Lipschitzstrati12cationinpowerboundedominimal12eldsYimuYinjointworkw - PDF document

1 Lipschitzstraticationinpower-bounde
Lipschitzstraticationinpower-boundedo-minimaleldsYimuYin(jointworkwithImmiHalupczok)SingularLandscape:aconferenceinhonorofBernardTeissier1 StraticationLetXRnbeasubset.AstraticationofXisafamilyX=(X0X1


Xd=X)ofsubsetsofXsuchthatdimXiifor0id,Xi:=XinXi�1,calledthei-thskeleton,iseitheremptyoradierentiablesubmanifoldofRnofdim

2 ensioni(notnecessarilycon-nected),andeac
ensioni(notnecessarilycon-nected),andeachconnectedcomponentofXiiscalledastratum,ForeachstratumS,clSS[Xi�1isaunionofstrata. ProjectionstotangentspacesForeachpointa2Xi,letPa:Rn�!TaXiandP?a:=id�Pa:Rn�!T?aXibetheorthogonalprojectionsontothetangentandthenormalspacesofXiata. Verdier'sconditionLetX=(Xi)beastraticationofX.Foreveryiandeverya2Xithereare&#

3 15;an(open)neighborhoodUaXofa,
15;an(open)neighborhoodUaXofa,aconstantCasuchthat,foreveryji,everyb2Xi\Ua,everyc2Xj\UawehavekP?cPbkCakc�bk: IntermsofvectoreldsLetX=(Xi)beastraticationofX.AvectoreldvonanopensubsetUXisX-rugoseifvistangenttothestrataofX(X-compatibleforshort),visdierentiableoneachstratumofX,foreverya2Xi\UthereisaconstantCasuchthat,fo

4 reveryji,allb2Xi\Uandc2Xj
reveryji,allb2Xi\Uandc2Xj\Uthataresucientlyclosetoasatisfykv(b)�v(c)kCakb�ck: ConcerningVerdier'sconditionTheorem.(Verdier)Everysubanalyticsetadmitsastrati-cationthatsatisesVerdier'scondition.(Loi)Theaboveholdsinallo-minimalstructures.Theorem(Brodersen{Trotman).XisVerdierifandonlyifeachrugosevectoreldonU\Xicanbeextendedtoarugosevectoreldonaneighb

5 orhoodofU\XiinX.IngeneralVerdier'scondit
orhoodofU\XiinX.IngeneralVerdier'sconditionisstrictlystrongerthanWhitney'scondi-tion(b).Butwedohave:Theorem(Teissier).Forcomplexanalyticstratications,Verdier'sconditionisequivalenttoWhitney'scondition(b). ConcerningMostowski'sconditionMostowski'sconditionisa(much)strongerconditionthanVerdier'scon-dition.Theorem(Parusinski).XisLipschitzifandonlyifthereisacon-stantCsuchthat,foreveryXi�1WXi,

6 ifvisanX-compatibleLipschitzvectore
ifvisanX-compatibleLipschitzvectoreldonWwithconstantLandisboundedonthelaststratumofXbyaconstantK,thenvcanbeextendedtoaLipschitzvectoreldonXwithconstantC(K+L).Theorem(Parusinski).LipschitzstraticationsexistforcompactsubanalyticsubsetsinR.Mainingredientsoftheproof:local atteningtheorem,Weierstrasspreparationforsubanalyticfunctions,andmore. Theorem(Nguyen{Valette).Lipschitzstraticationsexistfor

7 allde-nablecompactsetsinallpolynomi
allde-nablecompactsetsinallpolynomial-boundedo-minimalstructuresontherealeldR.TheirprooffollowscloselyandimprovesuponParusinski'sproofstrat-egy;inparticular,itrenesaversionoftheWeierstrasspreparationforsubanalyticfunctions(vandenDries{Speissegger).Ontheotherhand,ourresultstates:Theorem.Lipschitzstraticationsexistforalldenableclosedsetsinallpower-boundedo-minimalstructures(forinstance,in

8 theHahneldR((tQ))).Ourproofbypasses
theHahneldR((tQ))).Ourproofbypassesallofthemachineriesmentionedaboveandgoesthroughanalysisofdenablesetsinnon-archimedeano-minimalstructuresinstead. o-minimalityDefinition.LetLbealanguagethatcontainsabinaryrelation.AnL-structureMissaidtobeo-minimalifisatotalorderingonM,everydenablesubsetoftheanelineisaniteunionofintervals(includingpoints).AnL-theoryTiso-minimalifeveryoneof

9 itsmodelsiso-minimal. Twofundamenta
itsmodelsiso-minimal. Twofundamentalo-minimalstructuresTheorem(Tarski).ThetheoryRCFoftherealclosedeld(essen-tiallythetheoryofsemialgebraicsets)R=(R;+;;0;1)iso-minimal.Theorem(Wilkie).ThetheoryRCFexpoftherealclosedeldwiththeexponentialfunctionRexp=(R;+;;0;1;exp)iso-minimal. Polynomial/powerboundedstructuresLetRbeano-minimalstructurethatexpandsarealclosedeld.Definition.Apower

10 functioninRisadenableendomorphismof
functioninRisadenableendomorphismofthemultiplicativegroupofR.(Notethatsuchapowerfunctionfisuniquelydeterminedbyitsexponentf0(1).)WesaythatRispower-boundedifeverydenablefunctioninonevariableiseventuallydominatedbyapowerfunction.Theorem(Miller).EitherMispowerboundedorthereisaden-ableexponentialfunctioninM(meaningahomomorphismfromtheadditivegrouptothemultiplicativegroup).Note:InR,power-boundedbecome

11 spolynomial-bounded. Examplesofpoly
spolynomial-bounded. Examplesofpolynomial-boundedo-minimalstructuresonRRCF.(Semialgebraicsets).RCFan:Thetheoryofrealclosedeldswithrestrictedanalyticfunctionsfj[�1;1]n.(Subanalyticsets).RCFan;powers:RCFanplusallthepowers(xrforeachr2R).FurtherexpansionsofRCFanbycertainquasi-analyticfunctions{certainDenjoy-Carlemanclasses,{Gevreysummablefunctions,{certainsolutionsofsystemsofdi

12 erentialequations. Mostowski'scondi
erentialequations. Mostowski'scondition(quantitativeversion)Fixa(complete)o-minimaltheoryT(notnecessarilypowerbounded).LetRbeamodelofT,forexample,R;R((tQ));R((tQ1))((tR2));etc.TheMostowskiconditionisimposedoncertainnitesequencesofpointscalledchains.Thenotionofachaindependsonseveralconstants,whichhavetosatisfyfurtherconditionsonadditionalconstants.InR,letXbeadenablesetandX=(Xi)adenablestrati&

13 #12;cationofX. Definition.Letc;c0;C0;C00
#12;cationofX. Definition.Letc;c0;C0;C002Rbegiven.A(c;c0;C0;C00)-chainisasequenceofpointsa0;a1;:::;aminXwitha`2Xe`ande0�e1�


�emsuchthatthefollowingholds.For`=1;:::;m,wehave:ka0�a`kc
dist(a;Xe`)Foreachiwithemie0,(exactly)oneofthetwofollowingconditionsholds:(dist(a0;Xi�1)C0
dist(a0;Xi)ifi2fe0;:::;emgdist(a0;Xi�1)c0
dist(a0;Xi)ifi=2fe0

14 ;:::;emg: Anaugmented(c;c0;C0;C00)-chain
;:::;emg: Anaugmented(c;c0;C0;C00)-chainisa(c;c0;C0;C00)-chaintogetherwithanadditionalpointa002Xe0satisfyingC00ka0�a00kdist(a0;Xe0�1): Definition.WesaythatthestraticationX=(Xi)satisestheMostowskiconditionforthequintuple(c;c0;C0;C00;C000)ifthefollowingholds.Forevery(c;c0;C0;C00)-chain(ai),kP?a0Pa1:::PamkC000ka0�a1k dist(a0;Xem�1):Foreveryaugmented(c;c0;C0;C00)-chain((ai);a00),k(Pa

15 0�Pa00)Pa1:::PamkC000ka0�a00k dist
0�Pa00)Pa1:::PamkC000ka0�a00k dist(a0;Xem�1):Mostowski'soriginaldenition(?):Definition.ThestraticationXisaLipschitzstraticationifforevery1c2RthereexistsC2RsuchthatXsatisestheMostowskiconditionfor(c;2c2;2c2;2c;C). PlayingwiththeconstantsProposition.ThefollowingconditionsonXareequivalent:(1)XisaLipschitzstratication(inthesenseofMostowski).(2)Foreveryc2R,thereexistsaC2Rsuchth

16 atXsatisestheMostowskiconditionsfor
atXsatisestheMostowskiconditionsfor(c;c;C;C;C).(3)Foreveryc2R,thereexistsaC2RsuchthatXsatisestheMostowskiconditionsfor(c;c;1 c;1 c;C).Note:(1))(2)and(3))(1)areeasy.But,atrstglance,(2))(3)ishardlyplausible,because(3)considersmuchmorechains.Toshowthat,wewill(already)need\nonarchimedeanextrapolation"oftheMostowskicondition. Nonarchimedean/nonstandardmodelsLetVRbeaproperconvexsubring.Fact.T

17 hesubringVisavaluationringofR.Definition
hesubringVisavaluationringofR.Definition.ThesubringViscalledT-convexifforalldenable(noparametersallowed)continuousfunctionf:R�!R,f(V)V:LetTconvexbethetheoryofsuchpairs(R;V),whereVisanadditionalsymbolinthelanguage.Example.SupposethatTispowerbounded.LetRbetheHahneldR((tQ)).LetVbetheconvexhullofRinR,i.e.,V=R[[tQ]].ThenVisT-convex.Ourproofisactuallycarriedoutinasuitablemodel(R;V)ofTconvex,usingami

18 xtureoftechniquesino-minimalityandvaluat
xtureoftechniquesino-minimalityandvaluationtheories. ValuativechainsLetvalbethevaluationmapassociatedwiththevaluationringV.Definition.Aval-chainisasequenceofpointsa0;:::;amwitha`2Xe`ande0�e1�


�emsuchthat,forall1`m,val(a0�a`)=valdist(a0;Xe`�1�1)=valdist(a0;Xe`)�valdist(a0;Xe`�1):Anaugmentedval-chainisaval-chaina0;:::;amtogetherwithonemorepoin

19 ta002Xe0suchthatval(a0�a00)�
ta002Xe0suchthatval(a0�a00)�valdist(a0;Xe0�1):Definition.Ifwereplace�withinthetwoconditionsabovethentheresultingsequenceiscalledaweakval-chain.Notethata\segment"ofa(weak)val-chainisa(weak)val-chain. ThevaluativeMostowskiconditionDefinition.ThevaluativeMostowskiconditionstates:forallval-chain(ai),if(ai)isnotaugmentedthenval(P?a0Pa1


Pam)val(a0�a1)�vald

20 ist(a0;Xem�1);if(ai)isaugmentedt
ist(a0;Xem�1);if(ai)isaugmentedthenval((Pa0�Pa00)Pa1


Pam)val(a0�a00)�valdist(a0;Xem�1):Note:weshouldusetheoperatornormabove,butval(M)=val(kMk)foramatrixM. ValuativeLipschitzstraticationDefinition.ThestraticationXisavaluativeLipschitzstrati-cationifeveryval-chainsatises(thecorrespondingclauseof)theval-uativeMostowskicondition.Proposition.Thefollowingar

21 eequivalent:(1)XisaLipschitzstratic
eequivalent:(1)XisaLipschitzstraticationinthesenseofMostwoski.(2)XisavaluativeLipschitzstratication.(3)Everyweakval-chainsatisesthevaluativeMostowskicondi-tion.Note:Thevaluative\(2))(3)"hereimpliesthequantitative\(2))(3)"statedbefore. Strategy/mainingredientsoftheconstructionLetXbeadenableclosedsetinR.WeshallconstructastraticationYofXsuchthatYisdenableinR,Yisavaluati

22 veLipschitzstraticationin(R;V).West
veLipschitzstraticationin(R;V).WestartwithanystraticationX=(Xi)ofXinR.ThedesiredstraticationisobtainedbyreningtheskeletonsXsoneaftertheother,startingwithXdimX.Inductively,supposethatXs+1;:::;XdimXhavealreadybeenconstructed.WereneXs:=Xn[�isXibyremovingclosedsubsetsofdimensionlessthansinthreesteps. ThethreestepsStepR1:WepartitionXsinto\specia

23 lcells"andremoveallsuchcellsofdimensionl
lcells"andremoveallsuchcellsofdimensionlessthans.Suchacellisessentiallyafunctionf:A�!Rn�sof\slowgrowth",moreprecisely,val(f(a)�f(a0))val(a�a0);foralla;a02A:Actually,wecannotcutXsintosuchcellsdirectly;butwecanachievesuchadecompositionmodulocertain\uniformrotation"chosenfromaxednitesetOoforthogonalmatrices,usingaresultofKurdyka/Parusinski/Pawlucki. StepR2(themainstep):Considerasequ

24 enceS=(S`)0`m,whereS`
enceS=(S`)0`m,whereS`Xe`forsomee0e1�e2�


�em=sandeveryS`isa\specialcell"(afterasinglerotationinO).ThereisasubsetZSSmofdimensionlessthanssuchthat,onceZSSmisremoved,certainfunctionsassociatedwithSsatisfycertainestimates.ThereareonlynitelymanysuchZS.Theseestimatesarealloftheformval(@if(x))val(f(x))�val(`(x))+correctionterms;where&#

25 16;`(x)isthedistancebetweenthetuplepr
16;`(x)isthedistancebetweenthetuplepre`(x)andthesubsetRe`npre`(X). StepR3:Thissteponlyperformscertaincosmeticadjustment.WekeepthenotationfromStepR2andremoveonemoresetfromSm(again,foreachchoiceofSandeachrotationinO)sothatestimatesforthefunctionsassociatedwithSinStepR2holdontheentireSm.ThisnishestheconstructionofXs.Theorem.TheresultingstraticationisavaluativeLipschitzstrat-icationofX