20 3.3Maintypesoftwin-begettingrelatorsW(A;B).Ourrstseriesofexamples(1through6)illustratestwobasicdichotomies:{thetwins(F;G)verifyingagivenrelationW(F;G)=1mayhaveaxedorfreetangencyratiop=q{oragain,theircontinuousparameterc(constructedfromtheleadingcoef-cients:see(3.22))maybexedorfree.Thesecondseriesofexamples(7through17)imposesadditionalsymme-triesontwins.Moreprecisely,foreachofthe10basicnitegroupsAutiofautomorphismsofGrfA;Bg,weconstructrelationsW(F;G)1whosesetofnon-trivialsolutions(F;G)isgloballyinvariantunderAuti.Ourthirdseriesofexamples(18through20)dealswithmoreexceptionalsituations,e.g.withrelationsW(F;G)=1whosegeneralsolution(F;G)dependsonseveralcontinuousparameterswithxedorvariablepositions(i.e.parametersmakingtheir\rstappearance"insidecoecientsofxedorvariabledeptht).Theyalsoillustraterelatedquestionssuchastheglueingorsplittingofrelators.3.4Basicexamples.Example1.Fixedratiop=q,nocontinuousparameter.IfwesetW:=UPVQwithP;Q2ZandU=U(A;B):= A3B=fAfAfA;Bggg(3.23)V=V(A;B):= B2A=fBfB;Aggg(3.24)thenforeachs2N,W(F;G)=1hasaunique(uptoconjugacy)twinsolution(F;G)withtangencyorders(p;q)=(s;2s)andxedinvariantsc;;.Proof:Letuslookforasolution(F;G)withtangencyordersp6=q.Wemaywritetheinnitesimalgenerators(F;G)intheform:F=(1+X1ttxt)( pxp@)with6=0(3.25)G=(1+X1ttxt)( qxq@)with6=0(3.26)22 whichrigidlyxestheinvariantc:c:=2�1=�p2 qQ Pu0 v0=+pQ (p+q)P=+Q 3P(3.40)Asforthecurrentcoecients(t;t),theirrstoccurenceinWtakesplaceatdeptht.Moreprecisely:W=(X1twtxt)(43 p4q3x1+5s)(3.41)withwt=wtt+wtt+earlierterms(3.42)wt=P3 p3qut+Q2 pq2vt(3.43)wt=P3 p3qut+Q2 pq2vt(3.44)Butdueto(3.29)wemayfactorouttheterm:R=�Pu03 p3q=�Qv02 pq2(3.45)Eventually,afterexplicitingu0,v0etc...andrecallingthat(p;q)=(s;2s),wendwt=R(�ut u0+vt v0)=R1 6(6+t+t2)(t�q)(3.46)wt=R(�ut u0+vt v0)=1 6R(6+t+t2)(t�p)(3.47)Thepresenceofacommonfactor(6+t+t2)alongsidetheindividualfactors(t�q)and(t�p)isnoaccident,butaconsequenceoftheidentities:(t�p)ut+(t�q)ut=t�M(3.48)(t�p)vt+(t�q)vt=t�N(3.49)whicharebutspecialcasesof(3.8).Thusintheendwehave:wtR(1=6)(6+t+t2)[(t�q)t�(t�p)t]+earlierterms(3.50)24 withM=p(1+q0)+q;N=q(1+p0)+p(3.55)Theinitialcoecientsare:u0=Tq;p[1+q0](3.56)v0=Tp;q[1+p0](3.57)whereofcoursem[n]denotesthesequence(m;:::;m)oflengthn.Thesub-sequentcoecientsut,vtretaintheirearlierexpression(3.33)(3.34)butwith:ut=XTq;p[q1];p+t;p[q2](forq1;q20;q1+q2=q0)(3.58)ut=Tq+t;p[1+q0](3.59)vt=Tp+t;q[1+p0](3.60)vt=XTp;q[p1];q+t;q[p2](forp1;p20;p1+p2=p0)(3.61)Buttheearlieridentities(3.48)(3.49)remaininforce(beingasimpleconse-quenceof(3.8))andleadtoadrasticsimplicationofutandvt:ut=t�M t�p�t�q t�pTq+t;p[1+q0](3.62)vt=t�N t�q�t�p t�qTp+t;q[1+p0](3.63)Heretoo,WmayvanishonlyifM=N,whichinviewof(3.55)imposesp=q=p0=q0.Sowemaywrite:(p;q)=(sp;sq);(p0;q0)=(s0p;s0q)(3.64)withs;s0inNandp;qcoprime.TheargumentthenproceedsasinEx-ample1,exceptthat(3.39)nowbecomes:Pu0( pq)( p)s0p+Qv0( pq)( q)s0q=0(3.65)sothattheinvariantc:=q�pisnowconstrainedonlyby:cs0=�Qv0 Pu0ps0p qs0q(3.66)26 Then,usingtheasymptoticpropertiesofthegammafunction,itisastraight-forwardexercisetocheckthat,foranyxedpair(p;q)andfor(p0;q0)=(p;q)thedivisor
(t)is�0forallt�0andalllargeenough.Thus,foranygiventangencyorders(p;q),wecanpointtoanexplicitrelationW(F;G)=1thatadmitstwins(F;G)withthattangencyorder.Example3.Fixedratiop=q,onecontinuousparameter.IfwesetW(A;B):=fU(A;B);V(A;B)gwithU,VasinExample1,thentherelationW(F;G)=1hastwinsolutionsforthetangencyratiosp=q=1=2.Thesetwinsstillhavethesameiterationresidues=15=8and=�7=12asinExample1,buttheynowdependonafreecontinuousparameterc:=2�1Proof:Startingfromanypairs(p;q)and(;),wetake(F;G)asinExample1andgetthesameexpressionfor(U;V).TheexpressionforW,however,doeschange:W=(X0twtxt)( pq)( p)2( q)(x1+M+N@)(3.72)withaninitialcoecientw0thatvanishesiM=N.ThusW=0impliesp=q=1=2asbeforebutleaves(;)andthusc:=2�1completelyfree.Asforthecurrentcoecientwt,itisnowoftheform:wt=D(t)((t�q)t�(t�p)t)+earlierterms(3.73)withthesamedivisorD(t)asinExample1,uptothetrivialfactoru0v0=R:D(t)=u0v0t1 6(6+t+t2)withu0=Tq;p;p;v0=Tp;q;q(3.74)Sotheonlychangeistheappearanceofthecontinuousparameterc(invariantundergeneralconjugacies)orthepair(;)(invariantunderidentity-tangentconjugacies)insteadoftherigidityinExample1.Asforthevalues(;),theyarespecialcasesof(3.76),(3.77)below.Example4.Fixedratiop=q=p0=q0,onecontinuousparameter.IfwesetW(A;B):=fU(A;B);V(A;B)gwithU,VasinExample2(i.e.U:=( A)1+q0BandV:=( B)1+p0A)thentherelationW(F;G)=1hastwinsolutionsforthetangencyratiosp=q=p0=q0.Thesetwinsstillpos-sessthesameiterationresidues(;)asinExample2(see(3.81)below),buttheynowdependonafreeparameterc:=q�p28 withasummationextendingtoallqi0suchthatq1+q2=q0(resp.toallpi0suchthatp1+p2=p0.Thusweget:(;)=(30=29;�7=12)for(p0;q0)=(1;2)(3.82)(;)=(50=27;�8=45)for(p0;q0)=(1;3)(3.83)(;)=(413=132;�625=564)for(p0;q0)=(2;3)(3.84)(;)=(1785=856;�69=1120)for(p0;q0)=(1;4)(3.85)Example5.Freeratiop=qonecontinuousparameter.IfwesetW(A;B):=fU(A;B);V(A;B)gwithU=U(A;B):= A B B BA=fAfBfBfB;Agggg(3.86)V=V(A;B):= B A B AB=fBfAfBfA;Bgggg(3.87)thentherelationW(F;G)=1hasatwinsolutionforeachp6=qandeach(;).Thatsolutionisunique(uptoidentity-tangentconjugacies),withit-erationresidues:=q(4p2+5pq+3q2) p2(p+7q);=3p2(p+q) q2(3p+q)(3.88)Proof:InthefreestructuresGrfA;BgandLiefa;bg,theelementsU,V,Whavealternance(2,3),(2,3),(4,6)andtheirLieimagesu,v,whavecornercom-ponents:u0= a b b ba;v0= b a b ab;w0=[u0;v0](3.89)Goingovertotheboundstructures~G0and~L0anddeningU,V,WandU,V,Wintheusualway,wend:U=(Xt0utxt)( p)2( q)3(x1+M@)(3.90)V=(Xt0vtxt)( p)2( q)3(x1+M@)(3.91)W=(Xt0wtxt)( p)4( q)6(x1+M+N@)(3.92)30 Asimplecalculationthenyields:D(t)=pq(p�q)2t2(t2+qt+6pq)(3.106)sothatthedivisorD(t)isalways6=0forpositiveintegers(p;q;t)(p6=q).Theupshotisthat,giveninitialdata(p;q)and(;)(p6=q;6=0),theinductiveresolutionoffwt=0gispossible,leadingto`intrinsicalseries'forFconor,Gconor,Hnorthathaveexactlytheform(3.10),(3.11),(3.18)andcarryonlycoecientsofdepthtinpN+qN.Tondtheexactvaluesoftheiterationresidues(;),wemustrstapply(1.20)tocalculatethetermsofdegree(2+1;3)and(2;3+1)inu(a;b):=logU(ea;eb)andv(a;b):=logV(ea;eb);thenplugthisintow(a;b):=logW(ea;eb);andlastlyreplace(a;b)by(lp;lq).Wend:=�1 21 p�q(u0v00�v0u00) D(p)=�1 2pdetu0v0u00v00 detu0v0upvp(3.107)=�1 21 p�q(u0v000�v0u000) D(q)=�1 2qdetu0v0u000v000 detu0v0uqvq(3.108)withu0;v0;D(t)asaboveand:u00=Tp;q;p;q;q;p+Tp;q;q;q;p;p(3.109)u000=3Tp;q;q;q;q;p(3.110)v00=Tq;p;p;q;p;q+Tq;p;q;p;p;q(3.111)v000=2Tq;p;q;q;p;q+Tq;p;q;p;q;q(3.112)Whicheventuallyleadstothevalues;mentionedin(3.85).Remark1:Weobservethat;,asindeedallthesecondaryinvariants(m;n);(m;n); (m;n)carriedbythe\intrinsicseries"(3.10),(3.11),(3.18),arehomogeneousfunctionsofdegree0in(p;q).Remark2:Forthetimebeing,wehavesetasidethecaseoftwinswith32 Similarly,for(d1;d2)=(2;4)=(even;even)wemaytake:U=U(A;B):= A B4A;V=V(A;B):= B3 A2B(3.117)sinceinthiscase:D(t)=4pq(p�q)2(p+q)6pq+qt+t22q+2p+t(3.118)againwithpositivecoecientsonly.Hereagainwend:6=0;6=0.Weskiptheproofs,sincetheyfollowaxactlythesamelinesasinExample5.Ofcourse,theaboveRemarks1and2stillapply.3.5Exampleswithbuilt-insymmetriesToagiventwin-generatingrelationW(F;G)=1,onemaynotaddanyindependentrelationW1(F;G)=1withoutforcingFandGtocommute(whichbyourdenitiontwinsareforbiddentodo).Onemaywell,however,imposeadditionalsymmetries.Moreprecisely,givenanyoneoftheten(uptoconjugacy)nitesubgroups4ofAutGrfA;Bg,onemaylookforrelationsR(A;B)=1thatgeneratetwinswhilebeinginvariantundertheactionofthesubgroupAutjinquestion:RT(A;B)=QT(A;B)(R(A;B))T(QT(A;B))�1(3.119)(8T2Autj;T=1)(HeretheexponentThastobeanintegerandaunitroot.SoT=1).Ineachofthesetencases,weshallrestrictourselvestorelationsR(F;G)1whosegeneraltwinsolution(F;G)dependsonlyonafreeratiop=q(6=1unlessstatedotherwise)andafreecontinuousparameterc=q�p(likeinExamples5or6).Andasalwaysinthissection,weshallstrivetopickthoseexampleswhichareeasiesttoconstruct,ratherthanthosewiththelowestwordcomplexity.Webeginwiththesixcyclicgroupsofautomorphisms. 4Theirlistwasgivenattheendofx2,alongwiththenotationfortheirelements.Werecallthatamongthese10nitesubgroups6arecyclic;2abeliannon-cyclic;and2non-abelian.34 ThecornercomponentandthedivisorofR(resp.R1)aretwicethoseofW(resp.W1).ThedivisorD(p;q;t)ofW,inturn,beingthesameasinExample5or6,doesnotvanishforp6=qandp;q;t2N.NeitherdoesthedivisorD1(p;q;t)ofW1since:w10(a;b)=[w0(a;b);a](3.124)HenceD1(p;q;t)=((d1�1)p+d2q+t)D(p;q;t)(3.125)Thesimplerinvariancerelation(3.120)(boughtatthecostofamorecomplexW1)impliesthattheLieimager1(a;b)ofR(A,B)carriesonlycomponentsofodd(global)degreein(a,b).Thecomponentoflowestdegreeisthecornercomponentw10(a;b),withdegreeexactly2d+1.Butthenexttwocompo-nents(ofdegree2d+2)vanish,andsodotheinvariants;whichstemfromthesecomponents.ThesameconclusionalsoholdsforthesolutionsoftherelationR1,buttheproofisslightlylessdirect.First,wenotethat(3.119)implies:r(�a;�b)=e� w(a;b)r(a;b)(3.126)So,ifweset:r2(a;b)=r(a;b)+r(�a;�b)(3.127)wehave:r2(a;b)=(1+e� w(a;b))r(a;b)(3.128)Since1+exp(� w),asanoperatoron Lie(a;b)isclearlyinvertible,theidentityr(a;b)0isequivalenttor2(a;b)=0(thoughr2(a;b)isnottheLieimageofanywordR2(A;B)).Now,inviewofitsdenition(3.125),r2(a;b)carriesonlycomponentsofeven(global)degreein(a;b).Sohereagainthetwocomponentsofr2(a;b)immediatelysuperior(indegree)tothecornercomponentnecessarilyvanish,sothat==0.Example10.InvarianceunderI1:(A;B)7!(A�1;B)LetW:=fU;Vgbeatwin-generatingrelationofalternance(d1;d2)in(A,B),withd1even(asinExample5)orodd(asinthesameExample,butwithAandBexchanged).Thenifweset:R:=WWI1(ford1even)(3.129)R1:=W(WI1)�1(ford1odd)(3.130)36 Thenon-proportionalityofu0;v0iseasilycheckedbyspecialising(a,b)to(lp;lq).Theinvariance(mod[a;b])under(3.136)isimmediate.Asforthedi-visorassociatedwithw0=[u0;v0],atediousbutstraightforwardcalculationyields:D(p;q;t)=12p2q2t(p�q)2(t+6p+6q)(t+p�q)(t�p+q)(3.139)sothatD(p;q;t)6=0forp6=qandt2pN+qN(p;q;t�0).Thesameholdsforthedivisorofr0,whichissimplythreetimesthatofw0.ThatleavesonlytheJ-invariance,whichasusualistheeasiestparttocheck.Indeed,from(3.135)weget:RJ=W�1RW(3.140)Remark:Althoughwehavepostponeddealingwithtwinswithequaltan-gencyorders(p=q)untilthe\systematic"investigationofx5,itmaybenotedthatintheaboveExample11(asalsoinExample15below)theactionofJandJ2exchangesbothtypesoftwins(i.e.p6=qandp=q).Example12.InvarianceunderK1:(A;B)7!(B�1;A).Ifwesetsuccessively:W:=f B2 A3B; A2 B3Ag(3.141)Q:=f A3B; B3Ag(3.142)W1:=fQ;Wg(3.143)R:=W1;WK11WK211WK311(3.144)thentheK1-invariantrelationR(F;G)1hasageneraltwin-solutionthatdependsonp=qandc.Proof:SinceK41=1,(3.144)givesRK1W�11RW1whichtakescaresoftheinvariance.Uptotheinnocuousfactor:T6p+6q+t;4p+4qTp;q;q;qTq;p;p;p�(2p+2q+t)pq(p�q)2(p+q)2(3.145)thedivisorassociatedwithW1coincideswiththedivisorofW,whichweal-readyencounteredinExample6andfound(see(3.114))tobenon-vanishing.ThereasonforbracketingWwithQisofcourseaquestionofparity:whereasWandWK1haveoppositecornercomponents(sincetheseareof38 Moreover,sinceWhasaneven(global)alternance2(d1+d2)=4d1=4d2,thetwofactorsWandWI0whichmakeupRcontributethesamecornercomponentandthesame(non-vanishing)divisor{namely(3.114)ifwetakeUasin(3.113).Example15.InvarianceunderAut3=fI;J;J2;S1;S1J;JS1g.WetakethesamewordWofalternance(10;10)asinExample11,butthistimeweset:W1=ffA;Bg;Wg(3.152)W2=W1(W1S1)�1=ffA;Bg;WgfWS1;fB;Agg(3.153)R=W2WJ2WJ22(3.154)ThentheAut3-invariantrelationR=1hasageneraltwin-solutionthatdependsonp=qandc.Proof:ThewholepointofbracketingWwithfA;BgisofcoursetogetawordW1withthesamedivisorasW(uptothetrivialfactorTq;pT10p+10q+t;p+q)butsuchthatW1and(W1S1)�1havethesamecornercomponents(ratherthanoppositeonesasinthecaseofW).ThisyieldsaW2verifyingWS12=W�12andwithtwicethecornercomponentofW1.MoreoversincethecornercomponentsoffA;BgandWareinvariantundertheaction(3.136)ofJ,sotooarethoseofW1andW2.TheupshotisthatthecornercomponentanddivisorofRareexactlysixtimesthoseofW1,whichinturnareessentiallythesameasthoseofW.Asfortheinvariance,itisenoughtocheckitfortwogeneratorsofAut3,e.g.JandS1.DuetoWS12=W�12,thedenitionofRyields:RJ=W�12RW2;RS1=W�12R�1W2(3.155)Example16.InvarianceunderAut4=fI;I0;I1;I2;S1;S2;K1;K2g.IfwetakeawordW=W(A;B)suchthat:(*)Wbetwin-generating(**)WS1=W�1(***)Wbeofalternance(d1;d2)in(A,B)withd1=d2=odd40 Despiteitsforbiddingcomplexity,thisexampleisthe`simplestofitskind',atleastaslongasweinsistonworkingwithmulticommutatorsanddemandacertainsymmetryinAandB.Butifwedroptheserequirements,wemayproduceaslightlysimplerexample:Example17.InvarianceunderAut4:simplerexample.Ifweset:U:=f( A3Bf( AB);B�1gg(3.165)V:=f( A2Bf( A2B); A(B2)gg(3.166)W1:=UV(3.167)W2:= B2W1(3.168)W:=W2(WS12)�1(3.169)R:=WWK1WI0WK2(3.170)thentheAut4-invariantrelationR=1hasageneraltwinsolutionthatdependsonp=qandc.Proof:ItisenoughtocheckthatthenewWstillmeetsallthreeconditions(*)(**)(***)ofExample16.Clearly,ifwesetbn:= anb,wendforthecornercomponentsofU,V,W1,W2:u0=� b3 b2b;v0=2 b2 b2b1(3.171)w10=u0+v0;w20= b b(u0+v0)(3.172)Next,wecheckthatw10(andsow20)vanishunderallrealisations(a;b)7!(lp;lq).ThenwecalculatethedivisorattachedtoW1:D1(p;q;t)=�(p�q)2qt2(t�p+q)(3.173)andtheoneattachedtoW2:D2(p;q;t)T3p+5q+t;q;qD1(p;q;t)(t+3p+4q)(t+3p+5q)D1(p;q;t)(3.174)42 Proof:ThedivisorDl(p;q;t)ofRlisequaltothedivisorD(p;q;t)ofWmultiplied:(i)bythetrivial(i.e.t-independent)factorsTp;q(nj);p(mj�1)contributedbythewordsWj(j=1;


;l).(ii)bytheelementary(i.e.t-ane)factors(t�jp�jq)(j=1;


;l)whichweintroducedbybracketingRj�1withWj.ThereforeDl(p;q;tj)0fortj:=jp+jq(j=1;


;l).Undernormalcircumstances,thisshouldpreventtheexistenceoftwin-solutions,buthereithastheoppositeeectofenlargingtheirnumber,byintroducinglnewfreeparameterscj.Indeed,foranyinitialconditions(p;q)and(;),therelationRl(F;G)1isequivalenttothesystem:logRj�1cjlogWj(mod:termsofdepthtj+1)(j=1;


;l)(3.180)which,foranygivenchoice(c1;


;cl)inCl,clearlyadmits(uptoconjugacy)auniquesolution(F;G).ThecoecientsofF;Gwithdepthtt1arecalculatedinductivelyex-actlyasifweweredealingwiththesolerelationR0(F;G):=W(F;G)=1.Thenthecoecientswithdeptht1tt2arecalculatedfromlogR0=c1logW1;thosewithdeptht2tt3fromlogR1=c2logW2;etc...;andlastlythoseofdepthtltfromlogRl�1=cllogWl.Actually,theconclusionwouldremainunchangedifthe(j;j),insteadofforminganincreasingsequence,werepairwisedistinctandcomparable(forthenaturalorderonN2),whileofcourseremainingpositive.Fornon-comparablepairs,however,therewouldoccursomeslightchanges,sincetheorderofthesequenceformedbythezerostjofthedivisor's`elementaryfactors'woulddependonthetangencyratiop=q.Example19.Onemovableparameter:LetR1andR2betwotwin-generatingrelationsofthetypeencounteredinExample9(see(3.120-121)),withinvarianceunderI0:Rj(A�1;B�1)=(Rj(A;B))�1(j=1;2)(3.181)Thenifweset:R(A;B):=R2(A;R1(AB))(3.182)44 withr i(a;b)denotingthelowest-degreehomogeneouscomponentinthese-riesri(a;b):=logRi(ea;eb)andwitha='@;b= @for,say,'givenand unknown.Heretoo,each`factor'r i(a;b)contributesitsownparame-ters,butthemultibracketontheright-handsideof(3.188)introducesr�1additionalparameterswhichhavetheeectofconnecting(undercontinuousdeformations)theseparatesolutions.Thus,whereasthesolutionsofdierentialequationsmaybeseamlessly`welded'together,thesolutionsofcompositionequationscanonlybe`glued'5.Thisre ectsaverybasicdierencebetweenthetwoclassesofproblems.Norisitduetothefactthatwearesolvingour(dierentialorcomposition)equationsinringsofpowersseries:thedierencepersists,undiminished,whenwegoovertotransserialsolutions(seex8infra).4SomeLietheory.Active/passivesubalge-bras.Divisorsanduniversalkernels.4.1Theactive/passiveltrationofLie[a;b].Thetwentyexamplesoftwinsreviewedintheprevioussectionareeasytoconstructbutsomewhatatypicalinsofarasallofthemverifyrelationsmadeupofsuitablyarrangedmulti-commutators.Beforeturningtothedescrip-tionoftruly`generic'twins,wemustinsertasectiondevotedtothenaturalltrationsthatariseonfreealgebraswhentheygetrepresentedasone-variabledierentialalgebras.Letusforsimplicitydealwiththetwo-generatoralgebraLie[a;b].Itadmitsanaturalsequenceofdecreasingideals6:Lie[a;b]=Uker0Uker1Uker2:::Uker1=Pass(4.187)[Ukeri;Ukerj]Ukeri+j8i;j(0i;j1)(4.188)Thek-thidealUkerkisdenedasconsistingofallw(a;b)2Lie[a;b]whichvanishuptoorderk:fw2Ukerkg()fw(F;G)=O(k)8p;q;'; g(4.189) 5withonlyaweakinteractionstemmingfromtheperturbationR0.6withUkerstandingforuniversalkernel.46 tooneanotherandtoPass,withrespecttothenaturalscalarproductonLie[a;b]9.Thisagainwouldleadtoworthwhiledevelopmentswhich,how-ever,wouldbeadistractionfromourpresentinvestigation.Nextcomesthequestionofthedimensions,whichmakessenseonlyinsidespeciedhomogeneouscomponentsLie[a;b](d1;d2).Foranygivendegree(d1;d2),thecomponentsUker(d1;d2)iclearlybecomestationaryafteracertaincriticali:=icrit(d1;d2).Thisstationaryidealcom-ponentcoincideswithUker(d1;d2)1.OfcoursethecorrespondingcomponentsAct(d1;d2)ioftheactivealgebraturnempty.Foranygivenindexi,thedimensionsdim(Uker(d1;d2)i)andalso,lessob-viously,dim(Act(d1;d2)i),arenon-decreasing(butnon-convex)functionsofd1andd2.Remark3:Thespaces[Uker(d01;d02)i0;Uker(d001;d002)i00]areclearly(strict)sub-spacesofUker(d01+d001;d02+d002)i0+i00.Takingintoaccountthecorrespondingquo-tients10Act(d01;d001);(d02;d002)(i0;i00)wouldleadtoaconsiderablerenementofourac-tive/passiveltration,butweneednotgointothathere.Beforeestablishingtheexactformulasforthemaindimensions,weadducetwotables.Thersttableextendstoalldegrees(d1;d2)(15;15)anditsentriesarethree-numbercolumns:top:=dim(Act0(d1;d2))middle:=dim(Act(d1;d2))bottom: =dim(Pass0(d1;d2))WhenAct(d1;d2)doesn'treducetoitsrstcomponentAct1(d1;d2),thenum-berisenteredas.Thesecondtableextendstoalldegrees(4;4)(d1;d2)(14;14)and 9ietheoneinducedbythenaturalscalarproductontheenveloppingfreeassociativealgebraAss(a;b)10whichmaybejoinedtoformLiealgebras.48 12345678910111111111110000000000000000000021122334455000000000000000000003123445667800012346790000112345412456678990012581215212600013611172535513467889101100259152231405200139193659931376135689101011120038152336496786001619438615325539971467810111212130041222365274100132002113686182343603999814689101213141400615314974103142186003175915334368412742224915791011121415160072140671001421922570042593255603127424934589101589111213141617009265286132186257339005351373999992224458988691116810121314151618001133651111682433324450074819660415863720805016333121691112141516171800133982137214306426568008622708752423596613552287861317101213151617181900154899172265387535721001080364124135959286220574900214171012141517181920001856120208328477668897001210047817095192140253484380769151811131516181920210020671422543995898221116001412461823137335207105363512961350 4.2Dimensions.Throughoutthissection,weshallusethefollowingnotations:p(n):=nbofpartitionsofnwithpositivesummands(4.193)p(n):=1+p(1)+p(2)+


+p(n)(4.194)p(n;m):=nbofpartitionsofnwithmnon-negativesummands(4.195):=nbofpartitionsofnwithatmostmpositivesummands(4.196)P(n;m):=nbofpartitionsofnwithmpositivesummands(4.197)Clearly,p(;)andP(;)areexpressibleintermsofeachother;p(n;m)P(n+m;m);P(n;m)p(n;m)�p(n;m�1)(4.198)Letusrstgetallthemainstatementsandformulasoutoftheway(Prop.4.1through4.4)andthenproceedwiththeproofs.Proposition4.1Fullalgebra:ThedimensionsL(d1;d2):=dimLie[a;b](d1;d2)aregivenbytheclassicalfor-mula:L(d1;d2)=1 d1+d2Xjd1;jd2()((d1+d2)=)! (d1=d)!(d2=d)!(4.199)with(:)astheMobiusfunction.Proposition4.2Fullactivealgebra:ThedimensionsD(d1;d2):=dimAct(d1;d2)=codimUker(d1;d2)1aregivenby:D(d1;d2)=p(d1+d2�1;d1)+p(d1+d2�1;d2)�p(d1+d2�1)(4.200)=P(2d1+d2�1;d1)+P(d1+2d2�1;d2)�p(d1+d2�1)(4.201)=p(d1+d2�1;d1)�p(d1�1)(ifd2�d1�3)(4.202)=p(d1+d2�1;d2)�p(d2�1)(ifd1�d2�3)(4.203)Proposition4.3Leadingactivealgebra:ThedimensionsD(d1;d2)0:=dimAct(d1;d2)0=codimUker(d1;d2)0aregivenby:D(d1;d2)0=1+E((d1�1)(d2�1)) d1)+E((d1�1)(d2�1)) d2)(4.204)whereE(x)denotestheentirepartoftherealnumberx.52 Insymmetricfashion,itisalsoisomorphictothesubspaceofHd1+d2�1d1thatisorthogonaltotheconstraintsLag orPow withindicesoftheform =; oroftheform =(n1;:::;ns) withXnid2�2ands+Xnid1+d2�1(4.208)Bylinearity,theconstraintsLag andPow arewhollydeterminedbythe`constrainttensors'Lag  andPow  suchthat:Lag ; �=Lag  (4.209)Pow ; �=Pow  (4.210)4.4Lagrangianconstraints.Proposition4.6Tensorof`Lagrangianconstraints':Itisexplicitelygivenby:Lag  =Lag (m1;:::;mr)(n1;:::;ns) (4.211)X =1 


r lag1 (m1):::lagr (mr)(4.212)withasumextendingtoalldecompositions =1 


r of intorsub-partitionsi (someofwhichmaybeempty)andwithintegerslagi (mi)denedby:lag; (m):=1(8m2N)(4.213)andfornon-emptypartitions bymeansoftheidentity:(�1)m (m)(y):='�1(x)('0(x))m+X1s;1ni(4.214)lag(n1;:::;ns) (m)'knk�1('0(x))m+s�knkY1is(')(1+ni)(x)thatconnectsthesuccessivederivativesoftwofunctions'(x)and (y)linkedbythereciprocityrelation:fy=Zxdx1 '(x1)g()fx=Zydy1 (y1)g(4.215)54 Remark4:Letuswritedowntheexponentialsumslag (m)fortherstsevenpartitions,iefork k3:lag(;) (m)=1lag(1) (m)=(1�m)lag(2) (m)=(3 4�1 2m)+(�1)m1 4lag(1;1) (m)=(7 4�2m+1 2m2)+(�1)m1 4lag(3) (m)=(11 36�1 6m)+(�1)m1 4+(�2)m(�1 18)lag(2;1) (m)=(�23 9+29 12m�1 2m2)+(�1)m(�1+1 4m)+(�2)m(1 18)lag(1;1;1) (m)=(+13 4�49 12m+3 2m2�1 6m3)+(�1)m(3 4�3 4m)Form=0wemustpositlag (0):=0,butforsmall,positivevaluesofm(iesmallerthans�1+kk)theaboveformulas,oftheirown,yield0.4.5Powerconstraints.Proposition4.7Tensorof`powerconstraints':ItisexplicitelygivenbyPow ; =1andfornon-emptypartitionsby:Pow  =Pow (m1;:::;mr)(n1;:::;ns) (4.221)(ifrs)=0(4.222)(if1sr)=Xj(1�n1)mj(1)(1�n2)mj(2):::(1�ns)mj(s)(4.223)withasumrangingoverallr!=(r�s)!injectionsjofthesetf1;:::;sgintothesetf1;:::;rgandwithnmdenedintheusualwayforvanishingarguments.13Proofs:derivationofthepowerconstraints. 13ie00:=1;n0:=1(8n1);0n:=0(8n1)56 holdtruewhenever:rs+2;tk2f1;0;�1;�2;�3;:::g;X(1�tk)r�2whichestablishestheanalyticexpressionofthe`powerconstraints'.4.6Proofs.DerivationoftheLagrangianconstraints:Themodelweusedfor:Act:=Lie[a;b]=Uker1(4.233)wasobtainedbyspecialising(a;b)as(@y; (y)@y).Withequalright,wemighthavespecialised(a;b)as('(x)@x;@x).Ifwenowconsiderthechangeofvariablewhichtakesusfromthersttothesecondspecialisation,wend:x7!y=h(x);(@y; (y)@y)7!('(x)@x;@x)(4.234)'(x)=1=h0(x); h(x)=h0(x)=1(4.235)whichisreadilyseentoimply,foranypositivem: (m)@7!'(m)h h0@(4.236)=(�1)m('0)m+(�1)mXlag (m)'k k('0)m�s�k k'(1+ )withthesamecoecientslagasin(4.214).Moregenerally,wehave: ( )@7!(�1)k k('0)k k(4.237)+(�1)k kXLag  (m)'k k+1�r('0)k k�k k�s'(1+ )withtheusualnotations: := (m1;:::;mr); ( ):= (m1)::: (mr)(4.238) :=(n1;:::;ns) ; (1+ ):= (1+n1)::: (1+ns)(4.239)58 constraintsareexhaustive.Tollthisonelastgape,weintroduceonHtheltrationH=[Hp,whereHpdenotesthesubspaceofHgeneratedbyproductsofpLieelements,andwecheckthat,iftheexhaustivenesshypothesisisvalidforallcomponentssuchthatd1+d2d�1,thenthenextcomponentd1+d2=dhascodimension:codimHd1;d2p=Xnd1�2n+sd1+d2�pP(n;s)8p1(4.244)ButthisleavesnoscopeforHd1;d2(=Actd1;d2)tohaveadimensionsmallerthanDd1;d2,whichbyinductionestablishesthevalidityoftheexpressionforDd1;d2.Butsincethepowerconstraintsarealsomutuallyindependentand,ifexhaustive,alsoleadtothesameexpressionforDd1;d2,itmeansthatthey,too,mustbeexhaustive.Thiscompletestheproof.5Generic,low-complexityidentity-tangenttwins.5.1Finiteco-dimension.Thegeneralpicture.Thebasictoolsforinvestigatingtwin-begettingrelationsW(A;B)=1isnotthewordW(A;B)itself14butitsimagew(a;b)inthenaturalclosureofLie(a;b),denedintheusualway:w(a;b):=logW(eaeb)=Xm1;n1wm;n(a;b)2 Lie(a;b)(5.245)Wealsorequirepreciseinformationaboutthenatureofthehomogeneouscomponentswm;n(a;b)andtheirexactpositionwithrespecttotheUker-ltrationofLie(a;b).Soitisnaturaltoset:ActW:=f(m;n)2N2jwm;n(a;b)2Uker0nUker1gActiW:=f(m;n)2N2jwm;n(a;b)2UkerinUkeri+1g(0i)ActW=Act0W[Act1W[Act2W[:::(5.246)Let ActW, Act0W, Act1W...denotetheconvexhullsofthesesetsandletdActW,dAct0W,dAct1W...bethecorresponding(nite)setsofextremalpointsor`summits'. 14atanyrate,ifweareinterestedinpowerseriessolutionsonly.Butwhensearchingforgeneraltransserialtwins(seex8),thewordW(A;B)returnstotheforefront.60 for(m;n)=(3;2)andW(A;B):=Ac1BAc2B�1Ac3BAc1B�1Ac2BAc3B�1(5.247)w2;3(a;b)=1 2c1c2c3[[a;b];[a[a;b]]](5.248)withc1;c2;c32Z;c1+c2+c3=0(5.249)Supposelastlythattheedgelinkingthetwosummitshasslope�p0 q0:=�n1�n2 m1�m26=�1.Thesimplestinstanceofthissituationis:W(A;B):=AB3AB�3ABA�3B�1(5.250)w(a;b)=[a[a[a;b]]]+3[b[b;a]]+


=logW(ea;eb)(5.251)HHHHHHHHHH123411122311111111p q=1 2 - 6codim=45.3Fixedratiop=qandonecontinuousparameter.LetusnowexaminethegenericcounterpartofExamples3and4inx3.4.SupposedActWhasasummit(m0;n0)whichisalsoindAct0W.Supposefurtherthatthehomogeneouspolynomialwm0;n0(p;q)hasoneorseveralpos-itiverationalrootsp=q(otherthanp=q=1)suchthatthelineDofslope�p=qdrawntroughthesummit(m0;n0)liesoutsidetheconvexhull ActW.Thismayoccurwithany(m0;n0)(2;2)and6=(2;2).Thesimplestin-stanceofthissituationcorrespondsto(m0;n0)=(3;2)and:w(a;b)=+c1[b[a[a[a;b]]]]�c2[a[a[b[a;b]]]]+:::(5.252)withc1 c2=2 3p+2q p+q(5.253)Exercise:ndthesimplestwordsW(A;B)correspondingtothesimplestvaluesofp=q.62 HHHHHHHHHHHHHHHHHHHHQQQQQQQQQQQQQQQQQ@@@@@@@@@@@@@@146427121113531686815191342165379981519124152165368671211123414215316426637748951122334455111111111111p q=1p q=2 3;p q=1 2; - 6ABABA:codim=31B:codim=31Drawingalineofslopep=qthroughthesummit(4;3)withineitheroftheangulardomainsA;Bandcountingthedimensionsofthecellsbelowthatline,thenadding4,whichisthedimensionofAct04;3,onendsthesame`codimension'withAandB,namely31.Exercise:ndthesimplestwordsW(A;B)thatinduceanextremalw4;3(a;b)asaboveandforwhichallcomponentslyingbelowthebissectrixofA(orB)vanish.Remark1:additionalsymmetries.Theaboveconstructionsinx5.2,x5.3,x5.4providegenericanaloguestothetypicalsituationsexempliedinx3,inEx1-2,3-4,5-6respectively.Asfortheothersituationsreviewedinx3(Ex7through20)andwhichmostlyinvolvetwinswithextrasymmetriesorinvarianceproperties,theytoohavetheirgenericcounterparts.Moreprecisely,thereadermaysatisfyhimselfbygoingseriatimthroughthese16examplesthateachachievablesetofsymmetriescanalsobeachievedinnite`codimension'.Remark2:precautionswiththedivisors.Theconstructionsofthissection,asindeedtheearlierconstructionsofx3,doworkprovidedthecorrespondingdivisorD(t)doesn'tvanishonN.Sim-plecalculationsconrmthatthisisindeedthecasewithallthreelowest-complexityexamplesproducedinx5.2,x5.3,x5.4.Thereexist,however,ex-ceptionalcasesinwhichthepolynomialD(t)mayhavekrootsonN.Even64 Then,dependingonwhethertheresonancekis1,2,3et...,ourequation(6.256),whichmayberewrittenas:Gm1n1:::Gmrn?r=1withGmn:=FnGmF�n(6.258)admitsageneral1,2,3...-parametertwinsolutionoftheform:G=G1]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.107;&#x Td ;&#x[000;:=Hb1q1(b12C;q12N)(6.259)G=G1;b2]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.106;&#x Td ;&#x[000;:=Hb1q1Hb2q2K1;b2]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.107;&#x Td ;&#x[000;(bi2C;qi2N)(6.260)G=G1;b2;b3]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.107;&#x Td ;&#x[000;:=Hb1q1Hb2q2Hb3q3K1;b2;b3]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.21;&#x -1.;ć ;&#xTd [;(bi2C;qi2N)(6.261)etc:::withoperatorsHbqandKcorrespondingtopost-compositionbymappingsoftheform:hbiqi:x7!x(1+bixqi)�1=qi(bi2C;qi2N)(6.262)k1;b2]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.107;&#x Td ;&#x[000;:x7!x(1+O(xq1+q2))(6.263)k1;b2;b3]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.107;&#x Td ;&#x[000;:x7!x(1+O(xq1+q2))ifq1q2q3(6.264)etc:::provided(fork3)thattheqi'sverifynodependencerelationoftheform:qi=Xj6=imjqjwithmj2NandXmj&#x-278;012(6.265)Fork=1thesolution(6.259)isofthetypewhichwehavedismissedas`elementary'inx1.1.Fork2,however,thesolutions(6.260),(6.261)etcaregenuinetwinsassoonastwoofthefreecomplexparametersbiarechosen6=0.Proof:Straightforward:applyCampbell-Hausdortorephraseequation(6.258)intermsoftheinnitesimalgeneratorsGmnofthefactorsGmnandobservethat,wheneverallparametersbibutonevanish,the`correctivefac-tor'H1;b2]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.107;&#x Td ;&#x[000;;H1;b2;b3]TJ;&#x/F30;&#x 5.9;ݶ ;&#xTf 1;�.20; -1;&#x.107;&#x Td ;&#x[000;:::reducestotheidentityoperator.6.2Simpleexamples.Thesimplestpossible`resonant'divisoris:
W():=(�aq1)(�aq2)=m0+m1+m22(6.266)a2N;1q1q2;m0=aq1+q2;m1=�aq1�aq2;m2=1(6.267)e:g:a=2;q1=1;q2=2;m0=8;m1=�6;m2=1(6.268)66 Hereagain,W6,unlikeW5,forcesare ection-symmetryoftype(6.274).Fromtheformalviewpoint,thevalueofadoesn'tmatter,aslongaisnotaunitroot,butfromtheviewpointofanalysis,asweshallseeinamoment,thepicturechangescompletelydependingonwhetheralieson,oroutside,theunitcircle.6.3Twinsoftangency(0;0).PickanywordW(A;B)leadingtotwinsoftangencyorder(0;q).Thentheword:W(A;B):=W(A;fA;Bg)(6.285)admitsageneraltwinsolutionoftangencyorder(0;0).Toseethis,normaliseFbysettingf(x):=axfortherightvalueofa.20Next,solveW(F;G0)=1withrespecttoG0asabove.ThenendbysolvingG0=fF;GgwithrespecttoG,whichisalways(uniquely)possibleforanychoiceofb:=g0(0),providedbisnotaunitroot.6.4Simpleexamples.WiththewordWasin(6.285)andanyoneofthewordsWinx6.2.6.5Pre-identity-tangent`twins':type(0;0).Letpbeprime5andlete1;e2betwodistinctunitroots:ep1=ep2=1;e16=1;e26=1;e1e26=1Thenthesystemf:=x7!e1x+O(x2)(6.286)g:=x7!e2x+O(x2)(6.287)Fp=Gp=(FG)p=1(6.288)iseasilyseentopossessageneralnon-elementary21solutionthatinherently22dependsoninnitelymanyparameters.Similarresultsholdfornon-prime 20iefora`resonant'rootofthedivisor
W().21intheusualsenseofnotbeingreducible,evenunderajointramiedchangeofco-ordinates,toapairofhomographies.22ieafternormalisingF;Gunderajointchangeofco-ordinates.68 operatorslt:=x1+t@x:p:=Lie(a)!Z;w7!p:w(6.290)pi:=Lie(a)!Z[t];w7!pi:w(1ir)(6.291)withp:w:=w(lp1;:::;lpr):x x1+p1+


+pr(6.292)pi:w:=@w(lp1;:::;lpi+lt+pi:::;lpr):xjj=0 x1+t+p1+


+pr(6.293)Ifw2Lie(a)ishomogeneousofdegreed1ina1,d2ina2,...,thenitsp-degreeissimplydenedasp-deg(w):=Ppidi.Clearly,ifwisofp-degreeP,thesetwoidentitieshold:(t�P)p:w=rX1(t�pi)pi:w(6.294)pi:w=dip:w+O(t)ast!0(6.295)Now,xamulti-integerp:=(p1;:::;pr)andashorter(oneelementless!)sequencew:=(w1;:::;wr�1)ofhomogeneouselementsofLie(a),ofvariousoridenticalp-degrees,itdoesn'stmatter,butallsubjecttotheorthogonalitycondition:p:w1=p:w2=


=p:wr�1=0(6.296)Inviewof(6.294),(6.295)therelation:D(t):=(�1)i (t�pi)det[p1:w;:::;[pi:w;:::;pr:w](6.297)denesafunctionD(t)=Dw(t;p)thatisnotonlyindependentofi24butalsopolynomialint.Thisfunctionistheexactgeneralisationforsiblingsofthe`divisors'whichwecameupagainstwhendiscussingtwins.Proposition6.1Siblings.Considerasystemoftype(6.289)andxamulti-integerp.Assumethat 24theterm[pi:wisomittedin(6.297)andofcoursepj:wstandsfor[pj:w1;:::;pj:wr�1]70 wouldamountstoimposingthreeindependentrelations.However,whendealingwithidentity-tangentmappingsFi,imposingthepair-wisecommu-tationofthethreewords:V1:=V(F1;F2;F3);V2:=V(F2;F3;F1);V3:=V(F3;F1;F2)(6.305)amountstoimposingonlytwoindependentrelations,e.g.:W12(F1;F2;F3):=fV1;V2g=1(6.306)W23(F1;F2;F3):=fV2;V3g=1(6.307)This`transitiveness'propertyofcommutationforidentity-tangentmappingsissimplyduetothefactthatF;GdocommuteifandonlyiftheirlogarithmsF:=log(F)=f@x;G=log(G)=g@xcoincide,asoperators,uptomultiplicationbyaconstant.Thuswehaveoursystemof2relations(6.326),(6.307)withthreeun-knownsFi,andwecaneasilyseethanitadmitsageneralsiblingsolutionwithfreetangencyorders(p1;p2;p3)byobservingthat:logV(F1;F2;F3)=�2p1p1(p1�p2)(p2�p3)(p3�p1)x1+2p1+2p2+2p3@x+termsofhigherdegreeandthenreasoningasinx3.4,Example5or6.7Analyticnatureoftwins.7.1Multiplierjaj6=1:convergence.Letusstartwiththecasewhich,fromtheanalyticviewpoint,issimplest:thatofamultiplierjaj6=1.Werstrequireanauxiliarylemma.The`sandwichequation':Considertheequation:Hm1K1Hm2K2:::Hmr�1Kr�1HmrKr=1(7.308)h(x)unknownax(7.309)ki(x)givenbixandanalytic(7.310)m1+


+mr6=0mi2Z(7.311)am1+


+mrb1:::br=1(7.312)72 smalloforder(l+1)ln0 n0kwith :=infj ij1fori=1::landwithk&#x]TJ/;༗ ;.9;Ւ ;&#xTf 1;.54; 0 ;&#xTd [;1denotingthelowesttangencyorderforthenewunknownhandthenewdatabi.ButifwechooseH analyticandcloseenough(ietangenttoahighenoughorder)toaformalsolutionofH0,wecanmakekaslargeaswewishandinparticularensurethatjl kj1.Fixingsuchak,weseethattheanalyticoperatorsPn(1)convergenormallytoananalyticoperatorP1(1)whichnecessarilycoincideswithH0.WhichmeansthattheformalH0wasanalyticintherstplace.QED.Applicationtotwins:TheschemeclearlyappliestoallfourexamplesW1(A;B):::W4(A;B)abovesincem2=1.Itwouldapplyequallywelltoanyexampleconstructedfromamonicpolynomial26
W()withatleastone`resonant'rootjaj&#x]TJ/;༗ ;.9;Ւ ;&#xTf 1;.62; 0 ;&#xTd [;1.Itap-plies,infact,withminoradaptations,tomostsituationsinvolvingmultipliersjaj6=1.Aninterestingasideisthis:oncefhasbeennormalisedtof(x)=ax,whatcanbesaidofganditsnaturalRiemannsurface?Thelatterappearstopossess,foralmostallvaluesoftheparametersbi,ahighlyfractalboundary.7.2Multiplierjaj=1:divergence.Whenaisontheunitcirclebutnotaunitroot,weshouldexpectgenericdivergenceoftheformaltwinssince,inasuitablez-chart(z1),solvingthetwinequationreducestosolvinganinnitesequenceofaneequations:P'n= nwithP'(z):=Xck'(akz+bk)andz1(7.321)where,atthen-thinductivestep, nisknownand'nunknown27.Now,evenforananalytic28input n,thesolution'nisgenericallydivergentandnon-summable.Ofcourse,compensationwithintheseriesP'ncannotberuledoutohand,butwhenthesolutiondependsonacontinuousparameter,asimpleargumentshowsthatwemusthavedivergenceforalmostallvaluesofthatparameter. 26ieapolynomialwithintegercoecientsandaleadingtermwithunitcoecient.27ThesumPinthedenitionofPisnitebutitneverreducestoasingleterm.28atinnity.74 R+isfreeofsingularitiesineitheroftheBorelplanesf1gandf2gconju-gatetothemultiplicativeplanefz1gandfz2g.Asaconsequencethereexistsaprivilegedrealaccelero-summation.30P4:Theprivilegedsumthusobtainedcoincideswiththeoneproducedbythe`geometric'methodsketchedinx8.4.P5:Theinvariantsassociatedwiththez1-andz2-resurgenceregroupnat-urallytoformageometricobject,the\shadowtwins",consistingoftwoformalseries31alsoconnectedbyonerelation.These\shadowtwins"carryalltheobstructionstotheanalyticityoftheoriginaltwins(f;g).P6:Whenevertwinsdependononeorseveralcontinuousparameters,theintrinsictwinfunctionsareguaranteedtobenon-analytic(iestrictlyresur-gent)except(atmost)foradiscretesetofparametervalues.P7:Itwouldseemreasonabletoconjecturethenon-analyticityofallgen-uine32identity-tangenttwins.7.5Identity-tangenttwins:autonomousdierential-dierenceequationsanddoubleresurgence.Thephenomenonofdoubleresurgenceinthe`intrinsictwinfunctions'issimplesttounderstandincaseswithafreeratiop=qandafreecontinuousparameter ==,likeinEx5andEx6ofx3.4.Tofurthersimplify,weimposesymmetriesthatmaketheiterativeresidues;ofF;Gvanish33sothatFandGmaybeseparatelynormalisedto:Fnor=exp(�xp+1@x)Gnor=exp(�xq+1@x)( :==)(7.322) 30consistingincalculatingonR+boththeaccelerationintegralinthe1-planeandtheLaplaceintegralinthe2-plane.31theyarenotpowerseries,though,and,unliketheoriginaltwins,theycarrytranscen-dentalratherthanrationalcoecients32ienon-elementary,inthesenseofx1.133see(1.19).76 withandF:=logF=(x)@xFn:=logFn=n(x)@x=gn(x) @xgn(x)@xm;n:=[m;n]:=m0n�0mnSwitchingovertothe`criticalvariable'z2:=x�qthatnormalisesGweget:G:=logG=@z2)g(z2):=z2+1F:=logF='(z2)@z2Fn:=logFn='n(z2)@z2='(z2+n)@z2'm;n:=['m;'n]='(z2+m)'0(z2+n)�'0(z2+m)'(z2+n)Freezingto=1theparameterassociatedwithGandexpandingeverythinginpowersoftheparameterassociatedwithF,wend:'(z2)='1&#x]TJ/;༩ ;.97;
T; 6.;ֆ ;� Td;&#x [00;(z2)+2'2&#x]TJ/;༩ ;.97;
T; 6.;ֆ ;� Td;&#x [00;(z2)+:::withP2'1&#x]TJ/;༩ ;.97;
T; 6.;և ;� Td;&#x [00;=0(7.327)P2'&#xk-27;=earlierterms(k=2;3:::)(7.328)withabilineardierence-dierentialoperatorP2oftheform:P2'(z2):=Xmcm;n['(z2+m);'(z2+n)]andandcoecientscm;nanein:=p=qandeasilycalculablefromthelineari-sations(7.325),(7.326):c�2;0=c0;2=�1 2(�1)c�1;0=c0;1=2(2+1)c�1;1=�3(+1)Secondlinearisation:IfwenowlineariseinBi.e.inG,thepictureremainsmuchthesame,with(F;;';;z2;;P2)and(G; ; ;;z1;�1;P1)exchangingplaces,butwith78 withadierence-dierentialoperatorP2oftheform(nitesum):P1 (z1):=Xn1;n2;n3cn1;n2;n3[ (z1+n1)[ (z1+n2); (z1+n3)]]whosecoecientscn1;n2;n3areanein:=p=qandeasilydeduciblefromthelinearisations(7.329),(7.330):cn1;n2;n32(2+1)cn1;n2;n3+(�1)cn1;n2;n3Singularautonomousdierence-dierentialoperatorsP:Thehomogeneousequations(7.327)or(7.331)whichstarttheinductioncaneasilybeshowntoadmitauniqueformalsolutionoftheform:'1&#x]TJ/;༩ ;.97;
T; 6.;ֆ ;� Td;&#x [00;(z2)=z1�2(1+X 2n()z�2n2)(:=p=q)(7.333) 1&#x]TJ/;༩ ;.97;
T; 6.;ֆ ;� Td;&#x [00;(z1)=z1�1 1(1+X2n()z�2n1)(�1:=q=p)(7.334)thatisdivergent-resurgentinitssingle\criticalvariable",z2orz1,andalwayspossessesacountableinnityofnon-zeroalienderivatives.Thesameholdsforthenon-homogeneousequations(7.328)or(7.332)thatcontinuetheinduction,aslongastheirright-handsidesarethemselvesresurgent.Here,the`invariants'or`resurgencecoecients'or`Stokesconstants'whichentertheresurgenceequationsastheironlytranscendentalingredi-ent,areparticularlyinterestingentirefunctionsof,ofso-calledautarkictype34.7.6Identity-tangenttwins:canonicalaccelero-summability.Thus,thingsarefairlyunproblematicaslongasweexpandour`intrinsictwin-relatedfunctions'inpowersofthefreeparameterandconsidereachcontribution'&#xk-27;or &#xk-27;inisolation.Butthemomentweattempttosumallthesecontributions,asharpdissymmetrymakesitselffeltbetweenthe 34iewithanasymptoticscompletelydenedbyanitesetof{equallyautarkic{entirefunctions.Actually,theclassofautonomousdierence-dierentialequationsandtheirinvariantswouldwarrantaspecialinvestigation,butthereisnoroomforthathere.80 allorders,andcombinestheoriginal,resurgence-bearingvariablezwithacountableinnityofso-calledpseudo-variablesZ!1;:::;!r.Therestrictionisobtainedtherefrombyjettisoningthetruevariableandretainingonlythepseudo-variables.Fordetails,seeforinstance[E9],x2.4.Applyingthistoapairoftwins(f;g)39linkedbyarelation(7.336),wegetrstamixedobject(7.337),andthena`pure'one,the`shadowtwins',linkedbyarelation(7.338)formallyidenticalto(7.336),andwhichconcentratesalltheobstructionstotheanalyticityof(f;g).twins:1=W(f;g)=)(7.336)1=W(display(f);display(g))=)(7.337)shadowtwins:1=W(restrict(f);restrict(g))(7.338)8Transserialtwins.8.1Remindersabouttransseries.Thissectionissomethingofanaside,andweshallbeextremelysketchy.Letz+1.TheformaltrigebraTorR[[[z]]]consists,veryroughly,ofthenaturalcompletionofR[[z�1]]underthebasicoperationsf+;;;@gandtheirinverses.Itselements,theso-calledtransseries,mayinvolveextremelyintricateconcatenationsofexponentialsandlogarithms,butalwaysadmita(unique)distinguishedor`canonical'representationobtainedby{expellingallinnitesimalsfrominsidetheexponentials{expellingallsumsfrominsidethelogarithms40Thereisanatural,totalorderonthetrigebraoftransseries.Inparticular,inthedistinguishedrepresentation,eachtransseriesappearsintheguiseofatransnite,well-orderedsumoftransmonomialswhich,despitetheir`atom-icity',maycarryinnitelymanycoecients,withacomplexarborescentstructureonthem.Somuchfortheformaltransseries.ThegeometriccounterpartisthetrigebraRfffzgggofso-calledanalysablegermsat+1.Theseareinone-to-onecorrespondancewithasmall(orhuge,dependingonhowyoulookat 39expressedinanychartthatmakesthemsimultaneouslyresurgent.40usinginbothcasesthefunctionalequationsandTaylorexpansionsofexpandlog.82 essentiallyone,ieisomorphicunderthefullstruxturef+;;;@;g.Thisisthegoodnews,orthe`positiveside'oftheindiscernibilitytheorem.Thetrigebrasoftransseriesandanalysablefunctionswereintroducedinthelate90stosolvetheso-calledDulacproblemaboutthenitenessoflimit-cycles.See[E5],[E6],[E7],[E8].Theemphasistherewassquarelyontheanalyticside,anddevelopmentsabouttheformalconstruction{iethetrigebraoftransseries{werekepttoaminimum.AfarmoresophisticatedtheoryofformaltransserieswassubsequentlydevelopedbyvanderHoeven(see[H1],[H2],[H3]),withspecialattentiontothealgorithmicresolutionofdierential,functionaletcequations.Thepresentsectionx8,incidentally,benetedfromexchangeswehadwithvanderHoeven.Lastly,forcomple-mentsabouttheindiscernibilitytheorem,thefast/slowfunctions(suchasthetrans-orultra-exponentialsandtheirreciprocals)andthewholesubjectof`universalfast/slowasymptotics',wereferto[E5],chap.7-10.8.2Transserialtwinsofexponentialortransexponen-tialtype.Switchingfrompowerseriestotransseries,especiallyofthetransexponentialsort,bringssignicantchangestothetypologyoftwins.Thereappearsawholenewclassoftwin-begettingrelationswhichmaybebecalled`regular'or`orderly'andwhichhadnotrueequivalentinthemorerestrictivesettingofpowerseries.Considerforinstancethefollowingseriesof(F;G)-relations,whereFhasbeennormalisedtotheunitshiftT.Theyconsistofaprincipal,highly-alternate,(1+k)-shrinking42factorfT;Tn1:


:Tnk:Ggandaperturba-tive,evenmorealternate,(2+k)-shrinkingfactor.Wk(T;G).twinrelationnatureofGpar:nb:1=W0(T;G)fT;Ggtranslations1param.1=W1(T;G)fT;Tn1:Ggdilatations2param.1=W2(T;G)fT;Tn1:Tn2:Ggexponentials3param.1=W3(T;G)fT;Tn1:Tn2:Tn3:Ggtransexp.ofstr.14param.:::::::::::::::::::::::::::::::::::::::::::::1=Wk(T;G)fT;Tn1:


:Tnk:Ggtransexp.ofstr.k�2kparam. 42anoperatorG7!P(G)isk-shrinkingif,8m�0,it`shrinks'anytransexponentialofstrengthm+ktosomethingthatliesinthetransexponentialgrowthrangeofstrengthm.84 onthesegraphsandappearinginthepropersuccession.44hi:=kii


k11(8.343)Z1:=(h0(z);h1(z))=(z;h1(z))Z2:=(h1(z);h2(z)):::Zr�1:=(hr�2(z);hr�1(z))Zr:=(hr�1(z);hr(z))=(hr�1(z);z)Functionalcontinuationandperiodicadjustment:Forany(~f0;~g0)wemayalwaysconstructsuchasquarewithanapproximate( f0:=T; g0),with f0:=Tand g0smooth,evenanalytic,exceptpossiblyattheextremalpointsz=aandz=b.Provided g0iscloseenoughtoaleadingsectionof~g0(itselftakenlargeenoughtodetermineallthenitelymany,discreteorcontinuousparametersonwhichthegeneraltwinsolution(f;g)depends)thereclearlyexistsauniquefunctionalcontinuationof g0(thatof f0istrivial)overthewholeinterval[b;+1[,with g0everywheresmooth(oranalytic)exceptpossiblyatasequencesof`images'z3;z4;z5:::ofz1;z2andwiththeconsecutiveimagesoftherpointsZiretainingtheirproperorderinsidethesuccesiveimagesofthe`fundamentalsquare'.Anasymptoticanalysisof g0at+1willrevealanoscillatorypartin-terferingwiththepropertransserialpart.Buttheseparasiticaloscillationsmayalwaysbeuniquelycorrectedbyconjugation g07!g0=h�1 g0hwithasuitable1-periodicmappingh,ieonethatcommuteswiththeunitshiftT.Andnotonlydoesthisuniqueh-conjugationspiritawaytheparasiticaloscillations;italsohastheautomaticeectofrestoringsmoothnessatthepointsz1;z2;z3:::.Thepair(f0;g0)thusobtainedmayberegardedastheexactgeometriccounterpart,or`sum',oftheformalpair(~f0;~g0).Thereexits,however,asignicantdierencedependingonwhethertheformalpair( f0; g0)doesordoesnotinvolvetransexponentialsEn.Ifitdoesn't,then(f0;g0)isdeterminedabsolutelyandreal-analyticon[:::;+1[,thoughofcourseusuallynotat+1. 44thatorderisalwaysunambiguouslydeterminedbytheformaltwin(~f0;~g0),duetothefullorderthatexistsontransseries.86 structureaswellasthenaturaltopology.Embedding-inducedordersonfreegroups:AnyisomorphismAi7!fiofther-generatorfreegroupGr(A)intoasub-groupofthegroupITendowedwithitsnaturalorderorsomeexoticI;,clearlyinducesatotalgrouporder(withleft-andright-stability)onGr(A).Similarly,foranysystemffigoftransserialstwinsorsiblings,suchanimbeddinginducesatotalorderonasuitablequotientGr(A)=fW(A)g.Whilethereprobablyexist,onthefreeGr(A)orits`bound'quotients,othergroupordersthanthoseobtainableintheabovemanner,thishugeclassofembedding-inducedgroupordersisnonethelessquiteinteresting,ifonlybecauseitpointstothreeratherstrangedichotomies:{rst,thedichotomybetweennatural/exoticorders,dependingonwhethertheunderlyingembeddingisintofIT;gorfIT;I;g.{second,thedichotomybetweenconvergent/divergentorders,dependingonwhetherallffigcan/cannotbechosensimultaneouslyconvergent.{third,thedichotomybetweensummable/non-summableorders,dependingonwhetherallffigcan/cannotbechosensimultaneouslysummable45.Ultraexponentialarbitration:Inthecaseoftwinsorsiblingsffig,theskilfuladditiontothemofinnites-imaltermsoftype1=E1,iesmallbeyondalltransexponentialorders,canalwaysrestoreindependence,andinduceonGr(A)grouporderswhichsome-timescanbeobtainedinnootherway.Thus,theorderonthetwo-generatorgroupGr(A1;A2)correspondingtotheindependentpairff1(z):=E1(z);f2(z):=z+1gor,whatamountstothesame46,tothepairff1(z):=z+1;f2(z):=z+1=E1(z)g,isspecictothat(very)particularembedding,andcapableofaremarkablecombinatorialinterpretation. 45thisthirddichotomyisquitedistinctfromthesecondone!46indeed,bothpairsff1;f2gandff1;f2gareconjugateundera\superfast"E1.Note,however,thatreplacingE1byE1intheaboveembeddingwouldmakenodierencetotheorderinducedonGr(A1;A2).Indeed,althoughE1andE1are\distinguishable"whenusedjointlyinthesamerelation{thelatterismuchfastergrowingthantheformerandallitsniteiterates{theyarenonetheless\undistinguishable"whenoccuringinisolation!88 Areidentity-tangenttwinsalwaysdivergent?Whatisthearithmeticnature47oftheirresurgenceinvariants?Whatautarkyrelationsdotheseverify48?Doalltransserialformaltwinspossessareal-analytic(asopposedtomerelycohesive)geometricrealisation?Someavenuesforexploration:ExtendtheUker-ltrationtothecaseofr-generatorfreealgebrasandhigherdimensionaldierentialrepresentations.Calculatethedimensionsthatgowiththeseltrations.Investigatethehigher-dimensionalanalogueoftwinsandsiblings.Explorethepotential49ofrepresentationsoffreeor`nearlyfree'groupsintogroupsof(oneormanydimensional)germmappings.REFERENCES.[Ce]D.Cerveau,Unelistedeproblemes,inEcuacionesDiferenciales,pp455-460,Univ.Valladolid,1997[Co]S.D.Cohen,Thegroupoftranslationsandrationalpowersisfree,Ph.D.thesis,BowlingGreenStateuniversity,19[E1]J.Ecalle,Lesfonctionsresurgentes,Vol.1,Algebresdefonctionsresurgentes.Publ.Math.Orsay(1981).[E2]J.Ecalle,Lesfonctionsresurgentes,Vol.2,Lesfonctionsresurgentesappliqueesal'iteration.Publ.Math.Orsay(1981).[E3]J.Ecalle,Lesfonctionsresurgentes,Vol.3,L'equationdupontetlaclassicationanalytiquedesobjetslocaux.Publ.Math.Orsay(1985).[E4]J.Ecalle,Theaccelerationoperatorsandtheirapplications.Proc.In-ternat.Cong.Math.,Kyoto,1990,vol.2,Springer,Tokyo,1991,p1249-1258.[E5]J.Ecalle,IntroductionauxfonctionsanalysablesetpreuveconstructivedelaconjecturedeDulac.Actual.Math.,Hermann,Paris,1992. 47presumablytranscendental48whentherearefreecontinuousparameters.49forthe\wordproblem",thedescriptionofallpossiblegrouporders,etc90