On he No mal Beha iou o Pa ially Ellip ic Lowe
Dimensional To i o Hamil onian Sys ems

Angel Jo ba and Jo di Villanue a
Oc ob e 16 h, 1996
Depa amen de Ma ema ica Aplicada I
Uni e si a Poli ecnica de Ca alunya
Diagonal 647, 08028 Ba celona, Spain.
E-mails:
[email p o ec ed]
,
[email p o ec ed]
Abs ac
The pu p ose o his pap e is o s udy he dynamics nea a educible lowe dimen-
sional in a ian o i o a ni e-dimensional au onomous Hamil onian sys em wi h
`
deg ees o eedom. We will o cus in he case in which he o us has (some) ellip ic
di ec ions.
Fi s , le us assume ha he o us is o ally ellip ic. In his case, i is shown
ha he diusion ime ( he ime o mo eaway om he o us) is exp onen ially big
wi h he ini ial dis ance o he o us. The esul is alid, in pa icula , when he
o us is o maximal dimension and when i is o dimension 0 (ellip ic poin ). In
he maximal dimension case, ou esul s coincide wi h p e ious ones. In he ze o
dimension case, ou esul s imp o e he exis ing b ounds in he li e a u e.
Le us assume now ha he o us (o dimension
, 0

< `
) is pa ially
ellip ic (le us call
m
e
o he numb e o hese di ec ions). In his case we show ha ,
gi en a xed numbe o ellip ic di ec ions (le us call
m
1

m
e
o his numb e ),
he e exis a Can o amily o in a ian o i o dimension
+
m
1
, ha gene alize he
linea oscilla ions co esp onding o hese ellip ic di ec ions. Mo eo e , he Leb esgue
measu e o he complemen a y o his Can o se (in he equency space
R
+
m
1
)
is p o en o be exp onen ially small wi h he dis ance o he ini ial o us. This is
aso o Can o ian cen al mani old" heo em, in which he cen al mani old is
comple ely lled up byin a ian o i and i is uniquely dened.
The p o o o hese esul s is based on he cons uc ion o sui able no mal o ms
a ound he ini ial o us.
2
Con en s
1In o duc ion 3
2 Summa y 4
2.1 No a ion and o mula ion o he p oblem . . . . . . . . . . . . . . . . . . . 4
2.1.1 Reducibili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Linea no mal b eha iou o he o us . . . . . . . . . . . . . . . . . 5
2.1.3 Semino mal o m: o mal desc ip ion . . . . . . . . . . . . . . . . . 6
2.2 Resul s and main ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Semino mal o m: bounds on he emainde . . . . . . . . . . . . . 8
2.2.2 Ellip ic o i a e e y s icky . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Can o amilies o in a ian o i . . . . . . . . . . . . . . . . . . . . 9
3 No mal o m and eec i e s abili y 11
3.1 No a ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Bounding he emainde o he no mal o m . . . . . . . . . . . . . . . . . 12
3.3 Eec i e s abili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Es ima es on he amilies o lowe dimensional o i 23
4.1 Nondegene acy condi ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1 Nondegene acy o he in insic equencies . . . . . . . . . . . . . . 24
4.1.2 Nondegene acy o he no mal equencies . . . . . . . . . . . . . . . 26
4.2 Main heo ems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 P o o o Theo em 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 P elimina ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3.2 The i e a i escheme . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.3 Con e gence o he i e a i escheme . . . . . . . . . . . . . . . . . . 37
4.3.4 Bounds on he measu e . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Basic lemmas 40
6 Acknowledgemen s 44
Re e ences 44
A. Jo ba and J. Villanue a
3
1In o duc ion
The s udy o he solu ions close o an in a ian ob jec is a classical sub jec in Dynamical
Sys ems. He e we will add ess he p oblem o desc ibing he phase space nea an in a ian
o us o a Hamil onian sys em. To x he no a ion, le us call
H
o a eal analy ic
Hamil onian wi h
`
deg ees o eedom, and le us assume i has an in a ian
-dimensional
o us, 0


`
. No e ha we a e including he wo limi cases, ha is, when i is an
equilib ium p oin and when i is a maximal dimensional o us.
To s a he discussion, le us assume ha he o us has some ellip ic di ec ions, his
is, ha he linea ized no mal ow con ains some ha monic oscilla o s. A na u al ques ion
is i hese oscilla ions p e sis when he nonlinea pa o he Hamil onian is added. I he
o us is o ally ellip ic, ano he na u al p oblem is he (nonlinea ) s abili y a ound his
o us.
The e a e known answe s o hese ques ions in some conc e e cases. I
= 0 ( he o us
is an equilib ium p oin ) and i is o ally ellip ic, KAM heo y says ha he e is plen yo
maximal dimension in a ian o i a ound he poin (see 6]): he complemen a y o he
se o in a ian o i has measu e exp onen ially small wi h he dis ance o he poin . I
is well known ha i
`
=2, he maximal dimensional o i spli he ene gy le els
H
=
h
in disconnec ed comp onen s. This is he basis o p o e he nonlinea s abili y o he
poin . Un o una ely, i
` >
2, he in a ian o i do no sepa a e he ene gy le els. In
his case i is gene ally b elie ed ha some diusion can ake place in he phase space (see
2]). Ne e heless, i is s ill p ossible o gi e lowe bounds on he diusion ime, ha a e
exp onen ially big wi h he dis ance o he poin ( hey ollow immedia ely om 6]).
I
=
`
( he o us has maximal dimension) we can no sp eak ab ou no mal b eha iou
since he e a e no a ailable di ec ions". The nonlinea s abili y has b een s udied in 14]
and 13] (among o he s), whe e i is shown ha he diusion ime is also b ounded by an
exp onen ially big (wi h he dis ance o he ini ial o us) quan i y. In 13] i is also ela ed
his exp onen ially big s abili y ime wi h he densi y o in a ian maximal dimensional
o i a ound he ini ial one, by showing ha he o al measu e o he gaps be ween he
in a ian o i nea by is no bigge han an exp onen ially small quan i y wi h esp ec o
he dis ance o he ini ial one. In ac , in 13] i is p o ed ha unde an ex a s eepness
condi ion he diusion ime is, a leas , sup e exp onen ial. This condi ion co esp onds
o he classical quasi-con exi y hyp o hesis used o ob ain global" and exp onen ially big
s abili y ime o a p e u b ed in eg able Hamil onian sys em wi h esp ec o he size o
he p e u ba ion (see 5] and e e ences he ein).
In his wo k we will conside hese p oblems, wi hou any s eepness condi ion, o a
lowe dimensional o us. The wo limi cases men ioned ab o e a e included, and he esul s
ob ained can b e summa ized as ollows: o a o ally ellip ic o us, weha e ob ained lowe
bounds o he diusion ime. They ag ee wi h he bounds o 13] in he case
=
`
bu ,
o he case
=0, hey a e b e e han he ones di ec ly de i ed om 6]. Mo eo e , we
show he exis ence o quasip e io dic solu ions ha gene alize he linea oscilla ions o he
no mal ow o he comple e sys em. I he o us has no mal b eha iou o he kind some
cen es"

some saddles" we ob ain, o any combina ion o cen es, a Can o amily o
in an o i a ound he ini ial one, by adding o he ini ial se o equencies new ones ha
come om he nonlinea oscilla ions asso cia ed o he chosen cen es. Those in a ian o i
ha e he same no mal b eha iou as he ini ial one (o cou se, skipping he cen es ha
gi e ise o he amily). This esul is a so o Can o ian cen al mani old" heo em,
4
No mal Beha iou o Lowe Dimensional To i
in which we ob ain an in a ian mani old pa ame ized on aCan o se and comple ely
lled up byin a ian o i. We no e ha we ob ain a Can o ian cen al submani old" o
eachcombina ion o cen es, and ha i is uniquely dened.
The p o o s a e based on he cons uc ion o sui able no mal o ms. The es ima es on
he di ussion ime a e ob ained b ounding he emainde o his no mal o m, while he
exis ence o amilies o lowe dimensional o i is p o ed by applying aKAM scheme o
his emainde .
The pap e has been o ganized in he ollowing way: Sec ion 2 summa izes he main
ideas and esul s con ained in he wo k. Sec ion 3 con ains he de ails conce ning he
no mal o m and he bounds on he diusion ime. Sec ion 4 is de o ed o he exis ence
o amilies o o i nea he ini ial one and, nally, in Sec ion 5, we ha e included some
basic lemmas used along he pap e .
2 Summa y
He e we ha e included a echnical desc ip ion o he p oblem, he me ho dology used in
he p o o s and he esul s ob ained. Weha e ommi ed he echnical de ails o he p o o s
in o de o simpli y he eading.
2.1 No a ion and o mula ion o he p oblem
Le
H
b e a Hamil onian sys em o
`
deg ees o eedom dened on
R
2
`
,ha ing an in a ian
-dimensional iso opic o us ( ha is, he canonical 2- o m o
R
2
`
es ic ed o he angen
bundle o he o us anish), 0


`
,wi h a quasip e io dic ow gi en by he ec o o
basic equencies ^
!
(0)
2
R
. We assume, om he iso opic cha ac e o he o us, ha we
can in o duce (wi h a canonical change o co o dina es)
angula a iables
^

desc ibing
he ini ial o us. Hence, he Hamil onian in hese co o dina es akes he o m
H
(
^
 x
^
I y
)= ^
!
(0)
>
^
I
+
1
2
z
>
B
(
^

)
z
+
H
1
(
^
 x
^
I y
)

whe e
z
>
= (
x
>
y
>
). He e,
x
,
y
a e
m
-dimensional eal ec o s, and
^

,
^
I
b elong o
R
,
+
m
=
`
. O cou se,
^

,
x
a e he p osi ions and
^
I
,
y
he esp ec i e conjuga e momen a. As
^

is an angula a iable, we assume ha
H
depends oni ina2

-p e io dic way. Mo eo e ,
we will use
u
>
o deno e he scala p o duc o wo ec o s.
We also supp ose ha he Hamil onian
H
can be ex ended o a eal analy ic unc ion
dened on he se
D
m
(

0
R
0
) gi en by
D
m
(

0
R
0
)=
(
^
 x
^
I y
)
2
C

C
m

C

C
m
:
j
Im
^

j

0

j
z
j
R
0

j
^
I
j
R
2
0
g

(1)
whe e
j
:
j
deno es he inni y no m o a complex ec o (we will use he same no a ion o
he ma ix no m induced). The die en scaling o he a iables
z
and
^
I
in
D
m
(

0
R
0
)
is mo i a ed by he deni ion o deg ee o a monomial o he Taylo expansion (wi h
esp ec o
z
and
^
I
, see (10)) used along he pap e :
deg

h
ls
(
^

)
z
l
^
I
s

=
j
l
j
1
+2
j
s
j
1

(2)
A. Jo ba and J. Villanue a
5
wi h
l
2
N
2
m
,
s
2
N
, and whe e
j
k
j
1
is dened as
P
j
j
k
j
j
. The eason o coun ing
wice he exp onen
s
will b e clea la e (i is mo i a ed, basically,by he p op e ies o he
Poisson b acke ).
We assume he ini ial in a ian o us is gi en by
z
=0and
^
I
=0. Hence, we can ake
B
(
^

) as a symme ic 2
m
-dimensional ma ix, wi h eal co ecien s ha dep end on
^

in
analy ic and 2

-p e io dic way. Mo eo e , he Taylo expansion o
H
1
a ound
z
=0,
^
I
=0
b egins wi h e ms o deg ee a leas h ee.
2.1.1 Reducibili y
We will assume ha he no mal a ia ional ow a ound his o us (gi en by he ma ix
J
m
B
(^
!
(0)
), whe e
J
m
is he canonical 2- o m o
C
2
m
) can be educed o cons an co e-
cien s wi h a eal linea change o a iables ha dep ends quasip e io dically on
^

, ha ing
^
!
(0)
as a ec o o basic equencies (quasip e io dic Flo que educ ion).
1
The hyp o hesis
do es no seem o be e y es ic i e in ou con ex , since all he pa ially ellip ic o i
ob ained by KAM echniques ha e educible no mal ow (see, o ins ance, 7], 15], 8],
11]). This p op e y allows o cons uc a canonical change o co o dina es ha ans o ms
he ma ix
B
(
^

) o cons an co ecien s. Hence, we will assume ha
B
is a eal symme ic
ma ix, indep enden om
^

,and ha he ini ial Hamil onian in hose Flo que a iables
lo oks like:
H
(
^
 x
^
I y
)= ^
!
(0)
>
^
I
+
1
2
z
>
B
z
+
H
2
(
^
 x
^
I y
)

(3)
whe e
H
2
b egins wi h e ms o deg ee a leas h ee.
2.1.2 Linea no mal beha iou o he o us
We also assume ha he ma ix
J
m
B
has die en eigen alues, gi en by he complex ec o

2
C
2
m
, ha akes he o m

>
=(

1
:::
m

;

1
:::
;

m
) ( his s uc u e comes om
he canonical cha ac e o he sys em). We no e ha in his case, die en eigen alues also
means nonze o eigen alues. We will e e o hose eigen alues as he no mal eigen alues
o he o us. We ema k ha i

j
=
i
(wi h

2
R
n
0
g
and
i
=
p
;
1) is an eigen alue,
hen

j
+
m
=
;
i
. The ec o s o
R
2
m
ha a e combina ion o eigen ec o s co esp onding
o (couples o ) eigen alues o his o m a e called he ellip ic di ec ions o he o us.
The s udy o he b eha iou o he ini ial o us in hose di ec ions is he main issue in
his pap e . Mo eo e , he e may be o he eigen alues wi h eal pa die en om ze o,
ha dene he hyp e b olic di ec ions o he o us. They can be g oup ed in one o hese
wo ollowing o ms:
1. i

j
=

2
R
n
0
g
, hen

j
+
m
=
;

,
2. i

j
=

+
i
(wi h
 
2
R
n
0
g
), hen, om he eal cha ac e o he ma ix
B
,
we can ake

j
+1
=

;
i
,and hence,

j
+
m
=
;

;
i
and

j
+
m
+1
=
;

+
i
.
The imagina y pa s o he eigen alues a e usually called no mal equencies o he o us.
Fo easons ha will b e clea la e , i is e y con enien o pu he ma ix
J
m
B
in diag-
onal o m. This is p ossible wi h a complex canonical change o basis, ha ans o ms he
1
I he o us is educible excep by an small emainde , i is s ill p ossible o de i e simila esul s (by
adding a pe u ba i e pa ame e ). See 9] and 11] o he main ideas and ela ed esul s.

6
No mal Beha iou o Lowe Dimensional To i
ini ial eal Hamil onian sys em in o a complex one. Thus, he complexied Hamil onian
has some symme ies b ecause i comes om a eal one. As his symme ies a e p ese ed
by he ans o ma ions used along he p o o s, he nal Hamil onian can be ealied. In
ac , complexica ion is no necessa y, bu i simplies he p o o s. Ne e heless, in he
p o o s we ha e no w i en explici ly he p ese a ion o hose symme ies. This is be-
cause he de ails a e e y edious and cumb e some and, on he o he hand, he in e es ed
eade should no ha e p oblems in w i ing hem (i is a e y s anda d me ho dology).
Fo u he uses, we deno e by
Z
>
=(
X
>
Y
>
) hose complex (canonical) a iables, and
by
B

he complex symme ic ma ix such ha
J
m
B

=diag(

).
2.1.3 Semino mal o m: o mal desc ip ion
Now we ake a subbundle o
R
2
m
,
G 
R
2
m
,in a ian by he ac ion o he ma ix
J
m
B
,
and such ha i only con ains eigen ec o s o ellip ic yp e. We pu 2
m
1
= dim(
G
)
(we ecall ha his dimension is always e en) and we call ~
!
(0)
2
R
m
1
o he ec o o
no mal equencies asso cia ed o his subbundle. As
G
will b e xed along he pap e , we
in o duce some no a ion ela ed o i . Fi s , we assume ha he  s
m
1
eigen alues o

a e he ones asso cia ed o
G
, ha is,

j
=
i
~
!
(0)
j
,
j
= 1
:::m
1
. We also deno e by
^

2
C
2(
m
;
m
1
)
he ec o ob ained skipping om

he 2
m
1
eigen alues asso cia ed o
G
.
This in o duces in a na u al way he decomp osi ion
X
>
= (
~
X
>

^
X
>
),
Y
>
= (
~
Y
>

^
Y
>
),
ob ained aking apa he  s
m
1
comp onen s om he las
m
;
m
1
.Mo eo e , we dene
~
Z
>
= (
~
X
>

~
Y
>
) and
^
Z
>
= (
^
X
>

^
Y
>
). A simila no a ion can be used o any ec o
l
2
N
2
m
, spli ing
l
>
= (
l
>
X
l
>
Y
), whe e
l
X
and
l
Y
a e he exp onen s o
X
and
Y
in he
monomial
Z
l
(
Z
l
=
X
l
X
Y
l
Y
). Then, we in o duce
!
(0)
2
R
+
m
1
as
!
(0)
>
=(^
!
(0)
>

~
!
(0)
>
),
and we ask o a Diophan ine condi ion o he ollowing o m,
2
j
ik
>
!
(0)
+
l
>
^

j

j
k
j

1
 k
2
Z
+
m
1
n
0
g
 l
2
N
2(
m
;
m
1
)

0
j
l
j
1

2

(4)
b eing
>
0 and
>
+
m
1
. This non esonance condi ion allows o cons uc ( o mally)
a semino mal o m ela ed o he chosen
G
.I we exp ess he Hamil onian in e ms o
he a iables
Z
, his semino mal o m is done by emo ing om
H
he monomials o he
ollowing o m (see (11) o he no a ions):
h
lsk
exp (
ik
>
^

)
Z
l
^
I
s
 l
2
N
2
m
 s
2
N
 k
2
Z

j
k
j
1
+
j
l
X
;
l
Y
j
1
6
=0

j
^
l
j
1

2

(5)
whe e
^
l
is he pa o
l
ha co esp onds o
^

. A e his no mal o m p o cess, using he
p ese a ion o he symme ies ha come om he complexica ion, we can ew i e his
( o mal) semino mal o m, in e ms o sui able eal a iables, in he ollowing o m:
H
(
^
 x
^
I y
)=
!
(0)
>
I
+
1
2
^
z
>
^
B
^
z
+
F
(
I
)+
1
2
^
z
>
Q
(
I
)^
z
+
O
3
(^
z
)

(6)
whe e, o simplici y,wedono change he name o he Hamil onian, and whe e we ex end
he decomp osi ion in o duced ab o e o he a iables (
x y
). He e, he ma ix
^
B
is a eal
2
The Diophan ine condi ion can b e elaxed when
j
l
j
1
=2and
l
>
^

only in ol es hyp e b olic eigen alues.
In his case, he esul s a e p o ed using a combined me hod based on a xed p oin scheme o he
hyp e b olic di ec ions and a New on me ho d o he emaining ones. This echnique allows o ha e
mul iple hype b olic eigen alues.
A. Jo ba and J. Villanue a
7
symme ic ma ix ob ained by p o jec ing
B
on he di ec ions gi en by he eigen alues
co esp onding o he eigen ec o s
^

.
I
is a compac no a ion o
I
>
=(
^
I
>

~
I
>
), whe e he
ac ions
~
I
can b e aken as
~
I
j
=
1
2
(
x
2
j
+
y
2
j
),
j
=1
:::m
1
,i wecho ose he eal no mal o m
a iables (
x y
) asso cia ed o he conside ed ellip ic di ec ions in adequa e (and s anda d)
way (see (44) in he p o o o Theo em 2). O cou se,
F
=
O
2
(
I
)and
Q
=
O
1
(
I
).
Now, we p o ceed o desc ib e he no mal b eha iou o he o us de i ed om his semi-
no mal o m. I is no dicul o check ha weha e he ollowing ( o mal) quasip e io dic
solu ions o he canonical equa ions o (6):
^

(
) =

^
!
(0)
+
@
F
@
^
I
(
I
(0))
!
+
^

(0)

^
I
(
) =
^
I
(0)

~
x
j
(
) =
q
2
~
I
j
(0) sin

~
!
(0)
+
@
F
@
~
I
j
(
I
(0))
!
+
~

j
(0)
!

(7)
~
y
j
(
) =
q
2
~
I
j
(0) cos

~
!
(0)
+
@
F
@
~
I
j
(
I
(0))
!
+
~

j
(0)
!

^
z
(
) = 0
:
Tha is, we ob ain a 2(
+
m
1
)-dimensional in a ian mani old (^
z
= 0) olia ed by a
con inuous (
+
m
1
)-dimensional amily o (
+
m
1
)-dimensional in a ian educible o i,
pa ame ized by
I
(0). The selec ion o he pa ame e
I
(0) is na u al, as
I
1
:::I
+
m
1
a e  s in eg als o he Hamil onian (6) es ic ed o he in a ian mani old ^
z
=0. We
ema k ha he o i o he amily collapse o lowe dimensional ones when any o he
~
I
j
(0)
b ecome ze o. In pa icula , i we ake
I
(0) = 0 we eco e he ini ial
-dimensional one.
In ac , o e e y 0

m
2

m
1
weha e, o his semino mal o m,

m
1
m
2

die en (
+
m
2
)-
dimensional amilies o (
+
m
2
)-dimensional in a ian o i. They a e asso cia ed o e e y
in a ian eal subbundle con ained in
G
. The skele on o hese amilies comes om he
na u al
-dimensional amily o
-dimensional o i con aining he ini ial one. This amily
is asso cia ed o he neu al di ec ions o he o us ( he neu al di ec ions a e conjuga ed
o he angen ones), an i is ob ained aking
G
= 0 in ou no a ion. Mo eo e , we also
ema k ha in (7) we only ha e eal o i when all he
~
I
j

0. This comes di ec ly om
he deni ion o
~
I
as a unc ion o he eal no mal o m a iables. To explain his ac le
us gi e he classical example o a 1-dimensional p endulum nea he ellip ic equilib ium
poin , 
x
+ sin(
x
)=0. The linea (no mal) equency a he equilib ium p oin is1. Mo ing
he ene gy le el in he eal phase space we ob ain p e io dic o bi s wi h equency smalle
han 1. I one wan s p e io dic o bi s wi h equency bigge han 1, one is o ced o ex end
he phase space om
R
2
o
C
2
, keeping he ime in
R
.This same phenomenon happ ens
when we s udy he no mal ellip ic di ec ions o a o us. I is imp o an o no e ha , o
us, a
s
-dimensional complex o us is a map om
T
s
o
C
2
`
. Hence, we will use he wo d
dimension" o e e o he eal dimension.
2.2 Resul s and main ideas
A basic esul in his pap e is he quan i a i e e sion o he semino mal o m, i we
only kill he monomials like (5) up o some ni e o de . F om he es ima es on his
semino mal o m, we deduce (unde ce ain nondegene acy condi ions) ha he no mal
8
No mal Beha iou o Lowe Dimensional To i
beha iou o he ini ial o us desc ib ed in Sec ion 2.1.3 is co ec " in he sense o he
classical KAM ideas: he ma jo i y" o hese o i eally exis (bu sligh ly de o med) in
he ini ial Hamil onian sys em. Mo eo e , we also deduce he long ime eec i e s abili y
o any eal a jec o y close o a o ally ellip ic o us. In he ollowing sec ions we p esen
he explici desc ip ion o hose esul s, and we explain he main ideas used in he p o o s.
2.2.1 Semino mal o m: b ounds on he emainde
We s a wi h he Hamil onian (3), whe e he no mal owis educed o cons an co e-
cien s. Then, we p e o m a ni e numbe o (semi)no mal o m s eps, by using sui able
canonical ans o ma ions ha emo e he monomials (5) up o a ni e deg ee. This allows
o show he con e gence o he p o cess on he se
D
m
(

1
R
), whe e

1
is indep enden
om
R
and
R
is small enough. By selec ing he o de up o which he semino mal o m
is done as a sui able unc ion o
R
, i is p ossible o ob ain a emainde o he semino mal
o m which is exp onen ially small wi h
R
.This is con ained in Theo em 1.
2.2.2 Ellip ic o i a e e y s icky
Nowle us assume ha he ini al o us has all he no mal di ec ions o ellip ic yp e. In
his case we can ake
G
=
R
2
m
, hewhole se o no mal di ec ions.
Then, using he no mal o m explained ab o e, one can w i e he ini ial Hamil onian
as an in eg able one plus an exp onen ially small p e u ba ion. Hence, i is e y na u al
o ob ain exp onen ially big es ima es o he diusion ime: he ime needed o a eal
a jec o y o go away om he se
D
m
(
 R
) ( o a p ecise deni ion o going away" see
Theo em 2) is bigge han
T
(
R
)=
cons :
exp
0
@
cons :

1
R

2

+1
1
A

(8)
b eing he cons an s on he deni ion o
T
(
R
) indep enden om
R
.As usual, we call he
exp onen
2

+1
he s abili y exp onen .
Le us compa e his esul wi h p e ious ones. In he case in which he ini ial o us is o
maximal dimension, no e ha he no mal a iables
z
>
=(
x
>
y
>
) a e missing e e ywhe e.
So, he se
D
m
(
 R
) (see (1)) eads
D
`
0
(
 R
)=
(
^

^
I
)
2
C
`

C
`
:
j
Im
^

j
 
j
^
I
j
R
2
g
:
To compa e wi h 13] we mus edene
R
2
as
R
,in o de o ha e he same uni s. Then,
he s abili y exp onen in (8) coincides wi h 13].
I he ini ial o us is an equilib ium p oin , he a iables
^

and
^
I
a e he ones ha a e
missing. Hence,
D
m
(
 R
) b ecomes
D
0
`
(0
R
)=
(
x y
)
2
C
`

C
`
:
j
(
x y
)
j
R
g
:
Hence, no escaling is necessa y o compa e he diusion ime o (8) wi h he one de i ed
om 6]: he imp o emen is ha he exp onen
1

+1
in 6] he e b ecomes
2

+1
. We no e
ha his imp o emen is no only on he diusion ime, bu also on he measu e o he
des oyed o i (see Rema k 10).
A. Jo ba and J. Villanue a
9
2.2.3 Can o amilies o in a ian o i
I is clea om Sec ion 2.1.3 ha compu ing he semino mal o mal o m asso cia ed o
G
a ound he ini ial o us, up o ni e o de , and skipping he non-in eg able emainde ,
hose ellip ic di ec ions dene a unique (
+
m
1
)-dimensional amily o (
+
m
1
)-dimensional
o i a ound he ini ial
-dimensional one. When we app oach he ini al o us, he in insic
equencies o he o i o he amily can b e selec ed such ha hey end o
!
(0)
.
In his case we will show ha when we add he emainde o he semino mal o m
mos o hese o i s ill p e sis in he comple e sys em
H
,ha ing also educible no mal
ow. The no mal eigen alues o hese o i a e close o he eigen alues
^

j
( ha a e he
ones no ela ed wi h
G
). O cou se, due o he die en small di iso s in ol ed in he
p oblem, we can no p o e he p e sis ence o all he in a ian o i p edic ed by he no mal
o m.
The hyp o heses needed a e usual in KAM me ho ds. The  s one is a non- esonance
condi ion in ol ing he equencies ^
!
(0)
and he no mal ones

, ha dep ends on he
conc e e selec ion o
G
and i is explici ly gi en in (4). The second hyp o hesis is a
nondegene acy condi ion, asking ha all he equencies a y wi h he ac ions. No e
ha , in gene al, we ha e mo e equencies (
+
m
) han ac ions (
+
m
1
,
m
1

m
). This
in o duces he classical lack-o -pa ame e s p oblem when wo king wi h lowe dimensional
o us, ha needs a sp ecial ea emen ( o ela ed esul s, see 4], 16] and 11]). The idea
ha weha e used he e is o cho ose a sui able (
+
m
1
)-dimensional se o pa ame e s, and
o ask o he exis ence o lowe dimensional o i asso cia ed o some o he alues o hese
pa ame e s. He e, he na u al pa ame e is he ec o o in insic equencies
!
2
R
+
m
1
o
he in a ian o i. To use his pa ame iza ion we need a ypical nondegene acy condi ion
on he equency map om
I
o
!
, his is, ha his map be a (lo cal) dieomo phism
a ound
I
= 0. This condi ion can be explici ly o mula ed compu ing he no mal o m
o Sec ion 2.1.3 up o deg ee 4 and i is gi en in (50). The con ol o he emaining
m
;
m
1
no mal equencies (no mal o he (
+
m
1
)-dimensional amily o o i) is mo e
dicul , since he e a e no ee pa ame e s o con ol hem. No e ha hose equencies
a e unc ions o he in insic ones. Then, he idea is o elimina e all he equencies o
which he Diophan ine condi ions needed o cons uc in a ian o i a e no sa ised.
This will lead us o elimina e alues o
!
o: a) con ol he in insic equencies
!
and b)
con ol he no mal ones as a unc ion o he in insic ones. Tocon ol he measu e o he
se o in isic equencies o which he asso cia ed no mal ones a e close o esonance, we
use he same kind o me ho d o 11]: we ask o a ex a se o nondegene acy condi ions
o he dep endence o hese no mal equencies wi h esp ec o he in insic ones. Those
condi ions a e gi en in (54). They ha e al eady b een conside ed in 12] and 7].
Wi h he o mula ion gi en ab o e, he esul is ha he measu e o he complemen a y
o he p ese ed o i is exp onen ially small: we in o duce
U
(
A
)=
n
!
2
R
+
m
1
:
j
!
;
!
(0)
j
A
o
A>
0

(9)
and le us dene
A
(
A
) as he se o equencies o
U
(
A
) o which we ha e educible
in a ian o i. Then, i
A
is small enough, we ha e
mes(
U
(
A
)
nA
(
A
))
mes(
U
(
A
))

cons :
exp
0
@
;
cons :

1
A

1

+1
1
A

16
No mal Beha iou o Lowe Dimensional To i
Then, he e exis s a cons an
$
, depending only on
,
m
,

,

0
,
^
N
4
,
^
N

,
^
S

,
^
T
3
and
^
T

,such ha he ol lowing bounds hold o he ans o med Hamil onian
H

!
G
1
,
j
N
(1)
;
N
j

4
R
4

$
^
S

R
p
+1


+2
+
R
2
p
;
1

2(

+2)
!

j
S
(1)
1
j

4
R
4

$
^
SR
p
+1

R
2


+1
+
R
4


+2
+
R
p
;
3


+2
+
R
p
;
2

2(

+2)
!

j
S
(1)
2
j

4
R
4

$
^
SR
p

R
2


+1
+
R
3


+2
+
R
p
;
1

2(

+2)
!

j
T
(1)
;
T
j

4
R
4

$
^
S

R
p
+1


+2
+
R
2
p
;
1

2(

+2)
!
:
Rema k 4
(A e y impo an one) I
p
is big enough and
 > R
, he dominan e m
in he bounds o
S
(1)
1
and
S
(1)
2
is gi en by he ac o
R
2
=

+1
. This wil l be he ac o o
dec easing o hose e ms du ing he no mal o m p ocess and i al lows o ake

o o de
R
2
=
(

+1)
, ha wil l p oduce he exponen
2
=
(

+1)
in
(36)
.As we ha e
2
=
(

+1)
<
1
, we
can deduce ha an adequa e selec ion o
p
is
p
=8
.This al lows o keep bounds like
(35)
du ing al l he i e a i e p ocess.
I we s a wi h a aw" Hamil onian (wi hou any p e ious s ep o no mal o m) he
dec easing ac o ob ained is o o de
R=

+1
, ha o ces us o selec

o o de
R
1
=
(

+1)
.
This p oduces a wo se exponen
1
=
(

+1)
in
(36)
. Fo ins ance, le us assume ha he
no mal o m has been done a ound an el lip ic equilib ium poin . He e he impo an issue
is o no e ha he bounds ob ained when kil ling deg ee
3
a e much wo se han he bounds
ob ained o he o he deg ees ( his has been obse ed nume ical ly in
17]
). Hence, o apply
he same bounds o al l he deg ees esul s in poo es ima es.
Rema k 5
The exponen
2
=
(

+1)
in Rema k
4
can be imp o ed in some e y degene a e
cases. Fo ins ance, le us conside a o al ly el lip ic o us, and we ake
G
=
R
2
m
. Le
q
be he lowes deg ee o he monomials o
N
co esponding o he ( o mal) no mal o m o
H
a ound he o us (o cou se,
q

4
). Then,

can be aken o o de
R
(
q
;
2)
=
(

+1)
, ha
p oduces he exponen
(
q
;
2)
=
(

+1)
in
(36)
.
P o o :
Du ing his p o o we will use die en cons an s $
j
,
j

0, ha will dep end only
on he same pa ame e s as he nal cons an $ o he s a emen o he lemma. Fi s ,
om he b ound (24) o Lemma 1, we ha e ha
j
G
1
j

1
R
1

$
0
^
SR
p
+1



j
G
2
j

1
R
1

$
0
^
SR
p



j
G
j

1
R
1

$
0
^
SR
p



whe e, as in Lemma 1,

j
=

;
j
and
R
j
=
R
exp (
;
j
). Then, o ob ain he b ounds o
he die en e ms o he ans o med Hamil onian, we only need o b ound he Poisson
b acke s ha app ea in (29){(33).
To ob ain p ecise es ima es, we will lo ok ca e ully in o he c i ical b ounds o he
die en pa ial de i a i es in ol ed, ha is, he ones asso cia ed o
N
4
and
T
3
. So, we
es ima e, sepa a ely, he con ibu ion o
N
4
,
N

,
T
3
and
T

, aking in o accoun ha
N
do es no dep end on
^

,
N
4
is a p olynomial o deg ee 4, and
T
3
only con ains e ms o
deg ee 3. Mo eo e , o b ound
e
S
2
(
T G
1
g
)we no e ha ( om he deni ion o
S
1
and
S
2
)

A. Jo ba and J. Villanue a
17
i only con ains e ms co esp onding o
@
@
^
Z
, and no o
@
@
^

o
@
@
^
I
.Thus, using he b ounds
on he Poisson b acke p o ided by Lemma 6 (see Rema k 11 o he case in which one
o he e ms has ni e deg ee), we ha e
j
e
S
2
(
T G
1
g
)
j

2
R
2

$
1
^
SR
p

R
2


+1
+
R
3


+2
!

j
T G
gj

2
R
2

$
2
^
S
R
p
+1


+2

j
N G
1
gj

2
R
2

$
3
^
SR
p
+1

R
2


+1
+
R
4


+2
!

j
N G
gj

2
R
2

$
4
^
SR
p

R
2


+1
+
R
4


+2
!

j
S G
gj

2
R
2

$
5
^
S
R
2
p
;
2


+2

j
T G
g
G
gj

3
R
3

$
6
^
S
R
2
p
;
1

2(

+2)

j
N G
g
G
gj

3
R
3

$
7
^
S

R
2
p

2

+3
+
R
2
p
+2

2(

+2)
!

and nally
j
H

j

4
R
4

$
8
^
S

R
2
p
;
2


+2
+
R
2
p
;
1

2(

+2)
!
:
F om ha , wi h a sui able deni ion o $ as a unc ion o $
0
{$
8
, he b ounds o he
s a emen o he lemma a e clea , i we ecall ha we ha e aken
p

6.
Now, we a e in condi ions o o mula e a quan i a i e esul ab ou pa ial educ ion
o semino mal o m" o he ini ial Hamil onian. Fo his pu p ose, we conside he Hamil-
onian
H
o (3), w i en as in (17) in e ms o he
Z
a iables. We assume ha
H
is
dened on
D
m
(

0
R
0
), o some 0
<
0
<
1 and 0
<R
0
<
1, wi h he ollowing b ounds:
j
N
j

0
R

^
NR
4
,
j
S
j

0
R

^
SR
3
and
j
T
j

0
R

^
TR
3
, o any 0
< R

R
0
, b eing
^
N
,
^
S
and
^
T
, p osi i e cons an s (indep enden om
R
). Then, we p o e he ollowing esul :
Theo em 1
We conside he Hamil onian
H
o
(17)
,wi h he hypo heses p e iously de-
sc ibed. We suppose ha he e exis s

0
>
0
and
>
+
m
1
such ha
j
ik
>
^
!
(0)
+
l
>

j

0
(
j
k
j
1
+
j
l
x
;
l
y
j
1
)

8
(
l s
)
2S 8
k
2
Z
wi h
j
l
x
;
l
y
j
1
+
j
k
j
1
6
=0
:
Then, o any
R>
0
smal l enough ( his condi ion on
R
depends only on
,
m
,

,

0
,

0
,
R
0
,
^
N
,
^
S
and
^
T
), he e exis s an analy ical canonical ans o ma ion
!
R
such ha
1.
!
R
;
Id
and
(!
R
)
;
1
;
Id
a e
2

-pe iodic on
^

.
2.
!
R
:
D
m
(3

0
=
4
R
exp (
;

0
=
4))
;!D
m
(

0
R
)

and
(!
R
)
;
1
:
D
m
(11

0
=
16
R
exp (
;
5

0
=
16))
;!D
m
(

0
R
)
:
18
No mal Beha iou o Lowe Dimensional To i
3. I we ake
(
^
 X
^
I Y
)
2D
m
(3

0
=
4
R
exp (
;

0
=
4))
and we dene
(
^


X


^
I

Y

)=
!
R
(
^
 X
^
I Y
)
, hen
j
^


;
^

j 

0
=
16
,
j
Z

;
Z
j 
R
0
exp (
;
1
=
2)
=
32
,
j
^
I

;
^
I
j 
R
2

0
exp (
;
1)
=
16
.Mo eo e , he same bounds hold o
(!
R
)
;
1
i
(
^
 X
^
I Y
)
2
D
m
(11

0
=
16
R
exp (
;
5

0
=
16))
.
4.
!
R
ans o ms
H
R
:=
H

!
R
= ^
!
(0)
>
^
I
+
1
2
Z
>
B

Z
+
N
R
+
S
R
+
T
R

decomposi ion analogous o
(17)
,wi h he bounds:
j
N
R
;
N
4
j
3

0
=
4
R
exp (
;

0
=
4)

cons :R
6
,
j
T
R
;
T
3
j
3

0
=
4
R
exp (
;

0
=
4)

cons :R
4
, whe e
N
4
and
T
3
we e in oducedin
(34)
, and can becompu ed wi h a no mal o m wi h espec o
S
up o deg ee
4
,and
j
S
R
j
3

0
=
4
R
exp (
;

0
=
4)

cons :
exp
0
@
;
cons :

1
R

2

+1
1
A
R
8

(36)
being he cons a s ha appea in he bounds o
N
R
,
T
R
and
S
R
, posi i e and inde-
penden om
R
.Mo eo e , o any
R
o which he esul holds,
H
R
is in no mal
o m wi h espec o
S
, a leas up o deg ee
8
.
Rema k 6
The dependence o
!
R
on
R
is no con inuous bu piecewise analy ic.
Rema k 7
F om he bounds p o idedbyLemma
2
o he i e a i e no mal o m p ocedu e
desc ibed in Lemma
1
, his exponen ial ly smal l bound seems o be he bes ha one can
ob ains by using his linea ly con e gen scheme.
P o o :
The p o o is done simul aneously o any0
<R

R
0
. The b ounds whe e
R
is no
w i en explici ly a e indep enden om
R
. All hese bounds and he die en condi ions
on he smallness o
R
will dep end only on he xed pa ame e s o he s a emen . The
main idea o his p o o is o use Lemma 1 ecu si ely,and o i e a e he b ounds p o ided
by Lemma 2 o
p
= 8 (see Rema k 4). Hence, o use his lemma, we need o pu he
ini ial Hamil onian in no mal o m wi h esp ec o
S
,up o deg ee a leas 8. Fo his
pu p ose, we cons uc ecu si ely he gene a ing unc ions
G
(0)
,
G
(1)
,
:::
,
G
(5)
,p o ided
by Lemma 1. Pu ing
H
(0)
=
H
, we can dene
H
(
n
+1)
:=
H
(
n
)

!
G
(
n
)
1
=
H
(
n
)
+
H
(
n
)
G
(
n
)
g
+
1
2!
H
(
n
)
G
(
n
)
g
G
(
n
)
g
+


(37)
o
n
= 0
:::
5. Le us conside  s he exp ession (37) as a o mal ans o ma ion.
F om he p op e y (14) o he Poisson b acke , and om he way in which he die en
G
(
j
)
a e selec ed in Lemma 1 (see Rema k 2), we can ensu e ha he non- esonan e ms
asso cia ed o
S
ha emain in
H
(6)
a e o deg ee a leas 9. Toshow ha his cons uc ion
is no only o mal, we a e going o p o e hewell dened cha ac e o he ans o ma ions
!
G
(
n
)
1
,
n
=0
:::
5, and o b ound
H
(
n
)
,
n
=1
:::
6. Fo his pu p ose, we expand
H
(
n
)
as
in (17), bu adding he sup e sc ip (
n
)" o
N
,
S
and
T
. We dene

0
=

0
192
, oin o duce

(0)
=

0
,
R
(0)
=
R
, and

(
n
)
=

(
n
;
1)
;
4

0
,
R
(
n
)
=
R
(
n
;
1)
exp (
;
4

0
),
n
=1
:::
6. Then,
we a e going o show ha aking



0
in Lemma 1, we ha e o
n
=0
:::
6, ha , i
R
is small enough,
j
N
(
n
)
j

(
n
)
R
(
n
)

^
N
(
n
)
(
R
(
n
)
)
4

j
S
(
n
)
j

(
n
)
R
(
n
)

^
S
(
n
)
(
R
(
n
)
)
n
+3

j
T
(
n
)
j

(
n
)
R
(
n
)

^
T
(
n
)
(
R
(
n
)
)
3
:
(38)
A. Jo ba and J. Villanue a
19
This is p o ed by (ni e) induc ion: assuming ha (38) holds o some
n
(0

n

5) and
using ha , i
R
is sucien ly small,
"
^
S
(
n
)
(
R
(
n
)
)
n
+1


+2
0

1 (39)
(" is p o ided by Lemma 1), we ha e
!
G
(
n
)
1

!
G
(
n
)
;
1
:
D
m
(

(
n
+1)
R
(
n
+1)
)
;!D
m
(

(
n
)
;
3

0
R
(
n
)
exp (
;
3

0
))
:
(40)
Then, he ac ha he successi e s eps inc ease a leas by one he deg ee o he no mal
o m wi h esp ec o
S
, makes e iden he es ima es o (38) o
n
+1. Fo mo e de ails
one can ew i e, wi h mino changes, he p o o o Lemma 2, using (38) ins ead o (35).
He e, he die en
R
-indep enden cons an s
^
N
(
n
)
,
^
T
(
n
)
and
^
S
(
n
)
,dened ecu si ely o
n
= 0
:::
6, dep end only on he same pa ame e s in ol ed in he o mula ion o he
Theo em. We ema k ha condi ion (39) o
n
=0
:::
5, imp oses only ani e numbe
o es ic ions on
R
.Le
R

0
he bigges alue o
R
o which hey hold.
The nex s ep is o con inue wi h he i e a i e no mal o m p o cess, bu using Lemma 2
(wi h
p
= 8) o bound
H
(
n
)
, 6

n

L
+1 (
L
will be de e mined b elow). This will be
done in an induc i eway,showing ha b ounds like (35) hold o each
H
(
n
)
,
n

6. Hence,
we add in (35) he sup e sc ip (
n
)" o
S
,
S
1
,
S
2
,
N

and
T

,andwe eplace
^
S
,
^
N

and
^
T

by
^
S
(
n
)
,
^
N
(
n
)

and
^
T
(
n
)

. All hese b ounds ha e b een aken on
D
m
(

(
n
)
R
(
n
)
), o some

(
n
)
,
R
(
n
)
ha will be de e mined below. Ini ially, o
n
=6, we can ake o ins ance
^
N
(6)
4
=
^
N
(6)
and
^
N
(6)

=
^
N
(6)
=
(
R

0
)
2
. The deni ion o he o he sup e -(6) cons an s
can be done simila ly. Be o e con inuing he i e a i e p o cedu e, we ema k ha , as he
ollowing s eps only a ec high o de e ms,
N
4
and
T
3
emain in a ian du ing all he
no mal o m p o cedu e. Then, Rema k 4 sugges s he deni ion



(
R
)=(
AR
)
2
=
(

+1)
,
whe e
A

1 will be de e mined la e (indep enden ly om
R
). F om his alue o

we
dene, ecu si ely,

(
n
+1)
=

(
n
)
;
4

,
R
(
n
+1)
=
R
(
n
)
exp (
;
4

), o
n

6. To p ese e he
p osi i eness o

(
n
)
, we es ic
n

L
(
R
), being
L
(
R
) he g ea es in ege o which we
ha e 4(
L
;
5)



0
=
8. This implies he ollowing es ic ion on
L
:
L

5+

0
32

1
AR

2

+1
:
(41)
Hence, we ake as
L
he in ege pa o (41). This implies
R
exp (
;

0
=
4)

R
(
n
)

R
i
6

n

L
+1. To apply Lemma 2, we assume ha o he cu en Hamil onian
H
(
n
)
,
6

n

L
, we ha e
^
S
(
n
)

^
S
(6)
,
^
N
(
n
)


^
N

and
^
T
(
n
)


^
T

, o some
^
N

and
^
T

o
b e p ecised la e ( hose b ounds a e necessa y o dene $ in Lemma 2, indep enden om
n
). I o he cu en alue o
n
we ha e
"
^
S
(
n
)
(
R
(
n
)
)
6


+2

1

(42)
hen, he canonical ans o ma ion !
G
(
n
)
1
gi en by Lemma 1 ac s like (40), eplacing

0
by

. The e o e, using Lemma 2, and ecalling ha
2

+1
<
1,
A

1and
R
(
n
)
< R <
1,
one ob ains he ollowing b ounds o he ans o med Hamil onian:
j
N
(
n
+1)
;
N
(
n
)
j

(
n
+1)
R
(
n
+1)

$
^
S
(
n
)

(
R
(
n
)
)
9
A
2
R
2
+
(
R
(
n
)
)
15

2
A
4
R
4
!

2$
^
S
(
n
)
A
2
(
R
(
n
)
)
6

20
No mal Beha iou o Lowe Dimensional To i
j
S
(
n
+1)
1
j

(
n
+1)
R
(
n
+1)

$
^
S
(
n
)
(
R
(
n
)
)
9

(
R
(
n
)
)
2
A
2
R
2
+
(
R
(
n
)
)
4
A
2
R
2
+
(
R
(
n
)
)
5
A
2
R
2
+
(
R
(
n
)
)
6

2
A
4
R
4
!


4$
^
S
(
n
)
A
2
(
R
(
n
)
)
9

j
S
(
n
+1)
2
j

(
n
+1)
R
(
n
+1)

$
^
S
(
n
)
(
R
(
n
)
)
8

(
R
(
n
)
)
2
A
2
R
2
+
(
R
(
n
)
)
3
A
2
R
2
+
(
R
(
n
)
)
7

2
A
4
R
4
!

3$
^
S
(
n
)
A
2
(
R
(
n
)
)
8

j
T
(
n
+1)
;
T
(
n
)
j

(
n
+1)
R
(
n
+1)

$
^
S
(
n
)

(
R
(
n
)
)
9
A
2
R
2
+
(
R
(
n
)
)
15

2
A
4
R
4
!

2$
^
S
(
n
)
A
2
(
R
(
n
)
)
4
:
We ake
A
= max
n
1

q
8$ exp (1)
o
,and hen, ecalling ha
R
(
n
+1)
=
R
(
n
)
exp (
;
4

), we
can dene induc i ely (
n

6),
^
S
(
n
+1)
=
(exp (4

))
9
exp (1)
^
S
(
n
)

^
N
(
n
+1)

=(exp (4

))
9

^
N
(
n
)

+
1
exp (1)
^
S
(
n
)
!

^
T
(
n
+1)

=(exp (4

))
9

^
T
(
n
)

+
1
exp (1)
^
S
(
n
)
!
:
Assuming
R
small enough such ha


1
=
72, we ob ain
^
S
(
n
)
=
^
S
(6)
exp ((6
;
n
)(1
;
36

))

^
S
(6)
exp ((6
;
n
)
=
2)

(43)
^
N
(
n
)


exp (36

(
n
;
6))

^
N
(6)

+
^
S
(6)
1
(exp (1)
;
1)
!

^
T
(
n
)


exp (36

(
n
;
6))

^
T
(6)

+
^
S
(6)
1
(exp (1)
;
1)
!
:
As we a e only in e es ed in hose b ounds o
n

L
+ 1, om he es ic ion on
L
in (41)
we can easily in o duce
n
-indep enden bounds
^
N

and
^
T

o
^
N
(
n
)

and
^
T
(
n
)

. Now,
assuming ha all he s eps a e well dened, i one pu s
n

L
(
R
) + 1 in (43), we ob ain he
exp onen ially small b ound o he s a emen o
^
S
(
L
+1)
. To jus i y ha we can each his
alue, we no e ha (42) holds o all he p e ious
n
,i we es ic
R
wi h "
^
S
(6)
R
3

1.
Then, o p o e he Theo em, we only ha e o in o duce !
R
= !
G
(0)
1

:::

!
G
(
L
)
1
,
and hence, (!
R
)
;
1
= !
G
(
L
)
;
1

:::

!
G
(0)
;
1
.I hose ans o ma ions ac as i has b een
said in he s a emen , he p o o is nished. Fi s , and om he domains o deni ion o
he die en canonical ans o ma ions !
G
(
n
)
1
(see (40), eplacing

0
by

i
n

6), we
deduce ha !
R
is dened on he domain gi en in he s a emen . Mo eo e , om he
b ounds o he die en componen s o !
G
(
n
)
1
;
Id
gi en by Lemma 1, and ema king ha
6

0
+(
L
;
5)



0
=
16, he nal bounds o !
R
;
Id
ollow immedia ly. We conside
now(!
R
)
;
1
. In his case, and using he same a gumen s on !
G
(
n
)
;
1
,one can check ha
i we dene

n
=11

0
=
16 +
n
0
,
R
n
=
R
exp (
;
5

0
=
16 +
n
0
) o
n
= 0
:::
6, and

n
=11

0
=
16 + 6

0
+(
n
;
6)

,
R
n
=
R
exp (
;
5

0
=
16 + 6

0
+(
n
;
6)

) o
n
=6
:::L
+1,
hen we ha e
!
G
(
n
)
;
1
:
D
m
(

n
R
n
)
;!D
m
(

n
+1
R
n
+1
)

o 0

n

L
. The p o o o his ac can b e done by combining he b ounds on !
G
(
n
)
;
1
;
Id
wi h he inequali y (27). Mo eo e , his allows o es ima e (!
R
)
;
1
;
Id
as i has b een
done wi h he case o !
R
.
A. Jo ba and J. Villanue a
21
3.3 Eec i e s abili y
An immedia e consequence o Theo em 1 is ha we can b ound he diusion sp eed a ound
a linea ly s able o us o a Hamil onian sys em. In his case, we ake
G
=
R
2
m
, and
hence,
S
=
N
. Then, we apply Theo em 1, wi hou aking in o accoun he e m
T
o
he decomp osi ion (17). In ac , in his case one can ew i e he p o o s o Lemmas 1,
2, and Theo em 1, in a simple o m (al hough he ac ual o mula ion also holds in his
pa icula case), o ob ain exp onen ially small b ounds o he emainde
S
R
.
Theo em 2
We conside he eal analy ic Hamil onian
(3)
dened on
D
m
(

0
R
0
)
o
some
0
< 
0
<
1
and
R
0
>
0
. We also assume ha al l he eigen alues o
J
m
B
a e o
el lip ic ype, and ha he e exis s

0
>
0
and
>`
such ha
j
ik
>
^
!
(0)
+
l
>

j

0
(
j
k
j
1
+
j
l
x
;
l
y
j
1
)

8
l
2
N
2
m
8
k
2
Z
wi h
j
l
x
;
l
y
j
1
+
j
k
j
1
6
=0
:
Le
R
2
(0
R
0
)
, and le us ake eal ini ial condi ions a
= 0
con ained in
D
m
(0
R
)
.
Then, we can dene
>
2
such ha , i
R
is smal l enough, he co esponding ajec o ies
belong o
D
m
(0
R
)
o any ime
0


T
(
R
)
,wi h
T
(
R
)=
cons :
exp
0
@
cons :

1
R

2

+1
1
A

being he cons an s in he deni ion o
T
(
R
)
independen om
R
.
Rema k 8
In he p oo , and only o echnical easons,

depends on

0
. Ne e heless,
we can ake

as close as we wan o
2
(by aking an ini ial

0
smal l enough, see he p oo
o de ails), bu his implies a educ ion on he se o al lowed
R
, and on he cons an s o
he s abili y ime.
The eason ha o ces o ake
>
2
is he no m used o he no mal a iables. I one
akes he Euclidean no m ins ead o he sup emum no m, he condi ion
>
2
is eplaced
by
>
1
.
P o o :
In o de o simpli y he p o o , we assume ha he ini ial eal a iables (
x y
) o
(3) co esp ond o he ones ha pu
B
in canonical eal o m, ha is,
z
>
B
z
=
m
X
j
=1

j
(
x
2
j
+
y
2
j
)

wi h

j
=
i
j
,
j
=1
:::m
.Mo eo e , we assume ha
R
0
<
1. We in o duce (
X Y
) o
deno e he complexied a iables
x
j
=
X
j
+
iY
j
p
2
 y
j
=
iX
j
+
Y
j
p
2
 j
=1
:::m
(44)
ha pu he ma ix
J
m
B
in he diagonal o m
J
m
B

.Then, we can w i e he Hamil onian
in hese a iables as
H
= ^
!
(0)
>
^
I
+
1
2
Z
>
B

Z
+
N
(
X
^
I Y
)+
S
(
^
 X
^
I Y
)


22
No mal Beha iou o Lowe Dimensional To i
whe e
N
can b e ew i en as a unc ion o
I
,
I
>
=(
^
I
>

~
I
>
), wi h
~
I
j
=
iX
j
Y
j
=
1
2
(
x
2
j
+
y
2
j
),
and
S
e ies
N
(
S
)=0. This co esp onds o he decomp osi ion (17) i one pu s
S
=
N
.
H
is dened on
D
m
(

0
R
0
=
p
2), wi h bounds o he ollowing o m:
j
N
j

0
R

^
NR
4
and
j
S
j

0
R

^
SR
3
, o any 0
< R

R
0
=
p
2. Now, we apply Theo em 1 and we ob ain,
o any
R
small enough, a canonical change !
R
such ha in he new co o dina e sys em
(
^
 X
^
I Y
)=!
R
(
^

R
X
R

^
I
R
Y
R
), we ha e
H
R
:=
H
!
R
= ^
!
(0)
>
^
I
R
+
1
2
(
Z
R
)
>
B

Z
R
+
N
R
(
X
R

^
I
R
Y
R
)+
S
R
(
^

R
X
R

^
I
R
Y
R
)

b eing
N
R
a unc ion o (
I
R
)
>
= ((
^
I
R
)
>

(
~
I
R
)
>
), wi h
~
I
R
j
=
iX
R
j
Y
R
j
.
H
R
is dened on
D
m
(3

0
=
4
R
exp (
;

0
=
4)), wi h
j
S
R
j
3

0
=
4
R
exp (
;

0
=
4)

cons :
exp
0
@
;
cons :

1
R

2

+1
1
A
R
8
:=
M
(
R
)
:
The canonical equa ions o (
X
R

^
I
R
Y
R
)a e
_
X
R
j
=
@
H
R
@Y
R
j

_
Y
R
j
=
;
@
H
R
@X
R
j
 j
=1
:::m
_
I
R
j
=
@
H
R
@
^

R
j
=
@S
R
@
^

R
j
 j
=1
::: :
(45)
F om his, one ob ains (using ha
N
R
is in ac only a unc ion o
I
, and ecalling
`
=
+
m
),
_
I
R
j
=
i
@
H
R
@Y
R
j
Y
R
j
;
i
@
H
R
@X
R
j
X
R
j
=
i
@S
R
@Y
R
j
Y
R
j
;
i
@S
R
@X
R
j
X
R
j
 j
=
+1
:::`:
Wepu
I
R
j
(
^

R
X
R

^
I
R
Y
R
) o he exp essions on he igh -hand side o
_
I
R
j
,
j
=1
:::`
.
We use Lemma 5 o b ound hese exp essions. Then, o
j
=1
:::
one has
jI
R
j
j
0
R
exp (
;

0
=
2)

4
M
(
R
)
3

0
exp (1)
:
(46)
I we combine Lemma 5 wi h he inequali y (27), one ob ains o
j
=
+1
:::`
ha ,
jI
R
j
j
0
R
exp (
;

0
=
2)

2
M
(
R
) exp (
;

0
=
2)
exp (
;

0
=
4)(1
;
exp (
;

0
=
4))

16
M
(
R
) exp (
;

0
=
4)

0
:
(47)
Tocon inue he p o o , we pu (
x
R
y
R
) o he a iables ha come om he ealica ion"
o (
X
R
Y
R
), ha is,
X
R
j
=(
x
R
j
;
iy
R
j
)
=
p
2,
Y
R
j
=(
y
R
j
;
ix
R
j
)
=
p
2. In ac , as !
R
p ese es
he symme ies o
H
(due o he complexica ion o a eal Hamil onian, see Sec ion 2.1.2),
weha e ha he Hamil onian in he a iables (
x
R
y
R
) is eal analy ic. Towo k wi h hose
die en ep esen a ions o he a iables, we gi e he ollowing ema ks: (
i
) he se o eal
a iables (
x
j
y
j
)such ha
j
x
j
j

j
y
j
j
A
is con ained in he se o complex (
X
j
Y
j
) such
ha
j
X
j
j

j
Y
j
j
A
, (
ii
) he se o complex (
X
j
Y
j
)such ha
j
X
j
j

j
Y
j
j
A
, is con ained
in he complex se o (
x
j
y
j
) such ha
j
x
j
j

j
y
j
j
p
2
A
( his p op e y has been used o
say ha
H
is dened in
D
m
(

0
R
0
=
p
2)), (
iii
) he se o eal (
x
j
y
j
) such ha
I
j

A
2
,
A. Jo ba and J. Villanue a
23
is con ained in he se o eal (
x
j
y
j
)such ha
j
x
j
j

j
y
j
j
p
2
A
. Those ema ks a e used
when wo king wi h hese die en kind o a iables, and one wan s o con ol he size o
he co esp onding domains, when we change he a iable ep esen a ion.
Now, we ake eal alues o (
^

R
x
R

^
I
R
y
R
) as ini ial condi ions a
= 0. To p o e
he lowe bound o he s abili y ime, we conside a xed 0
<  <
1, and we es ic
o ini ial condi ions such ha , when exp essed in e ms o (
^

R
X
R

^
I
R
Y
R
), hey b elong
o
D
m
(0
R
exp (
;

0
=
2)
=
p
2). Then, we ha e ha he co esp onding ini ial ac ions
I
R
a e b ounded by
j
I
R
j
(0)
j 
R
2

2
exp (
;

0
)
=
2,
j
= 1
:::`
. Using his, we deduce om
he b ounds (46) and (47) ha , o he a jec o ies o he Hamil onian equa ions (45), we
ha e
j
I
R
j
(
)
j
R
2
exp (
;

0
)
=
2 o 0


T
(
R
), whe e we can ake
T
(
R
)=
R
2

0
exp (
;
3

0
=
4)(1
;

2
)
32
M
(
R
)
:
This bound comes om (47), ha is he wo s case. This is he exp ession o he
s abili y ime o he s a emen o he Theo em. To use he bounds (46) and (47) o
I
R
j
, we need ha hese a jec o ies exp essed in e ms o (
^

R
X
R

^
I
R
Y
R
) b elong o
D
m
(0
R
exp (
;

0
=
2)) up o ime
T
(
R
). As we ha e
j
I
R
j
(
)
j 
R
2
exp (
;

0
)
=
2, his
ollows om ema ks (
iii
) and (
i
). F om ha we deduce, using he b ounds o (!
R
)
;
1
;
Id
p o ided by Theo em 1 and ema k (
ii
), ha he co esp onding eal a jec o ies in e ms
o (
^
 x
^
I y
) a e con ained in
D
m
(0
R
1
), being
R
1
dened by
R
1
=max
8
<
:
p
2

R
exp

;

0
2

+
R
0
exp (
;
1
=
2)
32
!

s
R
2
exp (
;

0
)+
R
2

0
exp (
;
1)
16
9
=

:
Then, i we gi e o (
^
 x
^
I y
) a eal se o poin s such ha , exp essed in e ms o
(
^
 X
^
I Y
), hey b elong o he domain (!
R
)
;
1
(
D
m
(0
R
exp (
;

0
=
2)
=
p
2)), hen, he
a jec o ies o
H
wi h ini ial condi ions in his se emain in
D
m
(0
R
1
) o a ime span
T
(
R
). Wi h simila a gumen s as he ones used o dene
R
1
(using now ema k (
ii
)), one
can check ha his domain can be aken as
D
m
(0
R
2
), b eing
R
2
dened by
R
2
=min
8
<
:
R
exp (
;

0
=
2)
p
2
;
R
0
exp (
;
1
=
2)
32

s
R
2

2
exp (
;

0
)
2
;
R
2

0
exp (
;
1)
16
9
=

:
I one conside s he ow !
H
dened om
D
m
(0
R
2
)
R
2
`
o
D
m
(0
R
1
)
R
2
`
, o
0


T
(
R
), hen, pu ing
R

R
2
in he s a emen , and aking an
R
-indep enden
alue o

close enough o1such ha
R
2
>
0, we can dene

=
R
1
=R
2
.
4 Es ima es on he amilies o lowe dimensional o i
Le us conside he eal analy ic educed Hamil onian
H
o (3) and a xed subbundle
G
o
ellip ic di ec ions o
J
m
B
. In Theo em 1 weha e p o ed ha , unde s anda d Diophan ine
condi ions, one can pu
H
in no mal o m wi h esp ec o he se
S
(see (15) and (16)
o he deni ion), wi h an exp onen ially small emainde . I we w i e his semino mal
o m in e ms o he complexied a iables
Z
,and wi hou changing he name o he
Hamil onian, one has
H
=
!
(0)
>
I
+
1
2
^
Z
>
^
B

^
Z
+
F
(
I
)+
1
2
^
Z
>
Q
(
I
)
^
Z
+
T
(
^
 X
^
I Y
)+
R
(
^
 X
^
I Y
)
:
(48)
24
No mal Beha iou o Lowe Dimensional To i
To explain he no a ion used, le us ecall ha he die en esonan e ms dep end only on
^
I
and on he p o duc s
X
j
Y
j
,
j
=1
:::m
, bu , om he s uc u e o
S
, no all he p ossible
combina ions o hose monomials akeplacein
M
(
S
). Then, wein o duce
I
>
=(
^
I
>

~
I
>
),
wi h
~
I
j
=
iX
j
Y
j
,
j
=1
:::m
1
,andwi h his deni ion (48) can b e desc ib ed as ollows:
he symme ic ma ix
^
B

is dened om
B

skipping he 2
m
1
eigen alues asso cia ed o
G
,
J
m
;
m
1
^
B

= diag(
^

).
F
and
Q
co esp ond o he no mal o m wi h esp ec o
S
,wi h
he expansion o
F
s a ing a second o de wi h esp ec o
I
,and wi h
Q
(0) =0. I is
no dicul o check ha by cho osing he a iables
~
Z
in sui able o m (as i has b een
done in he p o o o Theo em 2),
F
is eal analy ic. Mo eo e ,
Q
is a symme ic ma ix
such ha
J
m
;
m
1
Q
is diagonal,
T2M
(
NnS
)(so
T 
O
3
(
^
Z
)) and
R2
M
(
S
).
We assume ha his no mal o m has b een done o a gi en (and small enough)
R
,as
in he o mula ion o Theo em 1. We only conside he
R
-dep endence when we gi e he
bounds o he die en e ms o (48). To ob ain hese b ounds, le us dene

1
=3

0
=
4,
whe e we ecall ha

0
is he wid h o he s ip o anali ici y, wi h esp ec o
^

, o he
ini ial Hamil onian. Then, Theo em 1 implies ha , o any
R
small enough, we ha e
jF j
0
R

^
F
R
4

jF
3
j
0
R

^
F
3
R
6

jQj
0
R

^
Q
R
2

jQ
2
j
0
R

^
Q
2
R
4

jT j

1
R

^
T
R
3

jRj

1
R

cons :
exp

;
cons :

1
R

2

+1

R
8

(49)
To de i e hese bounds on
D
m
(0
R
), we ha e conside ed he unc ions ha dep end on
~
I
as unc ions o
~
Z
. He e, we ha e spli
F
=
F
2
+
F
3
and
Q
=
Q
1
+
Q
2
.
F
2
and he
comp onen s o
Q
1
a e p olynomials on
I
o deg ees 2 and 1 esp ec i ely.
F
3
and
Q
2
con ain
he emaining e ms. We no e ha he deni ion o
F
2
and
Q
1
do es no dep end on he
o de o he semino mal o m.
This semino mal o m has b een o mally explained in Sec ion 2.1.3, and we will use
he no a ion ela ed o (7) o ep esen he no mal o m o i.
The main pu p ose o his sec ion is o s udy he p e sis ence o hose o i when we add
he emainde
R
. We no e ha , as
jRj
is exp onen ially small wi h
R
,we can exp ec ha
he o i o (7) will su i e, excep he ones co esp onding o a se o pa ame e s (
I
(0)) o
exp onen ially small measu e wi h esp ec o
R
. We will show ha his asse ion holds,
assuming ce ain s anda d nondegene acy condi ions on his amily o o i, ha ha e b een
explained in Sec ion 2.2.3 (condi ions ha , as we will see, can b e checked by compu ing a
no mal o m up o deg ee 4, ha is, om
F
2
and
Q
1
). As i is a mo e na u al pa ame e ,
he esul s will b e o mula ed in e ms o equencies ins ead o ac ions.
4.1 Nondegene acy condi ions
Be o e he igo ous o mula ion o he esul s, le us gi e in explici o m hese nondegen-
e acy condi ions.
4.1.1 Nondegene acy o he in insic equencies
The  s one is a s anda d nondegene acy condi ion on he dep endence o he equencies
wi h esp ec o he ac ions: we equi e
de
C 6
=0

C
=
@
2
F
2
@I
2
(0)
:
(50)
A. Jo ba and J. Villanue a
25
This allows o pa ame ize he o i o he amily by hei ec o o in insic equencies
(ins ead o
I
(0)). O cou se, weha e o b e close enough o he ini ial
-dimensional o us.
This asse ion is jus ied by he ollowing lemma:
Lemma 3
Le us assume ha
de
C 6
=0
. Then, i
R
is smal l enough, he e exis s a eal
analy ic ec o ial unc ion
I
(
!
)
,dened on he se

!
2
C
+
m
1
:
j
!
;
!
(0)
j
1
8
(
jC
;
1
j
)
;
1
R
2


(51)
such ha
@
F
@I
(
I
(
!
)) =
!
;
!
(0)

wi h
I
(
!
(0)
)=0
. Mo eo e , we ha e
jI
(
!
)
j
1
4
R
2
o any
!
in he se
(51)
,and i
!
(1)
,
!
(2)
belong in
(51)
, hen
jI
(
!
(1)
)
;I
(
!
(2)
)
j
2
jC
;
1
jj
!
(1)
;
!
(2)
j
:
O cou se, we a e s il l using he no a ion o Sec ion
4
.
P o o :
We ha e
F
(
I
) =
1
2
I
>
C
I
+
F
3
. Then, we ake a xed
!
in he se (51), and we
wan o sol e he equa ion:
I
(
!
)=
C
;
1

!
;
!
(0)
;
@
F
3
@I
(
I
(
!
))
!
:
(52)
Pu ing he sup e sc ip s (
k
+ 1)" and (
k
)" o
I
(
!
) in (52), we can conside his exp es-
sion as an i e a i e p o cedu e, using
I
(0)
(
!
) = 0 as he seed. I we assume
jI
(
k
)
(
!
)
j
1
4
R
2
,
hen, using Cauchy inequali ies, we ha e o
R
small enough,
jI
(
k
+1)
(
!
)
j jC
;
1
j

1
8
(
jC
;
1
j
)
;
1
R
2
+
^
F
3
R
6
3
4
R
2
!

1
4
R
2

whe e we ha e used he b ounds o (49) o
F
3
, ema king ha
jF
3
j
0
R
is abound o he
sup emum no m o
F
3
(
I
) i
j
I
j 
R
2
. Mo eo e , o ensu e con e gence, we ema k ha
using he main alue heo em one has,
jI
(
k
+1)
(
!
)
;I
(
k
)
(
!
)
j
(
+
m
1
)
^
F
3
R
6

3
8

2
R
4
jI
(
k
)
(
!
)
;I
(
k
;
1)
(
!
)
j
1
2
jI
(
k
)
(
!
)
;I
(
k
;
1)
(
!
)
j

i
R
is small enough. Clea ly, he limi unc ion is analy ic wi h esp ec o
!
,and om
he eal analy ic cha ac e o
F
,
I
is in ac eal analy ic. Taking
!
(1)
,
!
(2)
in he se
(51), one has
I
(
!
(1)
)
;I
(
!
(2)
)=
C
;
1
(
!
(1)
;
!
(2)
)+
C
;
1

@
F
3
@I
(
I
(
!
(2)
))
;
@
F
3
@I
(
I
(
!
(1)
))
!

and wi h he same a gumen s p e iously used, we ob ain o
R
small enough
jI
(
!
(1)
)
;I
(
!
(2)
)
j
2
jC
;
1
jj
!
(1)
;
!
(2)
j
:
32
No mal Beha iou o Lowe Dimensional To i
4.3.2 The i e a i e scheme
Now, we can desc ib e he i e a i e p o cedu e used o cons uc in a ian (
+
m
1
)-
dimensional o i. This p o cess is gi en by a sequence o canonical changes o a iables,
cons uc ed as he ime one ow o a sui able gene a ing unc ion
S
!
.The changes a e
cons uc ed o kill he e ms ha obs uc s he exis ence o an in a ian educed o us
wi h ec o o basic equencies gi en by
!
. As usual ( o o e come he eec o he small
di iso s), he changes a e chosen o p o duce a quad a ically con e gen scheme, ins ead
o he linea one o Lemma 1.
Fi s , we desc ib e a gene ic s ep o his i e a i e p o cess. Fo his pu p ose, we expand
he Hamil onian
H
(0)
in he ollowing o m
H
(0)
=
a
(

)+
b
(

)
>
^
Z
+
c
(

)
>
I
+
1
2
^
Z
>
B
(

)
^
Z
+
I
>
E
(

)
^
Z
+
1
2
I
>
C
(

)
I
+)(

^
X I
^
Y
)

(67)
whe e we do no w i e explici ly he
!
-dep endence and whe e weha e skipp ed he sup e -
sc ip (0)" in he die en pa s o he Hamil onian. F om his expansion, wein o duce
he ollowing no a ions: 
H
(0)
]
(
^
Z
^
Z
)
=
B
, 
H
(0)
]
(
I
^
Z
)
=
E
and
<H
(0)
>
=
H
(0)
;
). F om
he bounds on he e ms o he decomp osi ion (60), we ha e ha ~
a
,
b
,
c
;
!
,
B
;
^
B
(0)

,
C
;C
(0)
and
E
a e all
O
(
^
H
(0)
). No e ha i we a e able o kill he e ms ~
a
,
b
and
c
;
!
,
we will ob ain an in a ian o us wi h in insic equency
!
.Ne e heless, as we wan o
ha e simple equa ions a e e y s ep o he i e a i escheme ( his is, linea equa ions wi h
cons an co ecien s), we a e o ced o kill some hing mo e. Then, we ask he nal o us
o ha e educible no mal ow gi en by a diagonal ma ix. This is, we wan ha he new
ma ix
B
e ies
B
=
J
m
;
m
1
(
B
) whe e, o a (2
s
)-dimensional ma ix
A
(

) dep ending
2

-p e io dically on

, we dene
J
s
(
A
)=
;
J
s
dp(
J
s
A
). He e, dp(
A
) deno es he diagonal
ma ix ob ained aking he diagonal en ies o
A
. Mo eo e , we ha e o elimina e
E
o
uncouple he neu al" and he no mal di ec ions o he o us up o  s o de . Thus,
o each s ep o he i e a i e p o cess, we use a canonical change o a iables, gi en by a
gene a ing unc ion o he o m
S
(

^
X I
^
Y
)=

>

+
d
(

)+
e
(

)
>
^
Z
+
(

)
>
I
+
1
2
^
Z
>
G
(

)
^
Z
+
I
>
F
(

)
^
Z
whe e

2
C
+
m
1
,
d
=0,
= 0 and
G
is a symme ic ma ix, wi h
J
m
;
m
1
(
G
)=0. The
ans o med Hamil onian is
H
(1)
=
H
(0)

!
S
1
. We expand
H
(1)
in he same way as
H
(0)
in (67), keeping he same name o he new a iables, bu adding he sup e sc ip (1)"
o
a
,
b
,
c
,
B
,
C
,
E
and ). Then, we ask ~
a
(1)
=0,
b
(1)
= 0,
c
(1)
;
!
= 0,
E
(1)
= 0 and
B
(1)
=
J
m
;
m
1
(
B
(1)
). We will show ha his can be achie ed up o  s o de in he size
o
^
H
(0)
. Fo his pu p ose, we w i e hose condi ions in e ms o he ini ial Hamil onian
and he gene a ing unc ion, and hen, we ob ain he ollowing equa ions:
(
eq
1
) ~
a
;
@d
@
!
=0,
(
eq
2
)
b
;
@e
@
!
+
^
B
(0)

J
m
;
m
1
e
=0,
(
eq
3
)
c
;
!
;
@
@
!
;C
(0)


+

@d
@

>

=0,
(
eq
4
)
B

;J
m
;
m
1
(
B

)
;
@G
@
!
+
^
B
(0)

J
m
;
m
1
G
;
GJ
m
;
m
1
^
B
(0)

=0,

A. Jo ba and J. Villanue a
33
(
eq
5
)
E

;
@F
@
!
;
FJ
m
;
m
1
^
B
(0)

=0,
b eing
B

=
B
;
2
4
@H
(0)

@I
0
@

+

@d
@
!
>
1
A
;
@H
(0)

@
^
Z
J
m
;
m
1
e
3
5
(
^
Z
^
Z
)

E

=
E
;C
(0)

@e
@
!
>
;
2
4
@H
(0)

@I
0
@

+

@d
@
!
>
1
A
;
@H
(0)

@
^
Z
J
m
;
m
1
e
3
5
(
I
^
Z
)
:
To sol e hose homological equa ions, we expand hem in Fou ie se ies and we equa e
he co esp onding co ecien s, ob aining he o mal solu ions. The nex s ep is o de i e
bounds on hose solu ions. As we will use hese bounds in i e a i e o m, we wan o
make clea which exp essions change om one s ep o ano he , and which ones can be
b ounded indep enden ly om he s ep. Fo his pu p ose, we ake xed p osi i e cons an s
m
, ^
m
, ~
m
,

2
,

1
, ^

, ~

dened as wice he co esp onding ini al alues
m
(0)
, ^
m
(0)
, ~
m
(0)
,

(0)
2
,

(0)
1
, ^

(0)
, ~

(0)
and a xed

1
, 0
< 
1
< 
(0)
1
.In wha ollows,
^
N
will deno e an
exp ession dep ending only on
m
, ^
m
,

1
,

2
, ^

, he die en dimensions
,
m
,
m
1
, plus

and

0
.
^
N
will be edened du ing he desc ip ion o he i e a i e scheme o mee a
ni e numbe o condi ions. The idea is o p e o m he bounds on he i e a i e scheme
pu ing he sup e sc ip (0)" on he e ms ha change a e e y i e a ion. Hence, we
w i e he b ounds on
^
H
(0)
as
k
^
H
(0)
k
E
(0)

(0)
R
(0)

M
(0)
and
L
E
(0)

(0)
R
(0)
^
H
(0)
g
L
(0)
, wi h
M
(0)
(
R
)

M
(
R
) and
L
(0)
(
R
)

(
M
(
R
))
1
;

. Hence, using Lemma 5,
k
a
;

(0)
k
E
(0)

(0)

M
(0)

k
E
k
E
(0)

(0)

2(
m
;
m
1
)
M
(0)
(
R
(0)
)
3

k
c
;
!
k
E
(0)

(0)

M
(0)
(
R
(0)
)
2

k
B
;
^
B
(0)

k
E
(0)

(0)

(2(
m
;
m
1
)+1)
M
(0)
(
R
(0)
)
2

k
b
k
E
(0)

(0)

M
(0)
R
(0)

k
C
;C
(0)
k
E
(0)

(0)

(2(
+
m
1
)+1)
M
(0)
(
R
(0)
)
4

k
)
k
E
(0)

(0)
R
(0)

^

(0)
+
M
(0)
:
(68)
Mo eo e , we can use Lemma 11 o deduce ha he same b ounds hold o hei Lipschi z
cons an s on
E
(0)
, eplacing
M
(0)
by
L
(0)
,and ^

(0)
by ~

(0)
. Then, o p o e he con e gence
o he expansion o
S
,we need some kind o con ol on he die en small di iso s in ol ed.
Fo his pu p ose, we es ic he pa ame e
!
o he subse
E
(1)
(
R
)
 E
(0)
(
R
) o which
he ollowing Diophan ine es ima es hold: we say ha
!
2E
(1)
, i
!
2E
(0)
,and
j
ik
>
!
+
l
>
^

(0)
(
!
)
j

(0)
(
R
)
j
k
j

1
 k
2
Z
+
m
1
n
0
g
 l
2
N
2(
m
;
m
1
)

0
<
j
l
j
1

2

(69)
o ce ain

(0)
>
0. We exp ec he measu e o
E
(0)
nE
(1)
o be o o de

(0)
and, hence,
as we wan o ha e exp onen ially small b ounds o his measu e, we ake

(0)

(
M
(0)
)

.
Then, we p o ceed o b ound he solu ions o he die en homological equa ions. Fo his
pu p ose, we use Lemma 4. Mo e p ecisely, we dene

(0)
=(
M
(0)
)

,and we ake

(0)
as
a alue o

o use he die en es ima es p o ided by his lemma. In o de o simpli y
he p o o s, we assume

(0)
;
N
(0)


0
=
4, whe e
N
2
N
will b e a xed in ege ha will
be de e mined be o e he desc ip ion o he i e a i e scheme. Mo eo e , we also assume
ha (
M
(0)
)


R
(0)

1. Then, one can sol e (
eq
1
)
;
(
eq
5
)as ollows:
34
No mal Beha iou o Lowe Dimensional To i
(
eq
1
) Fo
d
,we ha e
d
(

)=
X
k
2
Z
+
m
1
n
0
g
a
k
ik
>
!
exp(
ik
>

)

ha implies,
k
d
k
E
(1)

(0)
;

(0)




(0)
exp (1)
!

k
~
a
k
E
(1)

(0)

(0)

^
N
(
M
(0)
)
1
;

;

:
(
eq
2
) Fo any
j
,1

j

2(
m
;
m
1
), we ha e
e
j
(

)=
X
k
2
Z
+
m
1
b
jk
ik
>
!
+
^

(0)
j
exp(
ik
>

)

and hence,
k
e
k
E
(1)

(0)
;

(0)


2

1
+



(0)
exp (1)
!

1

(0)
!
k
b
k
E
(1)

(0)

^
N
(
M
(0)
)
1
;
2

;

:
(
eq
3
) Taking a e age wi h esp ec o

, we ob ain

=(
C
(0)
)
;
1
0
@
c
;
!
; C
(0)

@d
@
!
>
1
A
:
Thus,
k

k
E
(1)
=
k
(
C
(0)
)
;
1
C
(0)

k
E
(1)
k
(
C
(0)
)
;
1
k
E
(1)
k
C
(0)

k
E
(1)


m
0
B
@
k
c
;
!
k
E
(1)

0
+






C
(0)

@d
@
!
>






E
(1)

0
1
C
A


m

k
c
;
!
k
E
(1)

(0)
+ ^
m
k
d
k
E
(1)

(0)
;

(0)
(

(0)
;

(0)
) exp (1)
!

^
N
(
M
(0)
)
1
;

;

:
Tosol e he equa ion o
, we dene
c

=~
c
;
~
C
(0)

;C
(0)

@d
@
!
>
+
C
(0)

@d
@
!
>

and hen, o any 1

j

+
m
1
, we ha e
j
(

)=
X
k
2
Z
+
m
1
n
0
g
c

jk
ik
>
!
exp(
ik
>

)
:
To b ound
,  s we ha e ha
k
c

k
E
(1)

(0)
;
2

(0)
 k
~
c
k
E
(1)

(0)
+
kC
(0)
k
E
(1)

(0)

k

k
E
(1)
+
k
d
k
E
(1)

(0)
;

(0)

(0)
exp (1)
!


^
N
(
M
(0)
)
1
;
2

;


and om he e
k
k
E
(1)

(0)
;
3

(0)




(0)
exp (1)
!

k
c

k
E
(1)

(0)
;
2

(0)

(0)

^
N
(
M
(0)
)
1
;
3

;
2

:
A. Jo ba and J. Villanue a
35
(
eq
4
) We dene
B

=
B

;J
m
;
m
1
(
B

), and hen, i
G
= (
G
jl
), 1

j l

2(
m
;
m
1
),
we ha e
G
jl
(

)=
X
k
2
Z
+
m
1
B

jlk
ik
>
!
+
^

(0)
j
+
^

(0)
l
exp(
ik
>

)
:
In his sum we ha e o a oid he indices (
j l  k
) o which
j
j
;
l
j
=
m
;
m
1
and
k
=0. In hese cases weha e i ial ze o di iso s, bu also he co ecien
B

jl
0
is 0.
Mo eo e , we ema k ha he ma ix
G
is symme ic. Then, o bound
G
, we ha e
o b ound
B

. Fi s , we ha e
k
B

;
^
B
(0)

k
E
(1)

(0)
;
2

(0)
k
B
;
^
B
(0)

k
E
(1)

(0)
;
2

(0)
+
+(2(
m
;
m
1
)+1)(
+
m
1
)
k
H
(0)

k
E
(1)

(0)
R
(0)
(
R
(0)
)
4

k

k
E
(1)
+
k
d
k
E
(1)

(0)
;

(0)

(0)
exp (1)
!
+
+24(
m
;
m
1
)
2
k
H
(0)

k
E
(1)

(0)
R
(0)
(
R
(0)
)
3
k
e
k
E
(1)

(0)
;

(0)

^
N
(
M
(0)
)
1
;
6

;


and om he deni ion o
B

and he no m used, he same bound holds o
B

.
Then,
k
G
k
E
(1)

(0)
;
3

(0)


1

1
+



(0)
exp (1)
!

1

(0)
!
2(
m
;
m
1
)
k
B

k
E
(1)

(0)
;
2

(0)


^
N
(
M
(0)
)
1
;
7

;
2

:
(
eq
5
)The die en comp onen s o
F
a e gi en by
F
jl
(

)=
X
k
2
Z
+
m
1
E

jlk
ik
>
!
+
^

(0)
l
exp(
ik
>

)

o
j
=1
:::
+
m
1
and
l
=1
:::
2(
m
;
m
1
). Thus,
k
E

k
E
(1)

(0)
;
2

(0)
k
E
k
E
(1)

(0)
+2(
m
;
m
1
)
kC
(0)
k
E
(1)

(0)
k
e
k
E
(1)

(0)
;

(0)

(0)
exp (1)
+
+4(
m
;
m
1
)(
+
m
1
)
k
H
(0)

k
E
(1)

(0)
R
(0)
(
R
(0)
)
5

k

k
E
(1)
+
k
d
k
E
(1)

(0)
;

(0)

(0)
exp (1)
!
+
+8(
m
;
m
1
)
2
k
H
(0)

k
E
(1)

(0)
R
(0)
(
R
(0)
)
4
k
e
k
E
(1)

(0)
;

(0)

^
N
(
M
(0)
)
1
;
7

;


and, hence,
k
F
k
E
(1)

(0)
;
3

(0)


2

1
+



(0)
exp (1)
!

1

(0)
!
2(
m
;
m
1
)
k
E

k
E
(1)

(0)
;
2

(0)


^
N
(
M
(0)
)
1
;
8

;
2

:
We use hese es ima es o b ound he ans o med Hamil onian
H
(1)
. Fo his pu p ose, we
dene
H
(0)

:=
H
(0)
S
g
=
H
(0)

1
+
H
(0)

2
,wi h
H
(0)

1
=

!
>
I
+
1
2
^
Z
>
^
B
(0)

^
Z
+
1
2
I
>
C
(0)
I
+
H
(0)

S


36
No mal Beha iou o Lowe Dimensional To i
and
H
(0)

2
=
^
H
(0)
S
g
. No e ha we a e spli ing he con ibu ions ha a e
O
1
(
^
H
(0)
) and
O
2
(
^
H
(0)
). Then, by cons uc ion o
S
,one has
H
(0)
+
H
(0)

1
=

(1)
+
!
>
I
+
1
2
^
Z
>
^
B
(1)

^
Z
+
1
2
I
>
C
(1)
(

)
I
+
H
(1)


wi h
^
B
(1)

=
J
m
;
m
1
(
^
B
(1)

) and
<H
(1)

>
=0. Hence,
H
(1)
akes he same o m as
H
(0)
in
(60) i we dene
^
H
(1)
=
H
(0)

!
S
1
;
H
(0)
;
H
(0)

1
=
Z
1
0

H
(0)

2
+(1
;
)
H
(0)

1
S
g


!
S
d :
(70)
To b ound he die en e ms o
H
(1)
, we use Lemma 6 o b ound he Poisson b acke s
in ol ed in he p e ious exp essions:
k
H
(0)

1
k
E
(1)

(0)
;
4

(0)
R
(0)
exp (
;

(0)
)

^
N
(
M
(0)
)
1
;
12

;
2


k
H
(0)

1
S
gk
E
(1)

(0)
;
5

(0)
R
(0)
exp (
;
2

(0)
)

^
N
(
M
(0)
)
2
;
24

;
4


k
H
(0)

2
k
E
(1)

(0)
;
4

(0)
R
(0)
exp (
;

(0)
)

^
N
(
M
(0)
)
2
;
12

;
2

:
Hence, o b ound
^
H
(1)
one only needs o con ol he eec o !
S
. To his end, we ema k
ha om he b ounds on he solu ions o (
eq
1
)
;
(
eq
5
), one has
k
S
k
E
(1)

(0)
;
4

(0)
R
(0)

^
N
(
M
(0)
)
1
;
9

;
2


(71)
whe e
S
is aken wi h esp ec o (

^
X I
^
Y
). I we assume ha
k
S
k
E
(1)

(0)
;
4

(0)
R
(0)

(
R
(0)
)
2

(0)
exp (
;
1)
=
2

(72)
hen, !
S
is well dened om
D
+
m
1
m
;
m
1
(

(0)
;
5

(0)
R
(0)
exp (
;

(0)
)) o
D
+
m
1
m
;
m
1
(

(0)
;
4

(0)
R
(0)
), o any
;
1


1, and o any
!
2 E
(1)
( his ollows om Lemma 9 and
(27)). Mo e p ecisely, we ha e ha
k
!
S
;
Id
k
E
(1)

(0)
;
5

(0)
R
(0)
exp (
;

(0)
)
k
S
k
E
(1)

(0)
;
4

(0)
R
(0)

(73)
o any
;
1


1. F om (71) weha e ha (72) holds i
^
N
(
M
(0)
)
1
;
12

;
2


1, condi ion
ha will ollow immedia ely om he induc i e es ic ions. Applying he b ounds (71),
(71) and (73) o (70) and using Lemma 7, we deduce
k
^
H
(1)
k
E
(1)

(0)
;
6

(0)
R
(0)
exp (
;
3

(0)
)

^
N
(
M
(0)
)
2
;
24

;
4

:
(74)
Mo eo e , he b ound on
H
(0)

1
p o duces
k

(1)
;

(0)
k
E
(1)

^
N
(
M
(0)
)
1
;
12

;
2


k
^
B
(1)

;
^
B
(0)

k
E
(1)

^
N
(
M
(0)
)
1
;
14

;
2


kC
(1)
;C
(0)
k
E
(1)

(0)
;
4

(0)

^
N
(
M
(0)
)
1
;
16

;
2


k
H
(1)

;
H
(0)

k
E
(1)

(0)
;
4

(0)
R
(0)
exp (
;

(0)
)

^
N
(
M
(0)
)
1
;
12

;
2

:
(75)
We ake
N

6, and we dene

(1)
=

(0)
;
N
(0)
,and
R
(1)
=
R
(0)
exp (
;
(
N
;
3)

(0)
).
Then, i is no dicul o ew i e he bounds on
H
(1)
as he ones on
H
(0)
, bu now on
D
+
m
1
m
;
m
1
(

(1)
R
(1)
). To i e a e his scheme, we only need o check ha he b ounds
assumed on
H
(0)
o dene
^
N
s ill hold on
H
(1)
. This is done in he nex sec ion.
A. Jo ba and J. Villanue a
37
4.3.3 Con e gence o he i e a i e scheme
Lo oking a he b ounds o he p e ious sec ion, we ake
>
0 small enough such ha , o
s
=2(1
;
16

;
2

), weha e
s>
1. Then, assuming
^
N

1, wedene
M
(1)
=(
^
NM
(0)
)
s
(no e ha his is a b ound o he no m o
^
H
(1)
in (74)). I he hyp o heses needed o
i e a e hold, we ob ain ecu si ely
M
(
n
)
= (
^
NM
(0)
)
s
n
, and hence, o
R
small enough,
we ha e lim
n
!1
M
(
n
)
= 0. Le us dene
E

(
R
)as he se o pa ame e s
!
o which
all he s eps a e well dened. We assume ha , o any
!
2 E

(
R
), he comp osi ion o
canonical ans o ma ions !

= !
S
(0)
1

!
S
(1)
1

:::
(b eing
S
(
n
)
he gene a ing unc ion
used a he
n
-s ep o he i e a i e p o cedu e) is con e gen . Then, he limi Hamil onian
H

=
H
(0)

!

akes he o m:
H

=


(
!
)+
!
>
I
+
1
2
^
Z
>
^
B

(
!
)
^
Z
+
1
2
I
>
C

(
 !
)
I
+
H


(

^
X I
^
Y!
)

wi h
< H


>
=0. This is, we ob ain o any
!
2 E

a Hamil onian wi h an (
+
m
1
)-
dimensional educible o us, wi h linea quasip e io dic ow gi en by
!
.
Le us p o e ha he induc i e bounds hold. Fi s , we check ha we can dene,
ecu si ely, cons an s
m
(
n
)
, ^
m
(
n
)
,

(
n
)
1
,

(
n
)
2
and ^

(
n
)
, eplacing he ini ial sup e - (0)"
ones, such ha hey a e also b ounded by
m
, ^
m
,

1
,

2
and ^

, esp ec i ely. Top o e ha ,
we no e ha he exp essions in he igh -hand side o (75) can b e b ounded by(
^
NM
(0)
)
s=
2
(we ema k ha he same b ound holds o (71)). Hence, i e a ing his bounds, we only
need o use ha he sum
X
n

0

^
NM
(0)

s
n
+1
2

(76)
is con e gen o
R
small enough (and in ac , ha i go es o ze o when
R
do es), o
jus i y hese
n
-indep enden b ounds. The same a gumen s can be used o p o e ha
k


k
E

<
+
1
. He e, we only check he b ound
m
(
n
)

m
, b ecause is he only one ha
do es no ollow di ec ly: no e ha one can dene
m
(1)
=
m
(0)
1
;
m
(0)
(
^
NM
(0)
)
s=
2

and hen, aking
R
small enough, weha e
m
(1)

m
. Hence, i e a ing his deni ion and
assuming
m
(
n
)

m
by induc ion, we ha e
m
(
n
)

m
(0)
n
;
1
Y
j
=0
1
1
;
m
(
^
NM
(0)
)
s
n
+1
=
2
:
Unde his induc i e hyp o esis, one can b ound
m
(
n
)
by an inni e p o duc ha i is
con e gen b ecause (76) do es. F om he e, he b ound
m
(
n
)

m
ollows immedia ely o
R
small enough. Finally, wi h he induc i e deni ions

(
n
+1)
=

(
n
)
;
N
(
n
)
and
R
(
n
+1)
=
R
(
n
)
exp (
;
(
N
;
3)

(
n
)
),
n

0, we need o check ha

(
n
)


0
=
4and
R
(
n
)

M
(
n
)
. We
ema k ha , as we ake

(
n
)
=(
M
(
n
)
)

, we ha e,
X
n

0

(
n
)

(
M
(0)
)

+
X
n

1
(
^
NM
(0)
)
s
n


2(
M
(0)
)


(77)
a leas o
R
small enough. Then, as
N
will be a xed numb e , he b ound on

(
n
)
is
clea , aking
R
small enough. Mo eo e , wealsoha e
R
(
n
)

R
(0)
exp (
;

0
=
4)
>R
(0)
=
2=

38
No mal Beha iou o Lowe Dimensional To i
M
(0)

M
(
n
)
. To jus i y his las inequali y, we only need o ake
R
small enough such
ha
M
(1)

M
(0)
. Unde his assump ion, he sequence
M
(
n
)
g
n

0
is clea ly dec easing.
Finally, o p o e he well dened cha ac e o he limi Hamil onian, i only emains
o check he con e gence o !

. To do ha we w i e, o simplici y, !
(
n
)
= !
S
(
n
)
1
and
we dene
(
!
(
n
)
= !
(0)

:::

!
(
n
)
, o
n

0. We also pu

0
n
=

(
n
)
;

0
=
8 and
R
0
n
=
R
(
n
)
exp (
;

0
=
8),
n

1. Then, using in induc i e o m he b ounds (73), (71) and (72), i
is no dicul o check ha om Lemma 8 we ha e
k
(
!
(
n
+1)
;
(
!
(
n
)
k
E


0
n
+2
R
0
n
+2


(1 +
^
"(
^
NM
(0)
)
s
2
;
2

)
k
!
(1)

:::

!
(
n
+1)
;
!
(1)

:::

!
(
n
)
k
E


0
n
+2
R
0
n
+2

whe e
^
"only dep ends on
,
m
,
m
1
,

0
and
^
N
. I e a ing his b ound and aking

small
enough, one ob ains o
R
small enough
k
(
!
(
n
+1)
;
(
!
(
n
)
k
E


0
n
+2
R
0
n
+2

n
Y
j
=0
(1 +
^
"(
^
NM
(0)
)
s
j
+1
2
;
2

)(
^
NM
(0)
)
s
n
+2
2

2(
^
NM
(0)
)
s
n
+2
2

whe e we ha e used again he con e gen cha ac e o he sum (76). F om his b ound, i
is clea ha i
p>q

0, hen
k
(
!
(
p
)
;
(
!
(
q
)
k
E


0
=
8
R
(0)
exp (
;
3

0
=
8)

X
j

q
2(
^
NM
(0)
)
s
n
+2
2

bound ha goes oze o as
p q
!
+
1
.This allows o check ha he limi canonical ans-
o ma ion !

go es om
D
+
m
1
m
;
m
1
(

0
=
8
R
(0)
exp (
;
3

0
=
8)) o
D
+
m
1
m
;
m
1
(

(0)
R
(0)
).
4.3.4 Bounds on he measu e
Then, weha e shown he exis ence o eal in a ian educible o i o a se o pa ame e s
!
2E

.I only emains o b ound he measu e o
E

o , equi alen ly, he measu e o he
complemen a y se . To do ha , we s a ecalling how
E

is cons uc ed. I e a ing he
deni ion o
E
(1)
om
E
(0)
,we dene
E
(
n
+1)
om
E
(
n
)
in he same way as i has b een done
in (69), eplacing

(0)

(
M
(0)
)

by

(
n
)

(
M
(
n
)
)

. Then, we ha e
E

=
n

1
E
(
n
)
. This
is,
E

is cons uc ed by aking ou , in ecu si e o m, he se o pa ame e s
!
o which
he Diophan ine condi ions (69), o mula ed on he eigen alues o he p e ious s ep and
dep ending on he size o he emaining p e u ba i e e ms, do no hold. Then, he se o
emo ed pa ame e s can b e ob ained as union o se s o which one o hose condi ions is
no sa ised a some s ep o he i e a i e p o cess.
To es ima e he size o he emo ed se s, we will use a Lipschi z condi ion wi h esp ec
o
!
o he die en eigen alues
^

(
n
)
j
o
B
(
n
)

, o
n

0. To his end, we will p o e
ha his kind o egula i y holds o he successi e ans o med Hamil onians. As his
condi ion holds o he ini ial one, we ha e o check, by induc ion, ha he canonical
ans o ma ions used p ese e his kind o dep endence. The key poin is o b ound he
Lipschi z cons an s o he die en solu ions o (
eq
1
)
;
(
eq
5
). To do i , we ecall ha
we ha e b ounds like he ones o (68) o he Lipschi z cons an s o he die en e ms o
he decomp osi ion (60) o
H
(0)
.Then, we only ha e o p o e ha hose bounds o he
Lipschi z cons an s, can be i e a ed in he same way as he b ounds on he no ms. To
see ha , we can use he die en esul s gi en in i em (
a
)o Lemma 11 o b ound he
A. Jo ba and J. Villanue a
39
Lipschi z cons an s o he solu ions o (
eq
1
)
;
(
eq
5
). We ema k ha , o he denomina o s
ha app ea sol ing hese equa ions, we ha e
L
E
(0)
ik
>
!
+
l
>
^

(0)
gj
k
j
1
+

(0)
1
j
l
j
1
:
Then, combining Lemma 11 wi h s anda d inequali ies o b ound he Lipschi z cons an s o
sums and p o duc s, i is no dicul o check ha one can i e a e b ounds o he ollowing
o m:
L
E
(1)

(1)
R
(1)
^
H
(1)
g 
~
N
(
M
(0)
)
2
s
1

L
E
(1)

(1)
R
(1)
^
B
(1)

;
^
B
(0)

g 
~
N
(
M
(0)
)
s
1

L
E
(1)

(1)
R
(1)
C
(1)
;C
(0)
g 
~
N
(
M
(0)
)
s
1

L
E
(1)

(1)
R
(1)
H
(1)

;
H
(0)

g 
~
N
(
M
(0)
)
s
1

ha a e analogous o he ones o (74) and (75).
~
N

1 dep ends on he same pa ame e s
as
^
N
, plus ~
m
, ~

and

1
. Mo eo e , aking

small enough, we ha e 2
s
1
>
1. He e, he
selec ion o
N
(used o dene

(1)
and
R
(1)
) is done dep ending on he numb e o imes ha
we need o use Cauchy es ima es o bound he die en no ms and Lipschi z cons an s.
I e a ing hose exp essions, i is no dicul o check (by induc ion) ha we can dene
induc i ely ~
m
(
n
)
, ~

(
n
)
and

(
n
)
1
o which he assumed
n
-indep enden bounds hold. The
deduc ion o hose Lipschi z b ounds is edious bu i only in ol es simple inequali ies.
Full de ails in a e y simila con ex can be ound in 10] o 11].
Le us pa icula ize hose b ounds on he eigen alues o
B
(
n
)

.I we expand
^

(
n
)
j
,
j
=1
:::
2(
m
;
m
1
),
n

0, as in (66), eplacing only he sup e sc ip (0)"by (
n
)", we
ha e ha
L
E
(
n
)
(

(
n
)
j
g
(
NR
, b eing
(
N
a p osi i e cons an indep enden om
R
,
j
and
n
.
To jus i y his asse ion, we no e ha i holds o
n
=0, and ha he con ibu ions ha
come om he nex s eps a e exp onen ially small wi h
R
.
Those b ounds on he Lipschi z cons an s o

(
n
)
j
plus he nondegene acy condi ions
(66) a e he key o con ol he measu e o
E
(
n
)
nE
(
n
+1)
. We conside he decomp osi ion
E
(
n
)
nE
(
n
+1)
=

l
2
Z
2(
m
;
m
1
)
0
<
j
l
j
1

2
l
^
X
6
=
l
^
Y

k
2
Z
+
m
1
n
0
g
R
(
n
)
lk

wi h
R
(
n
)
lk
(
R
)=
(
!
2E
(
n
)
(
R
):
j
ik
>
!
+
l
>
^

(
n
)
(
!
)
j
<

(
n
)
(
R
)
j
k
j

1
)
:
To es ima e he measu e o
R
(
n
)
lk
,we ake
!
(1)
and
!
(2)
in his se and hen, we ha e
j
ik
>
(
!
(1)
;
!
(2)
)+
l
>
(
^

(
n
)
(
!
(1)
)
;
^

(
n
)
(
!
(2)
))
j
<
2

(
n
)
j
k
j

1
:
Le us s a wi h he case
j
l
j
1
= 1. Then,
l
>
^

(
n
)
=
^

(
n
)
j
o some
j
= 1
:::
2(
m
;
m
1
).
Hence, he p e ious exp ession can be ew i en as
j
i
(
k
+
j
)
>
(
!
(1)
;
!
(2)
)+
(

(
n
)
j
(
!
(1)
)
;
(

(
n
)
j
(
!
(2)
)
j
<
2

(
n
)
j
k
j

1
:
40
No mal Beha iou o Lowe Dimensional To i
Assuming ha
!
(1)
;
!
(2)
is pa allel o
k
+Re(
j
), we ha e
j
!
(1)
;
!
(2)
j
2
=
j
(
k
+ Re(
j
))
>
(
!
(1)
;
!
(2)
)
j
j
k
+Re(
j
)
j
2

j
(
k
+
j
)
>
(
!
(1)
;
!
(2)
)
j
j
k
+ Re(
j
)
j
2


1
j
k
+ Re(
j
)
j
2

j
(

(
n
)
j
(
!
(1)
)
;
(

(
n
)
j
(
!
(2)
)
j
+
2

(
n
)
j
k
j

1
!


1
j
k
+ Re(
j
)
j
2

(
NR
j
!
(1)
;
!
(2)
j
+
2

(
n
)
j
k
j

1
!
:
b eing
j
:
j
2
he Euclidean no m o a eal ec o . Using ha Re(
j
)
6
= 0 (see (66)), we ob ain
ha he e exis s aposi i e cons an $
1
, indep enden om
j
,
k
and
n
,such ha
j
!
(1)
;
!
(2)
j
2

$
1

(
n
)
j
k
j

1

o
R
small enough. In ac , his bound can be ex ended o he case
j
l
j
1
= 2,
l
x
6
=
l
y
,
using ha Re(
j
1
j
2
)
6
=0 i
j
1
6
=
j
2
.This is abound o he wid h o a sec ion o
R
(
n
)
lk
by
a line in he di ec ion
k
+ Re(
j
). Then, he measu e o
R
(
n
)
lk
can be bounded by
mes(
R
lk
)

$
1

(
n
)
j
k
j

1

p
+
m
1
1
4
(
jC
;
1
j
)
;
1
R
2

+
m
1
;
1

whe e 2
p
+
m
1
1
8
(
jC
;
1
j
)
;
1
R
2
is a bound o he diame e o
E
(0)
(
R
). Then, we ha e
mes(
E
(
n
)
nE
(
n
+1)
)

$
2
R
2(
+
m
1
;
1)

(
n
)
X
k
2
Z
+
m
1
n
0
g
1
j
k
j

1

whe e $
2
do es no dep end on
n
and
R
. Using ha #
k
2
Z
+
m
1
:
j
k
j
1
=
j
g 
2(
+
m
1
)
j
+
m
1
;
1
and ha
>
+
m
1
we ob ain
mes(
E
(
n
)
nE
(
n
+1)
)

$
2
R
2(
+
m
1
;
1)

(
n
)
X
j

1
2(
+
m
1
)
j
+
m
1
;
1
;


$
3
R
2(
+
m
1
;
1)

(
n
)

b eing $
3
also indep enden om
n
and
R
. As

(
n
)
=(
M
(
n
)
)

,we deduce, using (77), ha
o
R

1 small enough,
mes(
E
(0)
nE

)

$
3
R
2(
+
m
1
;
1)
0
@
(
M
(0)
)

+
X
n

1
(
^
NM
(0)
)
s
n

1
A

2$
3
(
M
(0)
)

:
Taking in o accoun he b ound on he measu e o
W

1
8
(
jC
;
1
j
)
;
1
R
2

nE
(0)
(weha e shown,
om (61), ha i is o o de (
M
(0)
)
2

), one ob ains he exp onen ially small b ounds on
he measu e o des oyed o i. To nish he p o o , we dene
A
as

0
<R

R

E

(
R
), whe e
R

is he maximum alue o
R
o which he i e a i escheme con e ges.
5 Basic lemmas
In his sec ion, we gi e some basics esul s used o bound he no ms (12) and (13) and
he ela ed Lipschi z cons an s, as well as he exp essions and ans o ma ions in ol ed
in he die en p o o s. Simila lemmas app ea in 11].
A. Jo ba and J. Villanue a
41
Lemma 4
Le
(

)
and
g
(

)
be analy ic unc ions o
complex a gumen s dened on a
s ip o wid h
 >
0
,
2

-pe iodic on

, and aking alues in
C
. Le us deno e by
k
he
Fou ie coecien s o
,
=
P
k
2
Z
k
exp (
ik
>

)
. Then, we ha e:
(
i
)
j
k
jj
j

exp (
;j
k
j
1

)
.
(
ii
)
j
g
j

j
j

j
g
j

.
(
iii
)
Fo e e y
0
< <
,





@
@
j






;


j
j


exp (1)
 j
=1
::: :
(
i
)
Le
d
k
g
k
2
Z
n
0
g

C
, wi h
j
d
k
j

j
k
j

1
, o some
>
0
and


0
. I we assume ha
=0
, hen, o any
0
< <
, we ha e ha he unc ion
g
dened as
g
(

)=
X
k
2
Z
n
0
g
k
d
k
exp (
ik
>

)

sa ises he bound
j
g
j

;





exp(1)
!

j
j


:
All hese bounds can be ex ended o he case in which
and
g
ake alues in
C
n
1
o
M
n
1
n
2
(
C
)
.
P o o :
I ems
(i)
and
(ii)
a e easily e ied. P o o s o
(iii)
and
(i )
ollows immedia ely
using (23).
Lemma 5
Le
(
 x I  y
)
and
g
(
  x I  y
)
be analy ic unc ions on
D
m
(
 R
)
, and
2

-
pe iodic on

. Then,
(
i
)
I
=
P
(
ls
)
2
N
2
m

N
ls
(

)
z
l
^
I
s
,we ha e
j
ls
j


j
j
R
R
j
l
j
1
+2
j
s
j
1
.
(
ii
)
j
g
j
R
j
j
R
j
g
j
R
.
(
iii
)
Fo e e y
0
< <
and
0
<<
1
,we ha e o
j
=1
:::
and
k
=1
:::
2
m
:





@
@
j






;
R

j
j
R

exp(1)






@
@I
j





R

j
j
R
(1
;

2
)
R
2






@
@z
k





R

j
j
R
(1
;

)
R
:
As in Lemma
4
, al l he bounds hold i
and
g
ake alues in
C
n
1
o
M
n
1
n
2
(
C
)
.
P o o :
The p o o o
(i)
and
(ii)
is s aigh o wa d.
(iii)
is p o ed using i em
(iii)
o
Lemma 4 and applying Cauchy es ima es o he unc ion
P
(
ls
)
2
N
2
m

N
j
ls
j

z
l
^
I
s
.
Lemma 6
Le us conside
(
  x I  y
)
and
g
(
  x I  y
)
complex- alued unc ions, such
ha
and
g
a e analy ic unc ions dened on
D
m
(
 R
)
,
2

-pe iodic on

. Then, o
e e y
0
< <
and
0
<<
1
, we ha e:
j
 g
gj

;
R

j
j
R

exp (1)





@g
@I






;
R
+
j
j
R
R
2
(1
;

2
)





@g
@






;
R
+
2
m
j
j
R
R
(1
;

)





@g
@z






;
R
: