Linearization of Analytic and Non-Analytic Germs of Diffeomorphisms of (C,0)

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Linearization of Analytic and Non-Analytic Germs of Diffeomorphisms of (C,0)

Carletti, Timoteo; Marmi, Stefano

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2000

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Carletti, T & Marmi, S 2000, 'Linearization of Analytic and Non-Analytic Germs of Diffeomorphisms of (C,0)',

Bulletin SMF, vol. 128, pp. 69-85.

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TIMOTEOCARLETTI,STEFANOMARMI

Abstra t. WestudySiegel's enterproblemonthelinearizationofgermsof dieomorphismsinonevariable. Inadditionofthe lassi alproblemsofformal andanalyti linearization,wegivesuÆ ient onditionsforthelinearizationto belongtosomealgebrasofultradierentiablegerms losedunder omposition andderivation,in ludingGevrey lasses.

Intheanalyti asewegiveapositiveanswertoaquestionofJ.-C.Yo oz onthe optimalityofthe estimatesobtained bythe lassi almajorantseries method.

Intheultradierentiable aseweprovethatthe Brjuno ondition is suf- ient forthe linearization to belong tothe same lassof the germ. If one allowsthelinearizationtobelessregularthanthegermonendsnew arith-meti al onditions,weakerthan theBrjuno ondition. Webrie ydis ussthe optimalityofourresults.

1. introdu tion

InthispaperwestudytheSiegel enterproblem[He℄. Considertwosubalgebras A

1 A

2

of zC[[z℄℄ losed with respe t to the omposition of formal series. For examplezC[[z℄℄ , zCfzg (the usual analyti ase)or Gevrey{s lasses, s> 0(i.e. seriesF(z)= P n0 f n z n

su hthatthereexist 1 ; 2 >0su hthatjf n j 1 n 2 (n!) s

for all n 0). Let F 2 A 1

being su h that F 0 (0) = 2 C . We say that F is linearizableinA 2 ifthereexistsH 2A 2

tangenttotheidentityandsu h that

(1.1) FÆH=HÆR

where R

(z) = z. When jj 6= 1, the Poin are-Konigs linearization theorem assures that F is linearizable in A

2

. When jj = 1, = e 2i!

, the problem is mu hmorediÆ ult, espe ially ifonelooks for ne essary and suÆ ient onditions on whi h assure that all F 2 A

1

with the same are linearizable in A 2

. The onlytrivial aseisA

2

=zC[[z℄℄(formallinearization)forwhi honeonlyneedsto assumethatisnotarootofunity,i.e. !2RnQ.

Intheanalyti aseA 1

=A 2

=zCfzgletS

denotethespa eofanalyti germs F 2zCfzganalyti andinje tiveintheunitdiskD andsu hthatDF(0)=(note thatanyF 2zCfzgtangenttoR

maybeassumedtobelongtoS

providedthat thevariable z issuitably res aled). LetR (F)denote theradiusof onvergen eof the uniquetangentto the identity linearizationH asso iated to F. J.-C. Yo oz [Yo℄provedthattheBrjuno ondition(seeAppendixA)isne essaryandsuÆ ient for having R (F)>0for all F 2S

. Morepre isely Yo ozprovedthe following

Date:September20,2004.

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estimate: assume that = e 2i!

is a Brjuno number. There exists a universal onstantC>0(independentof)su hthat

jlogR (!)+B(!)jC

whereR (!)=inf F2S

R (F)andB istheBrjunofun tion(A.3). Thus logR (!) B(!) C.

Brjuno'sproof[Br℄givesanestimateoftheform

logr(!) C 0

B(!) C 00

whereone an hooseC 0

=2[He℄. Yo oz'sproofisbasedonageometri renormal-izationargumentandYo ozhimself askedwhether ornotwaspossibleto obtain C

0

=1bydire tmanipulationofthepowerseriesexpansionofthelinearizationH asin Brjuno's proof ([Yo℄, Remarque 2.7.1, p. 21). Using anarithmeti al lemma duetoDavie[Da℄(AppendixB)wegiveapositiveanswer(Theorem2.1)toYo oz's question.

We then onsider themore generalultradierentiable ase A 1 A 2 6=zCfzg. If one requires A 2 = A 1

, i.e. the linearization H to be as regular as the given germF,on e again theBrjuno onditionissuÆ ient. Ourmethods donotallow us to on lude that the Brjuno onditionis also ne essary, a statement whi h is in generalfalseasweshowin se tion2.3whereweexhibitaGevrey{like lassfor whi hthesuÆ ient ondition oin ideswiththeoptimalarithmeti al onditionfor the asso iated linear problem. Nevertheless it is quite interesting to noti e that given any algebraofformal powerseries whi his losed under omposition (as it should ifonewhishes to study onjuga yproblems)and derivation agermin the algebraislinearizablein thesame algebraiftheBrjuno onditionissatised.

If the linearization is allowed to be less regular than the given germ(i.e. A 1 isapropersubsetofA

2

)onends anewarithmeti al ondition,weakerthanthe Brjuno ondition. This onditionis alsooptimalifthesmall divisorsarerepla ed with their absolute values as we show in se tion 2.4. We dis uss twoexamples, in ludingGevrey{s lasses.

1

A knwoledgements. Weare gratefulto J.{C. Yo ozfor averystimulating dis- ussion on erningGevrey lassesandsmalldivisorproblems.

2. the Siegel enter problem

Ourrst stepwill be theformal solutionof equation (1.1)assuming only that F 2 zC[[z℄℄. Sin e F 2 zC[[ z℄℄ is assumed to be tangent to R

then F(z) = P n1 f n z n with f 1

=. Analogouslysin eH 2zC[[z℄℄is tangentto theidentity

H(z)= P 1 n=1 h n z n with h 1

=1. If is nota root of unity equation (1.1)has a uniquesolution H 2zC [[z℄℄tangent to theidentity: thepowerseries oeÆ ients satisfythere urren erelation

(2.1) h 1 =1; h n = 1 n n X m=2 f m X n1+:::+nm=n;ni1 h n1 :::h nm :

In[Ca℄itisshownhowtogeneralizethe lassi alLagrangeinversionformulato non{analyti inversionproblemsontheeldofformalpowerseries soasto obtain anexpli itnon{re ursiveformulaforthepowerseries oeÆ ientsofH.

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2.1. The analyti ase: a dire t proof of Yo oz's lower bound. Let S denote thespa eof germsF 2zCfzg analyti andinje tivein the unit disk D = fz 2 C; jzj < 1g su h that DF(0) = and assume that = e

2i!

with ! 2 RnQ. With thetopologyofuniform onvergen eon ompa tsubsetsofD, S

isa ompa tspa e. LetH

F

2zC[[z℄℄ denotetheuniquetangenttotheidentityformal linearization asso iated to F, i.e. the unique formal solution of (1.1). Its power series oeÆ ientsaregivenby(2.1). LetR (F)denotetheradiusof onvergen eof H

F

. FollowingYo oz([Yo℄,p. 20)wedene

R (!)= inf F2S

R (F):

Wewillprovethefollowing

Theorem2.1. Yo oz's lower bound.

(2.2) logR (!) B(!) C

where C is auniversal onstant (independent of !) and B is the Brjuno fun tion (A.3).

OurmethodofproofofTheorem2.1willbetoapplyanarithmeti allemmadue toDavie(seeAppendixB)toestimatethesmalldivisors ontributionto(2.1). This isa tuallyavariationofthe lassi almajorantseriesmethod asusedin[Si ,Br℄.

Proof. Let s(z) = P n1 s n z n

be the unique solution analyti at z = 0 of the

equations(z)=z+(s(z)) ,where (z)= z 2 (2 z) (1 z) 2 = P n2 nz n . The oeÆ ients satisfy (2.3) s 1 =1; s n = n X m=2 m X n1+:::+nm=n;ni1 s n1 :::s nm :

Clearlythereexist twopositive onstants 1 ; 2 su hthat (2.4) js n j 1 n 2 :

Fromthe re urren erelation (2.1) and Bieberba h{De Branges'sbound jf n

j n foralln2weobtain

(2.5) jh n j 1 j n j n X m=2 m X n1+:::+nm=n;ni1 jh n 1 j:::jh n m j:

We now dedu e by indu tion on n that jh n j s n e K(n 1) for n 1, where K is dened in Appendix B. If weassume this holds for all n

0

<n then the above inequalitygives (2.6) jh n j 1 j n j n X m=2 m X n1+:::+nm=n;ni1 s n1 :::s nm e K(n1 1)+:::K(nm 1) : ButK(n 1 1)+:::K(n m 1)K(n 2)K(n 1)+logj n jandwededu e that (2.7) jh n je K(n 1) n X m=2 m X n1+:::+nm=n;ni1 s n 1 :::s n m =s n e K(n 1) ;

asrequired. Theorem2.1thenfollowsfromthefa tthatn 1

K(n)B(!)+ 3

for someuniversal onstant >0(Davie'slemma,Appendix B).

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2.2. The ultradierentiable ase. A lassi alresultofBorelsaysthatthemap J

R : C

1

([ 1;1℄;R)!R[[x℄℄whi hasso iatestof itsTaylorseriesat0issurje tive. Ontheotherhand,Cfzg=lim

! r>0 O(D r ),whereD r =fz2C; jzj<rgandO(D r )

is the C{ve tor spa e of C{valued fun tions analyti in D r

. Between C[[z℄℄ and Cfzg one has many important algebrasof \ultradierentiable" power series (i.e. asymptoti expansionsat z=0offun tions whi h are\between"C

1

andCfzg). In this part we will study the ase A

1 or A

2

(or both) is neither zCfzg nor zC[[z℄℄butageneralultradierentiablealgebrazC[[z℄℄

(Mn) denedasfollows. Let(M n ) n1

beasequen eofpositivereal numberssu h that:

0. inf n1 M 1=n n >0; 1. Thereexists C 1 >0su hthat M n+1 C n+1 1 M n foralln1; 2. Thesequen e(M n ) n1 islogarithmi ally onvex; 3. M n M m M m+n 1 forallm;n1. Denition2.2. Letf = P n1 f n z n

2zC[[z℄℄;f belongstothealgebrazC[[z℄℄ (Mn) ifthere existtwopositive onstants

1 ; 2 su hthat (2.8) jf n j 1 n 2 M n for alln1:

The role of the above assumptions on the sequen e (M n ) n1 is the following: 0. assures that zCfzg zC[[z℄℄ (M n ) ; 1. implies that zC[[z℄℄ (M n ) is stable for derivation. Condition 2. means that logM

n

is onvex, i.e. that the sequen e (M

n+1 =M

n

)is in reasing; itimplies that zC[[z℄℄ (M

n )

n1

is an algebra,i.e. stable

bymultipli ation. Condition3.impliesthatthisalgebrais losedfor omposition: if f;g2zC [[z℄℄ (M n ) n1 thenfÆg2zC[[z℄℄ (M n ) n1

. Thisisaverynaturalassumption sin ewewillstudya onjuga yproblem.

Lets>0. A veryimportant exampleofultradierentiablealgebrais givenby the algebra of Gevrey{s series whi h is obtained hosingM

n = (n!)

s

. It is easy to he k that the assumptions 0.{3. are veried. Butalso more rapidly growing

sequen esmaybe onsidered su hasM n

=n an

b

witha>0and1<b<2. Wethenhavethefollowing

Theorem2.3. 1. If F 2 zC[[z℄℄

(M n

)

and ! is a Brjuno number then also the linearization H belongstothe samealgebrazC[[z℄℄

(Mn) . 2. If F 2zCfzgand ! veries (2.9) limsup n!+1 0 k (n) X k =0 logq k +1 q k 1 n logM n 1 A <+1

where k(n) isdened bythe ondition q k (n)

n<q k (n)+1

, then the linearization H 2zC[[z℄℄

(Mn) .

3. LetF 2zC[[z℄℄ (Nn)

,wherethe sequen e(N n

)veries 0,1,2,3andis

asymptoti- ally boundedby the sequen e(M n

)(i.e. M n

N n

for all suÆ iently largen). If ! veries (2.10) limsup n!+1 0 k (n) X logq k +1 q k 1 n log M n N n 1 A <+1

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where k(n) isdened bythe ondition q k (n)

n<q k (n)+1

, then the linearization H 2zC[[z℄℄

(M n

) .

Notethat onditions(2.9)and(2.10)aregenerallyweakerthantheBrjuno on-dition. ForexampleifgivenF analyti oneonlyrequiresthelinearizationH tobe Gevrey{sthenone anallowthedenominatorsq

k

ofthe ontinuedfra tion expan-sionof! toverifyq

k +1 =O(e

qk

)forall0<swhereasanexponentialgrowth rate of the denominators of the onvergentsis learly forbidden from the Brjuno ondition. If thelinearization is required only to belong to the lass zC[[z℄℄

(M n ) withM n =n an b

,witha>0and1<b<2,one anevenhaveq k +1 =O(e q k )for

all>0and1<<bandtheseries P k 0 logqk +1 q b k

onverges. Thiskindofseries

havebeenstudiedin detailin[MMY ℄.

Proof. Weonlyprove(2.10)whi h learlyimplies(2.9)( hoosingN n 1)andalso assertion1. ( hoosingM n N n ). Sin eitisnotrestri tiveto assume

1 1and 2 1injf n j 1 n 2 N n one an immediately he kbyindu tiononnthatjh

n j n 1 1 2n 2 2 s n N n e K(n 1) ,wheres n isdenedin (2.3). Thusby(2.4)andDavie'slemmaonehas

1 n log jh n j M n 3 + 1 n log N n M n + k (n) X k =0 logq k +1 q k

forsomesuitable onstant 3

>0.

Problem. Are the arithmeti al onditions stated in Theorem 2.3 optimal? In parti ularisittruethatgivenanyalgebraA=zC[[z℄℄

(Mn)

andF 2AthenH 2A ifandonly if! isaBrjuno number?

Webelievethatthisproblemdeservesfurtherinvestigationsandthat some sur-prisingresultsmaybefound. Inthenexttwose tionswewillgivesomepreliminary results.

2.3. A Gevrey{like lass where the linear and non linear problem have the samesuÆ ient arithemti al ondition. LetC[[z℄℄

s

denotethealgebraof Gevrey{s omplex formal power series, s > 0. If s

0 > s > 0 then zC[[z℄℄ s zC[[z℄℄ s 0 ;let A s = \ s 0 >s zC[[z℄℄ s 0 : ClearlyA s

isanalgebrastablew.r.t. derivativeand omposition. Thisalgebra an beequivalently hara terizedrequiringthatgivenf(z)=

P n1 f n z n 2zC[[z℄℄one has (2.11) limsup n!1 logjf n j nlogn s

ConsiderEuler'sderivative(see[Du℄,se tion4)

(2.12) (Æ f)(z)= 1 X ( n )f n z n ;

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with =e 2i!

. It a tslinearlyon zA s

and it isalinearautomorphism ofzA s if andonlyif (2.13) lim k !1 logq k +1 q k logq k =0 where,asusual,(q k ) k 2N

isthesequen eofthedenominatorsofthe onvergentsof !. Thisfa t anbe easily he kedbyapplying thelawof thebest approximation (LemmaA.3,Appendix A)andthe haraterization(2.11)to

h(z)=(Æ 1 f)(z)= X n2 f n n z n :

Note that the arithmeti al onditionlogq k +1

=o( q k

logq k

)is mu h weakerthan Brjuno's ondition.

We now onsider the Siegel problem asso iated to a germ F 2 A s

. Applying thethird statementofTheorem 2.3with N

n =( n!) s+ and M n =( n!) s+ for any positive xed > > 0 one nds that ifthe following arithmeti al ondition is satised (2.14) lim k !1 1 logq k k X i=0 logq i+1 q i =0

thenthelinearizationH F

alsobelongsto A s

. 2

Theequivalen eof(2.14)and(2.13)istheobje tofthefollowing

Lemma 2.4. Let ( q l

) l0

be the sequen e of denominators of the onvergents of !2RnQ. The following statements areallequivalent:

(1) lim n!1 1 logn P k (n) l=0 logq l+1 q l =0 (2) P k ( n) l=0 logql+1 q l =o( logq k ) (3) logq k +1 =o(q k logq k )

Proof. 1. =)2. is trivial( hoosen=q k (n)

).

2. =)3. Writingforshort kisteadofk( n)

1 logq k k X l=0 logq l+1 q l = logq k +1 q k logq k + 1 logq k k 1 X l=0 logq l+1 q l = logq k +1 q k logq k + o(logq k 1 ) logq k Sin elim k !1 o(logq k 1 ) logqk =0weget3. 2

InTheorem2.3weprovedthatasuÆ ient onditionwiththis hoi eofMnandNnis

limsup n!+1 0 k (n) X i=0 logq i+1 qi n log(n!) 1 A C<+1

whi h anberewrittenas

limsup n!+1 0 k (n) X i=0 logq i+1 q i ( )logq k (n) C 1 A =0

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3. =)1. Firstofallnote thatsin eq k (n)

n2. triviallyimplies1. Thus itis enoughtoshowthat3. =)2.

logq k +1 =o( q k logq k )means: 8>0 9^n( ) su h that8l>n^( ) logq l+1 q l logq l < Iflogq l+1 <aq l

forsomepositive onstantsaand<1then:

1 logq k k X l=0 logq l+1 q l a logq k 1 X l=0 1 q 1 l aC logq k

forsomeuniversal onstantC thanksto(A.2). Iflogq l+1 aq l and 1 2

<<1, onsider thede omposition: (2.15) 1 logq k k X l=0 logq l+1 q l = logq k +1 q k logq k | {z } 1 + 1 logq k ^ n() X l=0 logq l+1 q l | {z } 2 + 1 logq k k 1 X l=^n()+1 logq l+1 q l | {z } 3

if k 1 n^( )+1 otherwise the se ond and the third terms are repla ed by 1 logq k P k 1 l=0 logql+1 q l

. Thethirdterm anbebounded fromaboveby:

1 logq k k 1 X l=^n()+1 logq l+1 q l logq k X l=^n() +1 k 1 logq l (k 1 n^( )) logq k 1 logq k : Sin elogq j 2 e q 1 2 j

,from(A.1) andthehypothesislogq l+1 aq l weobtain: 1 logq k k 1 X l=^n ()+1 logq l+1 q l (k 1) aq k 1 2 e q 1 2 k 1 2 ea (k 1)e (k 2)( 1 2 )logG C 1 withC 1 = 2 ea e 1+( 1 2 ) logG ( 1 2 ) logG ,G= p 5+1 2 .

These ondtermof(2.15)isbounded by

1 logq k ^ n() X l=0 logq l+1 q l C 2 ( k 1)logG log2 C 2

ifk>k()>n(),^ forsomepositive onstantC 2

.

Puttingtheseestimatestogetherwe anbound(2.15) with:

1 logq k k X l=0 logq l+1 q l +C 1 +C 2

forall>0andforallk>k(),thus P k l=0 log l+1 ql =o( logq k )

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2.4. Divergen e of the modied linearization power series when the ar-tihmeti al onditionsof Theorem 2.3are not satised. InTheorem2.3we proved that if F 2 zC fzg and ! veries ondition (2.9) then the linearization H 2zC[[z℄℄

(M n

)

. Thepowerseries oeÆ ientsh n

ofH aregivenby(2.1).

Letusdenethesequen eofstri tlypositiverealnumbers( ~ h n ) n0 asfollows: (2.16) ~ h 0 =1; ~ h n = 1 j n 1j n+1 X m=2 jf m j X n1+:::+nm=n+1 m;ni0 ~ h n1 ::: ~ h nm : Clearly jh n j ~ h n+1 . Let ~

H denote the formal power series asso iated to the sequen e( ~ h n ) n0 (2.17) ~ H(z)= 1 X m=1 ~ h n 1 z n

Following losely[Yo℄, Appendi e2,in thisse tionwewillprovethatif ondition (2.9)isviolatedthen ~ H doesn'tbelongtozC[[z℄℄ (M n ) .

Notethatsin eitisnotrestri tivetoassumethat jf 2 j1onehas (2.18) ~ h n > n 1 X k =0 ~ h k ~ h n 1 k ~ h n 1 ;

thusthesequen e( ~ h n ) n0 isstri tlyin reasing.

Let ! be an irrational numberwhi h violates(2.9) and let U = fq j : q j+1 (q j +1) 2 g where (q j ) j1

are the denominators of the onvergents of x. Sin e

inf n 1 n logM n = > 1wehave: k (n) X qj62U;j=0 logq j+1 q j logM n n k (n) X qj62U;j=0 2log(q j +1) q j = ~<+1

wherek( n)is denedby: q k ( n)

n<q k (n)+1

.

Ontheotherhandlimsup n!1 P k ( n) j=0 logq j+1 qj logM n n = 1thus (2.19) limsup n!1 0 k (n) X q j 2U:j=0 logq j+1 q j logM n n 1 A =1

thisimplies thatU is notempty. FromnowontheelementsofU will bedenoted by: q 0 0 <q 0 1 <:::. Letn i =b q 0 i+1 q 0 i +1 .

Lemma2.5. The subsequen e ~ h q 0 i i0 veries: (2.20) ~ h q 0 i+1 1 j q 0 i+1 1j ~ h n i q 0 i :

Proof. Fromthedenition(2.16)andtheassumptionjf 2 j1itfollowsthat ~ h 2s 1 jf 2 j 2s 1 ~ h 2 s 1 ~ h 2 s 1

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thusforalli2ands1onehas (2.21) ~ h 2s 1 ~ h i s 1 2 : Choosings=q 0 i +1,i=n i

thisleadsto thedesiredestimate:

~ h q 0 i+1 2jf 2 j j q 0 i+1 1j ~ h q 0 i+1 1 2jf 2 j j q 0 i+1 1j ~ h ni(q 0 i +1) 1 ~ h n i q 0 i j q 0 i+1 1j :

Bymeansofthepreviouslemmawe annowprovethatlimsup n!1 1 n log ~ h n Mn = +1. Let i =n i q 0 i q 0 i+1 . Then 1 i 1 1 q 0 i +1 2

, whi hassures that Q

i0

i =

forsomenite onstant (dependingon!). Thenfrom(2.20) weget:

1 q 0 i+1 log ~ h q 0 i+1 M q 0 i+1 2 4 i+1 X j=1 logj q 0 j 1j q 0 j 1 q 0 i+1 logM q 0 i+1 3 5 + 4 whi hdivergesasi!1.

AppendixA. ontinued fra tions andBrjuno's numbers

Herewesummarizebrie ysomebasi notionson ontinuedfra tiondevelopment andwedenetheBrjunonumbers.

Forarealnumber!,wenoteb! itsintegerpartandf!g=! b! itsfra tional part. WedenetheGauss' ontinuedfra tion algorithm:

a 0 =b! and! 0 =f!g foralln1: a n =b 1 !n 1 and! n =f 1 !n 1 g

namelythefollowingrepresentationof!:

!=a 0 +! 0 =a 0 + 1 a 1 +! 1 =:::

Forshortweusethenotation!=[a 0 ;a 1 ;:::;a n ;:::℄. It is well known that to every expression [a

0 ;a 1 ;:::;a n ;:::℄ there orresponds a uniqueirrational number. Let us dene thesequen es (p

n ) n2N and (q n ) n2N as follows: q 2 =1,q 1 =0,q n =a n q n 1 +q n 2 p 2 =0,p 1 =1,p n =a n p n 1 +p n 2

Itiseasytoshowthat: pn q n =[a 0 ;a 1 ;:::;a n ℄.

Foranygiven!2RnQ thesequen e pn q n n2N satises (A.1) q n p 5+1 2 ! n 1 ; n1 thus (A.2) X k 0 1 q k p 5+5 2 and X k 0 logq k q k 1 e 2 5 4 2 3 4 1 ;

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LemmaA.1. forall n1then: 1 q n +q n+1 jq n ! p n j< 1 q n+1 .

Lemma A.2. If for someinteger r ands, j ! r s j 1 2s 2 then r s = p k qk for some integerk.

Lemma A.3. The law of best approximation: if 1 q q k , (p;q) 6= (p n ;q n ) and n 1 then jqx pj > jq n x p n j. Moreover if (p;q) 6= (p n 1 ;q n 1 ) then jqx pj>jq n 1 x p n 1 j.

Foraproofofthese standardlemmaswereferto[HW ℄. The growth rate of ( q

n )

n2N

des ribes how rapidly ! anbe approximated by rationalnumbers. Forexample! isadiophantinenumber[Si℄ifand onlyifthere existtwo onstants >0and 1su hthatq

n+1 q

n

foralln0.

Toevery!2RnQ weasso iate,usingits onvergents,anarithmeti alfun tion:

(A.3) B(!)= X n0 logq n+1 q n

Wesaythat!isaBrjunonumberorthatitsatisestheBrjuno onditionifB(!)< +1. TheBrjuno onditiongivesalimitationtothegrowthrateof(q

n )

n2N

. Itwas originally introdu ed by A.D.Brjuno [Br℄. The Brjuno ondition is weaker than theDiophantine ondition: forexampleifa

n+1 e

a n

forsomepositive onstant and for all n0 then ! =[a

0 ;a

1 ;:::;a

n

;:::℄ isa Brjuno numberbut is not a diophantinenumber.

Appendix B. Davie'slemma

In this appendix we summarize the result of [Da℄ that we use, in parti ular LemmaB.4. Let!2RnQ andfq

n g

n2N

thepartialdenominatorsofthe ontinued fra tionfor! intheGauss'development.

Denition B.1. Let A k = n n0jkn!k 1 8q k o , E k = max( q k ;q k +1 =4) and k =q k =E k . Let A k

be the set of non negative integers j su hthat either j 2A k orfor somej 1 andj 2 inA k ,withj 2 j 1 <E k ,onehasj 1 <j<j 2 andq k divides j j 1

. Forany non negativeintegerndene:

l(n)=max (1+ k ) n q k 2;(m n k +n) 1 q k 1 wherem n =maxfjj0j n;j2A k

g. Wethendene thefun tion h k (n) h k (n)= ( mn+ k n qk 1 if m n +q k 2A k l( n) if m n +q k 62A k Thefun tionh k

(n)hassomeproperties olle tedin thefollowingproposition

PropositionB.2. Thefun tion h k (n)veries; (1) (1+k)n q k 2h k (n) (1+k)n q k 1for alln. (2) If n>0andn2A k thenh k (n)h k ( n 1)+1. (3) h k (n)h k ( n 1) for alln>0. (4) h k (n+q k )h k (n)+1for all n. Nowwesetg k ( n)=max h k (n);b n q

(12)

PropositionB.3. Thefun tion g k

isnon negative andveries:

(1) g k (0)=0 (2) g k (n) (1+ k )n qk for alln (3) g k (n 1 )+g k (n 2 )g k (n 1 +n 2 )for all n 1 andn 2 (4) ifn2A k andn>0then g k (n)g k ( n 1)+1

Theproofofthesepropositions anbefoundin [Da℄. Letk(n)be dened by the onditionq

k (n)

n<q k (n)+1

. Notethat k isnon{ de reasing.

LemmaB.4. Davie's lemmaLet

K(n)=nlog2+ k (n) X k =0 g k (n)log(2q k +1 ):

The fun tionK( n)veries:

(1) Thereexistsauniversal onstant 3 >0su hthat K(n)n 0 k (n) X k =0 logq k +1 q k + 3 1 A ; (2) K(n 1 )+K(n 2 )K(n 1 +n 2 ) for alln 1 andn 2 ; (3) logj n 1jK(n) K(n 1).

Theproofisastraightforwardappli ation ofPropositionB.3.

Referen es

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[Lo℄ P.Lo hak: Canoni alperturbationtheoryviasimultaneousapproximation,Russ.Math. Surv.47,57 133,(1992)

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[Si℄ C.L.Siegel:Iterationofanalyti fun tions,AnnalsofMathemati s43(1942)807 812 [Yo℄ J.-C.Yo oz: Theoreme de Siegel, polyn^omes quadratiques et nombres de Brjuno,

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(Timoteo Carletti, Stefano Marmi) Dipartimento di Matemati a \Ulisse Dini", Viale Morgagni67/A,50134Floren e,Italy

E-mailaddress,TimoteoCarletti: arlettiudini.math.unif i.it E-mailaddress,StefanoMarmi:marmiudini.math.unifi. it