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Exponentially long time stability for non-linearizable analytic germs of (C^n,0)
Carletti, Timoteo
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Annales de l'Institut Fourier
Publication date:
2004
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Citation for pulished version (HARVARD):
Carletti, T 2004, 'Exponentially long time stability for non-linearizable analytic germs of (C^n,0)', Annales de
l'Institut Fourier, vol. 54, no. 4, pp. 989-1004.
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TIMOTEOCARLETTI
Abstra t. Westudy the Siegel{S hroder enter problem on the lineariza-tionofanalyti germsofdi eomorphismsinseveral omplexvariables,inthe Gevrey{s, s > 0 ategory. We introdu e a new arithmeti al ondition of Brunotypeonthe linearpartofthegivengerm,whi hensurestheexisten e of a Gevrey{s formallinearization. We usethisfa t to provethe e e tive stability,i.e. stabilityfor nitebutlongtime,ofneighborhoodsoftheorigin fortheanalyti germ.
1. Introdu tion
In this paper we onsider the Siegel{S hroder enter problem [He , CM, Ca℄in some lassofultradi erentiablegermsof(C
n
;0), n1;letus onsidertwo lasses of formal power series A
1 A 2 C n [[z 1 ;:::;z n ℄℄, losed w.r.t. to derivation and omposition, let F 2 A
1 and all DF(0) = A 2 GL(n; C ), we say that F is linearizable in A 2 if there exists H 2 A 2
, normalizedwith DH(0) = I,whi h solves 1 : (1.1) FÆH(z)=HÆR A (z); where R A
(z)=Az. Inthefollowingwewill assumeAto bediagonalwith eigen-valuesofunit modulus
1 ;:::; n ,thusA=diag( 1 ;:::; n ). IfbothA 1 andA 2
oin idewiththeringofformalpowerseriesthengeneri ally theformallinearizationholdsifandonlyifAisnon{resonant,namelyforall 2
N n su hthatj j= P 1in i
2,andforallj2f1;:::;ngthen
j
6=0(where weusedthestandardnotation
= 1 1 ::: n n ).
When F is a germ of analyti di eomorphisms de ned in a neighborhood of theoriginand wewanttosolve(1.1)in thesame lassofanalyti germs,wehave to onsider several ases. If Ais thePoin are domain, namely sup
1jn j j j <1 or sup 1jn j 1 j
j < 1, then Koenigs [Ko℄ and Poin are [Po℄ proved that every analyti germF 2Diff(C
n
;0)su hthatF(0)=0andDF(0)=A, isanalyti ally linearizable. WhenA isnotin thePoin aredomain,wesaythatitisin theSiegel domain;thequestionisharderandsomeadditionalarithmeti al onditionson(
j )
j areneeded(see[He ℄x17page158).
Date:July20,2004.
Keywordsandphrases. Siegel enterproblem,Gevrey lass,Bruno ondition,e e tive stabil-ity,Nekoroshevlikeestimates.
1
HereFÆHmeansthe ompositionofFandH;inthefollowingwewilldenotethe omposition ofF n-timeswithitself,byF
n
insteadofF Æn
Letp2 N
,p2andletusde nefornon{resonant 1 ;:::; n : (1.2) (p)= min 1jn inf 2N n 0<j j<p j j j;
wesay that A veri esaDiophantine ondition of type( ; ) if there exist > 0 and n 1su hthatforall 2
N n
nf0gwehave(j j) j j
. Siegel[Si ℄in 1942forthen=1 aseand thenSternberg[St ℄ andGray[Gr℄in thegeneral ase provedthatifAveri esaDiophantine onditionthenthelinearizationproblemhas ananalyti solution. Bruno[Br℄weakenedthearithmeti al onditionbyaskingthe onvergen eoftheseries
P k log 1 (2 k +1 ) 2 k
. Weremark thatintheonedimensional asetheBruno ondition
2
isoptimal,asprovedbyYo oz[Yo℄.
In[CM ℄authorsstudiedtheSiegel{S hroder enterprobleminthe aseofgeneral algebrasofultradi erentiablegermsof(C;0),in ludingtheGevrey ase. In[Ca℄the multidimensional aseis onsidered: ifA
1 =A
2
andAveri esaBruno ondition, theneveryF2A
1
withF(0)=0andDF(0)=Ais linearizablein A 2 ,whereasif A 1 isproperly ontainedinA 2
new onditionsweakerthanBrunoaresuÆ ientto ensurelinearizabilityin A
2 .
In this paper we onsider in detail the ase where A 1
is the ring of germs of analyti di eomorphismsat theoriginof n 1 omplexvariables, and A
2 is the algebra of Gevrey{s, s > 0, formal power series: the Gevrey{s linearization of analyti germs. Let ^ F = P f z , (f ) 2N n C n
beaformalpowerseries, then wesaythat it isGevrey{s[Ba,Ra℄,s>0,ifthereexisttwopositive onstantsC
1 ;C 2 su hthat: (1.3) jf jC 1 C sj j 2 (j j!) s 8 2N n :
Wedenotethe lassofallformalve torvaluedpowerseriesGevrey{sby C s
. It is losedw.r.t. derivationand omposition.
IntheGevrey{s asethearithmeti al onditionintrodu edin [CM,Ca℄will be alledBruno{s ondition,s>0: forshortA2B
s
ifthereexistsastri tlyin reasing sequen eofpositiveinteger(p
k ) k su hthat: (1.4) limsup j j!+1 0 2 ( ) X m=0 log 1 (p m+1 ) p m slogj j 1 A <+1; where( )isde ned byp ( ) j j<p ( )+1 .
Remark 1.1. This de nition re all the lassi al one of Bruno [Br℄, where rst one suppose the existen e of astri tly in reasing sequen e of positive integersu h that (1.4) holds, then one anprove (see [Br℄ xIVpage 222) that one an take an exponentially growing sequen e, e.g. p
k =2
k
. This holdsalso inour ase, in fa t we an prove that (1.4)isequivalentto:
limsup N!+1 N X l=0 log 1 (2 l+1 ) 2 l sN2log2 ! <+1: 2
Inthis aselet!2(0;1)nQ su hthat=e 2i!
andlet(q n
) n
bethedenominatorsofthe onvergents[HW℄to!,then the Bruno ondition isequivalentto the onvergen eof theseries P k 0 logq k +1 q .
Letusgiveasket hofthe proof ofthis laim. Takeasequen e(p k
)for whi h (1.4) holds, then we an nd sequen es of positive integer (m
k ) k and (l k ) k su h that: m 0 =0,l k 1,m k +1 =m k +l k and 2 m k p k <2 m k +1 <<2 m k +l k p k +1 <2 m k +l k +1 : The fun tion 1 (p) is in reasing, hen e: 1 (2 q+1 ) < 1 (p k +1 ) for all q = m k ;:::;m k +l k 1, andso: (1.5) m k +l k 1 X q=m k log 1 (2 q+1 ) 2 q <4 log 1 (p k +1 ) p k :
Takeany j j>1, the orresponding ( )and xN( )=m ( )
+l ( )
2. Then, dividing thesum of (1.4)intopie es fromm
h tom h +l h 1,for h=0;:::;( ), andusingestimate (1.5)for ea h pie e, weget:
2 0 2 ( ) X m=0 log 1 (p m+1 ) p m slogj j 1 A > N( ) X l=0 log 1 (2 l+1 ) 2 l 2slogj j:
Nowthe laimfollows remarkingthat 2slogj j sN( )2log2+ .
Forrealnumber,theBruno{s ondition anbeslightlyweakened(see[CM℄);let !2(0;1)nQ ,thentheBruno{s,s>0, onditionreads:
(1.6) limsup k !+1 0 k X j=0 logq j+1 q j slogq k 1 A <+1; where(q k ) k
arethedenominatorsofthe onvergentsto!. Weremarkthatinboth asesthenew onditionsareweakerthanBruno ondition,whi hisre overedwhen s=0. Indimensiononeweprovethat the set
S s B s isPSL(2; Z ){invariant(see remark3.1). Themainresultof[Ca℄in the aseofGevrey{s lassesreads:
Theorem1.2(Gevrey{slinearization). Let 1
;:::; n
be omplexnumbersofunit modulusand A=diag(
1 ;:::; n ); let D 1 =fz2 C n :jz i j<1;1ingbethe isotropi polydisk of radius 1 and letF :D
1 !
C n
be an analyti fun tion, su h thatF(z)=Az+f(z),withf(0)=Df(0)=0. IfAisnon{resonantandveri esa Bruno{s, s>0, ondition (1.4)(or (1.6)when n=1), then thereexistsa formal Gevrey{slinearization
^
H whi h solves (1.1).
The aim of this paperis to show that the Gevrey hara ter of theformal lin-earization angiveinformation on erningthedynami softheanalyti germ. Let F(z)= Az+f(z) beagerm of analyti di eomorphism verifying thehypothesis of Theorem 1.2, assume moreoverF not to be analyti allylinearizable. We will showthatevenifthereisnotSiegel disk,where thedynami sofF is onjugateto thedynami sofitslinearpart,wehaveanopenneighborhoodoftheoriginwhi h \behavesasaSiegel disk" under theiterates of F for nite but longtime, whi h resultsexponentiallylong: thee e tivestability[GDFGS ℄ofthe xedpoint.
Inthe aseofanalyti linearization,jH 1 i
(z)j,i=1;:::;n,(whi hiswellde ned suÆ iently lose to the origin be ause H is tangent to the identity) is onstant along theorbits, namelyitisa rstintegralandjF
m (z
0
)jisboundedforallmand suÆ ientlysmalljz
0 j.
Wewillprovethatanynon{zeroz 0
belongingtoapolydiskofsuÆ ientlysmall 1=s
the Gevreyexponent of the formal linearization, and we an nd analmost rst integral: afun tionwhi hvariesbyaquantityoforderrunder mK iterations, whi himpliesthatF
m (z
0
)iswellde nedandboundedformK. Morepre isely weprovethefollowing
Theorem1.3. Let 1
;:::; n
be omplexnumbersofunitmodulusandA=diag( 1 ;:::; n ); letF :D 1 ! C n
beananalyti andunivalentfun tion,su hthatF(z)=Az+f(z), with f(0)=Df(0)=0. If A isnon{resonantandveri es aBruno{s, s>0, on-dition (1.4)(or (1.6)whenn=1),thenforallsuÆ ientlysmall0<r
<1,there existpositive onstants A
;B
;C
su hthat for all 0<jz 0 j <r =2,the m{th iterate of z 0
by F is well de ned and veri es jz m j = jF m (z 0 )j C r , for all mK = j A 1 exp n B (r =jz 0 j) 1=s ok .
ThehypothesisonthedomainforF isanaturalnormalization onditionbeing thewholeprobleminvariantbyhomothety.
Inse tion3we ompareourstabilityresultwiththestrongerresultswhi h an beprovedusing Yo oz'srenormalizationmethod [PM2℄ inthe asen=1. More-overwe dis uss therelation between ourBruno{s ondition andthe arithmeti al ondition ofPerez{Mar o[PM1, PM2℄ ensuring that in thenon{linearizable ase the xedpointisa umulatebyperiodi orbits.
A knwoledgements. Iamgratefulto D.Sauzinforaverystimulatingdis ussion on erningGevrey lassesandasymptoti analysis.
2. Proof of the mainTheorem
Inthis partwe willproveour main result,Theorem 1.3. Theproof will be di-videdinto threesteps: rstweusetheGevrey{s hara teroftheformal lineariza-tion
^
H, given by Theorem 1.2, to nd anapproximate solutionof the onjuga y equation (1.1) up to a (exponentially) small orre tion (paragraph 2.1); then we proveaniterativeLemmaallowingusto ontrolhowthesmallerrorintrodu edin thesolutionpropagates(paragraph2.2). Finallywe olle talltheinformationsto on ludetheproof(paragraph2.3).
2.1. Determinationof anapproximate solution. WeapplyTheorem1.2: the formalpowerseriessolution
^
H belongstoC s
,aswellasitsinverse ^ H 1 whi hsolves (formally): (2.1) ^ H 1 ÆF(z)=R A Æ ^ H 1 (z): Sin e ^ H 1 = P h z 2C s
,thereexistpositive onstantsA 1 andB 1 su h that (2.2) jh jA 1 B sj j 1 (j j!) s 8j j1:
For any positiveintegerN we onsider the ve torial polynomial, sumof homoge-neousve tormonomialsofdegree1lN,de nedby: H
N (z)= P N l=1 P j j=l h z
andtheRemainder Fun tion:
(2.3) R N (z)=H N ÆF(z) R A ÆH N (z):
ThefollowingProposition olle tssomeusefulpropertiesoftheremainderfun tion.
Proposition 2.1. Let R N
(z) be the remainder fun tion de ned in (2.3) and let 2 N n ,then: 1) R N (0)=0if j jN.
2) Forall 0<r<1there existsa positive onstants A 2 andB 2 su h that if j jN+1,then: 1 ! z R N (0) A 2 r j j B sN 2 (N!) s :
3) Forall 0<r<1andjzj<r=2 thereexist positive onstants A 3 ;B 3 su h that: (2.4) jR N (z)jA 3 B sN 3 (N!) s jzj r N+1 :
Whereweusedthe ompa t notation 1 ! z = 1 1!::: n! j j 1 z 1 ::: n zn .
Proof. Statement1)isanimmediate onsequen eofthede nitionofR N
. Toprove2)weobservethatR
N
(z)isananalyti fun tion onD 1
,thenonegets byCau hy'sestimatesforall0<r<1andforallj jN+1:
(2.5) 1 ! z R N (0) 1 (2) n 1 r j j+1 max jzj=r jH N ÆF(z)j:
Re allingtheGevreyestimate(2.2)forH N
andtheanalyti ityofF weobtain:
(2.6) 1 ! z R N (0) A 2 B sN 2 (N!) s r j j ;
forsomepositive onstantsA 2
andB 2
dependingontheprevious onstants,onthe dimensionnandonF.
To prove 3) let us write the Taylorseries R N (z) = P j jN+1 1 ! z R N (0)z , thenthebound onderivatives(2.6)impliestheestimate(2.4)foralljzj<r=2and forsomepositive onstantsA
3 andB 3 . Thebound (2.4)onR N
(z)dependsonthepositiveintegerN,sowe an deter-mine thevalue of N for whi h theright hand side of (2.4) attains its minimum, that'sPoin are'sideaofsummationatthe smallestterm.
Lemma2.2(Summationatthesmallestterm). LetR N
(z)de nedasbeforeandlet 0<r
<1=2thenthereexistpositive onstantsA 4
;B 4
su hthatforall0<jzj<r wehave: (2.7) jR N (z)jA 4 exp n B 4 r jzj 1=s o ; where N =bB 4 ( r =jzj) 1=s
andbx denotes the integerpart ofx2 R . Proof. Letus x0<r <1=2,thenfor0<jzj<r
byStirlingformulaweobtain:
(2.8) jR N (z)jA 4 NB 1 3 (jzj=r ) 1=s Ns e sN ;
for somepositive onstantA 4
. The righthand side of (2.8) attainsits minimum at N = B 3 (r =jzj) 1=s
, evaluating the value of this minimum we get (2.7) with
2.2. Control of the \errors". Letus de neH (z)=H N (z)andR(z)=R N (z), being
Nthe\optimalvalue"obtainedinLemma2.2. WeremarkthatH (z)doesn't solve(2.1)butthe\error",R(z),isverysmall: exponentiallysmall. Wewillprove thatforinitial onditionsinasuÆ ientlysmalldisk,one aniterateanexponentially largenumberoftimeswithoutleavingadisk,say,ofdoublesize.
Lemma2.3 (Iterationlemma). Leta;b; andR bepositive realnumbers. Letus onsider the sequen e ofpositive number(
j ) j0 de nedby: 0 =R and j+1 = j +aexpf b= j g: LetK=bR a 1 expfb=(2R ) g , then j 2Rfor allj K.
Proof. Letusprovebyindu tiononj thatforall0j K wehave
(2.9)
j
R+jaexpf b=(2R )
g;
thenthe laimwill followfrom(2.9)andthede nitionofK,infa t foralljK:
j R+jaexpf b=(2R ) gR+R a 1 expfb=(2R ) gaexpf b=(2R ) g2R:
The basis of indu tion is easily veri ed; assume (2.9) for all j K 1, we will proveitforj=K. Byde nitionof (
j )
j
andtheindu tion hypothesis wehave:
K = K 1 +aexpf b= K 1 g R+(K 1)aexpf b=(2R ) g+aexpf b= K 1 g;
weremarkthatfrom(2.9)withj=K 1,usingK 1<R a 1 expfb=(2R ) g,we get K 1 2Randexpf b= K 1 gexpf b=(2R ) g. Thenwe on lude: K R+(K 1)aexpf b=(2R ) g+aexpf b=(2R ) g;
whi hendstheindu tion.
Letr
asinLemma2.2,de ne(z)=jH (z)jforall0<jzj<r
,thenLemma2.2 admits the following Corollary, whi h allows us to ontrol the fun tion (z) on onse utivepointsofanorbitofF(z).
Corollary2.4. Let0<r
<1=2,letr 1
betheradiusofthemaximalpolydiskwhere H (z) is invertible and let r
= min (r
;r 1
). Then there exist positive onstants A
;B
su hthat for all0<jzj<r wehave: (2.10) (F(z)) (z) A exp n B r (z) 1=s o :
Proof. Byde nition (F(z))=jHÆF(z)j and (z)=jR A
ÆH (z)j, sin ej j
j =1 for1jn,andA=diag(
1 ;:::; n ) ,therefore: (F(z)) (z) HÆF(z) R A ÆH (z) =jR(z)j;
andfromLemma 2.2weget:
(2.11) (F(z)) (z) A 4 exp n B 4 r jzj 1=s o :
Wewanttoexpressthis onditionintermsof(z)insteadofjzj,todothiswehave to onsiderthedistortionpropertiesofH (z)andofitsinverse. LetJ(z)=
z H (z) betheJa obianof H (z)and letJ
1 =max jzjr 1 jJ(z)j, where r 1
has beende ned previously. Let0<jzj<r andletus allz
0
=H (z), learlyjz 0
us allJ 2 =max jz 0 j<r2 j z 0 H 1 (z 0
)j,thenforany0<jz 0
j<r 2
thereexistsonlyone z su hthat z=H 1 (z 0 ),whi hsatis esjzjJ 2 jz 0 j=J 2 jH (z)j. Letr =min (r ;r 1
)thenfrom(2.11)forany0<jzj<r weget: (F(z)) (z) A exp n B r (z) 1=s o ; whereA =A 4 andB =B 4 J 1=s 2 .
2.3. Endofthe proof. Wearenowableto on ludetheproofofthemain Theo-rem1.3. Takeany0<jz
0 j<r
=2andletusde ne 0 =jz 0 j, m =(F m (z 0 ))for allpositiveintegermforwhi hF
m (z
0
)iswellde ned,byCorollary2.4wehave:
(2.12) m m 1 +A exp n B (r = m 1 ) 1=s o : Letus allR=jz 0 j,a=A ,b=B r 1=s
and =1=s,thenwe anapplyLemma2.3, being m m ,to on ludethat: (2.13) m r 8mK = j jz 0 jA 1 exp n B r 2jz 0 j 1=s ok :
ThisimpliesthatH (z m
)iswellde nedinthisrangeofvaluesofm,itisnot onstant andit evolvesonlyby
jH (z m )j jH (z 0 )j r . Re alling that z m =F m (z 0 )we alsohavejF m (z 0 )jJ 2 r and jz m j jz 0 j J 2 r forall0<jz 0 j<r =2andall mK .
This on ludetheproofbysetting A =2A r 1 , B =B (r =(2r )) 1=s and C =J 2 .
3. Onedimensional ase
Inthis paperwe provedthat anyanalyti germsof di eomorphismsof (C n
;0) withdiagonal,non{resonantlinearparthasane e tivestabilitydomain,i.e. stable up to nite but \long times", lose to the xed point, provided the linear part veri esa new arithmeti al Bruno{like ondition (1.4) depending on aparameter s>0. Remark 3.1 (Invarian eof S s>0 B s
, n=1under thea tionof PSL(2;Z)). The ontinuedfra tiondevelopment[HW,MMY ℄ofanirrationalnumber! givesusthe sequen es: (a k ) k 0 and(! k ) k 0
. Then we introdu e ( k ) k 1 de ned by 1 =1 and for all integerk 0:
k = Q k j=0 ! k , whi h veri es : 1=2< k q k +1 <1and q n n 1 +q n 1 n =1, where q k
's are the denominators of the ontinued fra tion development of !. We laim that ondition Bruno{s (1.6) is equivalent to the followingone: (3.1) limsup k !+1 0 k X j=0 j 1 log! 1 j +slog k 1 1 A <+1:
This an be proved by using the relations between l
and q l
, to obtain the bound, for allinteger k>0:
k X l 1 log! l + logq l+1 q l k X l 1 log l q l+1 + l 1 log l 1 + q l 1 q l l logq l+1 18;
where weused the onvergen eof series q l and q l logq l
(see[MMY ℄ page 272).
To prove the invarian e of S
s B
s
under the a tion of PSL(2;Z), is enough to onsider its generators: T! =!+1andS! =1=!. For any irrational !,T a ts triviallybeing
k
(T!)= k
(!)forallk,whereasforSwehave k
(!)=! k 1
(S!) for all k1. Let ! bean irrational andlet !
0 =!
1
,let usalso denote witha 0 quantitiesgiven by the ontinuedfra tionalgorithm applied to!
0
,thenusing (3.1) one an prove:
! 0 0 k X j=0 0 j 1 log! 0 j 1 +s! 1 0 log 0 k 1 1 A =C(!;s)+ k +1 X j=0 j 1 log! 1 j +slog k ; whereC(!;s)=! 0 log! 1 1 slog! 0 + P 1 l=0 l 1 log! 1 l
,from whi h the laim follows.
Letus onsideraslightlystrongerversionof theBruno{s ondition: !2 R nQ belongsto ~ B s if: (3.2) lim k !+1 k X l=0 logq l+1 q l slogq k ! <+1; where(q n ) n
arethe onvergentsto!.
Remark3.2. This new onditionisstrongerthanBruno{s,be ausethe existen e of the limit isrequired. One an onstru tnumbers ! whi h verify B
s but not ~ B s , asfollows.
Letus all for shorts k = P k l=0 logql+1 q l slogq k . Fix > >0,Æ>0andto simplify takes>(2+Æ)=log2. We laimthatone an hooselargeenoughpositive integers k 1 anda 1 ;:::;a k1+1 su hthat: s k s k 1 Æ and s k1 2( ;2 );
forinstan etakek 1 >( logq 1 )Æ 1 andindu tivelya l+1 e Æql ql 1 q l 2 2ql q l 1 q 1 l , for all l = 1;:::;k 1
. Then one an take suÆ iently many a l 's equal to one, a k 1 +1+l =1for l=1;:::;2k 2
+1,and verifythat:
s k1+1+2j s k1+1+2j 2 < Æ;8j=0;:::;k 2 and s k1+1+2k2 < :
Then one iterate taking asuÆ iently large blo k of largeenough a k
's, followed by a suÆ iently large blo k of a
k
= 1. The real number whose ontinued fra tion development is given by x =[0;a
1 ;:::;a
n
;:::℄, veri es by onstru tion Bruno{s, with s>(2+Æ)=log2, being(s
k )
k
bounded by2 , butitdoesn'tverify ~ B s ,in fa t (s k ) k
os illatesfromvalueslargerthan tovaluessmallerthan ,withoutrea hing any limit.
Letusintrodu etwootherarithmeti al onditions. LetusdenotebyB 0 s
theset ofirrationalnumberswhose onvergentsverify:
(3.3) lim k !+1 logq k +1 q k logq k =s:
And ase ond onditionas follows, let( m ) m1 and (s m ) m1
betwopositive se-quen esofrealnumberssu hthat:
P +1 m = <+1and P +1 s m =<+1,
thenwede ne onditionB ; by: (3.4) logq m+1 q m s m logq m + m 8m1:
Proposition 3.3. Let ! 2 (0;1)nQ and let s > 0 then we have the following in lusions:
1) let!2 ~ B s
,if ! isnotaBruno number then!2B 0 s ,otherwise !2B 0 0 . 2) Letsand!2B ; then!2 ~ B s ;
Proof. Toprovethe rststatementletuswritethefollowingidentity:
(3.5) k X l=0 logq l+1 q l slogq k =C+ k X l=2 logq l+1 q l s(logq l logq l 1 ) ; where C =(1 s)logq 1 + logq 2 q 1 . By ondition ~ B s
, this series onvergesand then itsgeneri termgoestozero,fromwhi hweget:
(3.6) lim k !+1 logq k +1 q k logq k =s 1 lim k !+1 logq k 1 logq k : Letusdenotebys 0
bevalueoftherighthand sideof (3.6),then learlys 0
2[0;s℄. Letussupposes
0
>0,butthenwehaveforallsuÆ ientlylargek: C 1 q k logq k logq k +1 C 2 q k ;
forsomepositive onstantsC 1
;C 2
,from whi hweget logq k logq k +1 !0,andfrom(3.6) we on ludethat s 0 =s. Ifs 0 =0,namely logq k logqk +1
!1,thenitiseasyto he kthat!isaBrunonumber. Letusprovethese ondstatement. Foranypositiveintegerk,usingthede nition ofB ; we an write: (3.7) k X l=0 logq l+1 q l slogq k k X l=0 s l logq l + k X l=0 l slogq k ;
forall0lkwehavelogq l
logq k
thentherighthandsideof (3.7)isbounded by: logq k s P k l=0 s l + P k l=0 l . Byhypothesis P k l=0 s l
s, for allk, then using logq k logq 1 weobtain: k X l=0 logq l+1 q l slogq k logq 1 s k X l=0 s l ! + k X l=0 l ;
thenpassingto thelimitonkwehave:
lim k !+1 k X l=0 logq l+1 q l slogq k logq 1 (s )+ <+1:
Remark 3.4. These new arithmeti al onditionsareweaker thanthe Bruno one, forinstan e onditionB
0 s
isveri edbynumbers! whosedenominators(q k
) k
satisfy a growth ondition like q
k +1 (q k !) s . Condition B ;
implies onvergen e of the series: P k 0 logq k +1 qklogqk .
Let us on lude re alling a stability result of Perez{Mar o [PM1, PM2℄ and ompareitwithourresult. In[PM2℄authorproved(TheoremV.2.1Annexe2xf) usingageometri renormalizations heme"alaYo oz"validintheonedimensional ase,astabilityresultthat anbestatedasfollows:
Theorem3.5 (Perez{Mar o,Contr^ole deladi usion). Let! 2(0;1)nQ andlet (q
k )
k
be the denominators of its onvergents. Let F be an analyti andunivalent fun tion de nedin the unit disk fz2
C
:jzj<1g su h thatF(z)=z+O(jzj 2
), where=e
2i!
. Thereexisttwopositive onstants C 1
;C 2
su hthat if:
(3.8) jzjC 1 e P k 1 j=0 logq j+1 q j ; thenfor all integer0mq
k wehave: (3.9) jF m (z)jC 2 e P k 1 j=0 logq j+1 q j :
The meaning of the Theorem is lear: if westart inside a disk of radius r =
C 1 e P k 1 j=0 logq j+1 q j
thenwe anapply F, up to q k
times, without leavingadisk of radius rC
2 =C
1
. To ompare thisresult with oure e tivestabilityresult we have tomakeexpli ittherelationbetweenr andq
k
,whi hgivethetimeof "stability". Using our Bruno{s ondition (3.2) we an say that C rq
s k 1 C 0 for some positive onstantsC ;C 0
. But from (3.3)weget logq k C 3 q k 1 logq k 1 for some positive onstantC 3
,namelythereexist positive onstantsC 0 3 ;C 4 su hthat: q k exp n C 0 3 r 1=s log C 4 r 1=s o :
We anthenrestateTheorem3.5asfollows: ifjzjr,thenjF m (z)jrC 2 =C 1 for allinteger0m expf
C 0 3 r 1=s log C 4 r 1=s
g, obtainingabetterestimate onthetime of e e tivestability.
This improvement has been obtained thanks to a good understanding of the geometryofthedynami s,ifonewouldliketoobtainthesebetterestimatesalsofor germsinhigherdimensions,oneshouldextendthePerez{Mar oideastounderstand thegeometryofdynami sofgermsinhigherdimension. This ouldbeverydiÆ ult whereasourresultsareeasytoadapttoanydimension.
Weendwithalast remarkrelatedagaintotheworkofPerez{Mar o.
Remark 3.6. Perez{Mar o proved in [PM1, PM2℄ that any non{analyti ally lin-earizable analyti germ, univalent in the unit disk, whose multiplier at the xed point,veri es the following arithmeti al ondition:
(3.10) X k 0 loglogq k +1 q k <+1;
hasasequen eofperiodi orbitsa umulatingthe xedpoint,whoseperiods,(q nk
) k
, make the Brunoseriesdiverging.
OurBruno{s onditionimplies (3.10),infa t from (3.2)weget: N X k =0 loglogq k +1 q k N X k =0 logC 3 q k + logq k q k + loglogq k q k ;
we anletN grow andusingstandardnumbertheoryresults on erningthe onver-gents, we obtain the Perez{Mar o ondition. Then we an suppose these periodi
orbits from atoo fast es ape", a situation similarto the one holding in the Neko-roshevTheoremfor Hamiltoniansystems[Ne℄,wherethe resonantweb on nesthe ow for exponentially long times. It would be very interesting to know whether a similar phenomenontakes pla einhigher dimension.
We on lude by pointing out that our method gives us a stability exponent de-pending on the Gevrey exponent and independent of the dimension: the bigger is the exponent, longer is the time interval of stability, we an always take s small enough tohaveavery longtimeofstability.
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(TimoteoCarletti)S uolaNormaleSuperiore,piazzadeiCavalieri7,56126Pisa,Italy, andAstronomieet
Syst
emesDynamiques,Institutde M
e anique C
eleste, 77Av. Den-fertRo hereau,75014Paris,Fran e