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Lipschitzstrati12cationinpowerboundedominimal12eldsYimuYinjointworkw


15Strati12cationLetX18RnbeasubsetAstrati12cationofXisafamilyXX018X11811118XdXofsubsetsofXsuchthat15dimXi20ifor020i20d1523XiXinXi01calledthei-thskeletoniseitheremptyoradi11erentiablesubmanifoldofRnofdi

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Document on Subject : "Lipschitzstrati12cationinpowerboundedominimal12eldsYimuYinjointworkw"— Transcript:

1 Lipschitzstrati cationinpower-bounde
Lipschitzstrati cationinpower-boundedo-minimal eldsYimuYin(jointworkwithImmiHalupczok)SingularLandscape:aconferenceinhonorofBernardTeissier1 Strati cationLetXRnbeasubset.Astrati cationofXisafamilyX=(X0X1Xd=X)ofsubsetsofXsuchthatdimXiifor0id,Xi:=XinXi�1,calledthei-thskeleton,iseitheremptyoradi erentiablesubmanifoldofRnofdim

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),visdi erentiableoneachstratumofX,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`kcdist(a;Xe`)Foreachiwithemie0,(exactly)oneofthetwofollowingconditionsholds:(dist(a0;Xi�1)C0dist(a0;Xi)ifi2fe0;:::;emgdist(a0;Xi�1)c0dist(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?a0Pa1Pam)val(a0�a1)�vald

20 ist(a0;Xem�1);if(ai)isaugmentedt
ist(a0;Xem�1);if(ai)isaugmentedthenval((Pa0�Pa00)Pa1Pam)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