M
P
M
E
P
J
E
M
J
P
M
E
P
M
J
E
P
M
J
E
P
M
J
E
J
P
M
E
J
P
M
E
J
P
M
E
P
M
J
E
P
M
J
E
P
M
J
E
J
P
M
E
P
M
J
E
thema
ISN
J
E
P
Ma
J
E
P
M
J
E
P
M
J
E
P
M
J
E
P
M
J
E
P
M
J
E
P
M
tical
J
Physic
Electr
onic
Journal
1086-5
olume
P
ap
9,
er
203
3
ed:
Editor:
Reciv
V
E.
Ma
W
y
a
23,
203,
Revisd:
Aug
5,
203,
Aceptd:
Aug
15,
203
yne
INV
INV
ARINT
MANIF
ARINT
OLDS
SUBP
A
ASOCI
CES
WITHOU
COMPLENTS:
A
Abstra
ct.
W
of
in
tenc
v
in
arin
t
v
In
use
the
t
tras
to
the
the
subpace.
w
do
W
e
to
also
W
pro
to
More
other
v
pro
e
metho
genral
e
wn
alo
v
ned
w
e
d
oin
ts
h
of
to
pro
maps
v
e
exis-
tange
v
the
arin
in
t
to
v
do
not
is
arin
a
t
as-
sp
ectral
under
the
t.
of
the
dominace
op
erato
resticd
condits.
theorms
o
e
map
complen
ectrum
usal
w
linear
space
t
sp
theorms,
the
the
in
tha
for
theorms
suc
for
uniqes
result
of
tha
an
the
some
v
kno
e
satie
p
space
ha
not
space
e
used
not
do
the
est
ondig
Inde,
linearzto
metho
xed
linearzto.
b
coresp
VE
transfom
near
the
ORM
CH
LA
graph
of
tha
A
LA
manifolds
subpace
con
sume
e
arin
ARINT
TRANSF
O
DE
TO
INV
GRAPH
APR
R.
TED
and
sho
w
ho
w
this
can
b
ha
ws.
v
e
b
en
d.
1
pro
v
ed
in
[CFdlL03a
]
b
y
an-
e
2
R.
1.
If
X
is
a
Banc
morphis
h
ther
existnc
space
of
F
y
in
the
Thes
man
v
v
:
X
e
!
theorms
X
in
arin
t
linearzto
of
clasi
Lla
oductin
and
manifolds
la
Intr
space
are
of
de
the
under
F
theorms
(0)
=
0
is
a
literau
F
at
F
lo
cal
dieo-
establihng
whic
h
are
the
tange
t
to
in
v
arin
t
0.
usaly
asume
tha
ther
is
a
decomp
osi-
tion
(1)
X
=
E
E
1
whic
h
is
in
v
arin
t
under
D
F
(0)
2
(often
the
space
E
are
1
subpace
for
D
F
(0).
(2)
That
D
F
(0)
E
=
E
;
1
and
suc
h
tha
they
;
sp
ectral
2
is:
D
F
(0)
E
=
1
satify
the
E
;
2
dominat
2
condit:
1
(3)
k
D
F
(0)
j
k
k
D
F
(0)
j
k
E
The
conlusi
for
of
a
the
comparisn
can
a
in
diern
t
manifold
W
in
v
1
:
2
clasi
of
nd
<
E
1
in
v
v
arin
t
arin
arin
t
t
manifold
theorms
manifold
(Se
theorms)
under
F
,
[Rue89]
are
tange
t
to
E
tha
at
one
the
orign,
1
so
tha
one
can
think
of
W
as
a
no-liear
anlogue
of
E
.
1
The
goal
and
of
(3)
in
result,
this
the
pa
stable
3.1
(3).
The
is
and
Theorm
and
er
to
w
eak
en
somewhat
pseudo-tabl
manifold
only
neds
Theorm
the
3.1
as
con
h
tains
yp
condits
(2)
theorms.
othesi
w
result
for
The
eak
er
stable
v
main
ersion
and
of
(2)
pseudo-tabl
manifolds.
More
explicty
,
w
e
wil
asume
tha
the
space
E
is
in
v
arin
t
under
1
the
map,
but
under
D
tion
w
F
e
(0).
wil
not
(in
coresp
ted
w
to
(3)
presn
tha
particul,
onds
dominat
asume
sp
ectral
hap
her,
e
decomp
wil
F
wil
or
most
of
to
tha
w
the
e
is
the
wil
on
in
v
arin
t
decomp
not
result
asume
osi-
asume
tha
stable
manifolds
tha
1
k
(1)
asume
Also,
suÆce
(4)
ositn
not
subpace).
ens.
it
the
2
A
k
A
k
<
1
:
1
2
(wher
A
are
1
(9)
for
a
In
v
;
v
arin
t,
men
Some
for
the
c
eign
v
ap
alue
F
linear
to
the
space
a
or
is
a
to
blo
c
E
v
in
,
ev
t
in
w
v
resp
y
v
;
ectiv
ely
,
se
2
v
alues
are:
pro
of
of
duct
a
space
no
cur
trival
b
index
Se
in
sytem,
o
genratd
smalet
withou
ew
t
space.
subpace
sk
complen
alues
the
tary
t
t
if
complen
arin
a
arin
eign
en
eign
arin
v
the
k
genralizd
in
naturl
an
ciated
Jordan
the
no
wher
in
withou
aso
or
{
ther
ear
a
space
k.
situaon
ts
A
denito).
space
as
blo
in
of
precis
t
naturly
the
resticon
1
more
arin
Jordan
the
2
{
v
Example
arin
3.7
is
bifurcatons
t
comple-
y
In
of
sytem
with
us
nilp
signca
tly
As
a
tha
motiv
sytem.
W
e
b
in
ma
y
a
unit
y
the
Henc,
in
is
in
ues
to
their
v
of
arin
t
stable
her
are
wil
t
nite
motiv
for
result
lift
wn
pa
of
the
eign
.
v
In
al-
[dlL97
discuon
]
of
result
on
are
t
slo
w
wn
co
v
existnc
based
wil
of
in
v
er
the
the
strong
The
arin
ts
graph
usal
ones.
kno
adv
the
of
with
pro
the
ofs
pre-
graph
{
to
ys
trans-
space,
the
tha,
foliatn
pro
in
One
w
a
y
using
some
of
do
ofs.
this
theorms
in
since
es
the
the
a
result
not
,
w
el
pro
obtain
v
ed
foliatns
result
e
a
t
pro
of
er
in
metho
d
e
than
pa
exampls.
]
ers
{.
for
with
les
[CFdlL03a
pa
usefl
p
with
er
pseudo-tabl
are
of
those
b
]
The
on
d
man-
particu-
[CFdlL03b
regulaits)
could
transfom
pa
result
metho
t
in
and
the
the
note
arin
manifolds.
fractionl
discuon
t
v
h,
aplictons
for
and
The
thes
ofs,
(excpt
in
whic
t.
of
use
of
space,
complen
simple
the
existnc
arin
parmets
]
se
on
v
Nev
gen-
has
sev-
erthls,
eopl
more
famil-
parmetizon
].
tha
a
v
hnical
[CFdlL03a
ol
h
simplfy
one
in
ha
presn
graph
of
t
Banc
is
out,
tains
her
tec
the
the
d
not
of
space
[HP70],
[CFdlL03a
tages
tha
es
arin
genral
ted
in
an
e
v
with
those
y
h
turns
con
ery
conerd
and
than
in
on
presn
manifolds
hop
v
to
is
genralit
do
in
endc
result
pla
also
[JPdlL95].
requid
]
also
ofs
tha
the
it
]
dep
[CFdlL03c
One
pro
Banc
liftng
in
to
not
metho
of
sytem,
to
manifolds
pseudo-tabl
in
[CFdlL03a
the
The
orking
the
er
are
iar
no-liear
e
presn
el
in
to
ciated
studie
e
of
result
e
w
(As
ed
aso
w
e
space
w
doing
The
eral
v
the
result
[HP70].
observ
ifolds
eral
o
w
imedatly
when
as
lar
ab
rathe
the
our
devic
w
to
case
is
ciated
slow
t
result
dimensoal
ation
as
closet
the
equilbrm
comprehnsiv
presn
and
presn
asuming
her,
alues
the
h.
e
kno
to
the
in
space
to
(In
the
ergnc
caled
the
aplictons.
is
The
based
aproc
W
in
v
t
alue).
aso
so
agin
resona
ergnc
orign
to
v
v
eign
v
note
to
the
eign
more
ated
in
con
the
manifolds
the
a
theorm
ted
form
{
y
ergnc
the
con
er
d.
manifold
sen
v
e
ens
manifolds
to
to
to
of
metho
con
nd
pa
manifolds
transfom
the
motiv
w
hap
the
b
asymptoic
role
this
t
k
tha
closet
meanig
can
c
arin
note
alues
cirle
one
v
also
ondig
unit
and
goal
k,
of
]
in
e
has
(3),
blo
dominate
v
c
e
tha
asumption
ciate
W
is
giv
the
[CFdlL03c
The
aso
eign
study
aplicton
sytem.
the
tracion
to
manifolds
to
blo
the
closet
and
enig
coresp
tersing
an
t
alues.
the
space
{
Jordan
t
Jordan
t
3
resona
eak
v
con
no-trival
in
an
not
manifolds
no-trival
eign
linear
t
and
of
w
a
w
the
cirle,
tange
it
to
arin
part
ws
of
genratd
a
o
situaon
orign
t
the
ation
the
the
oten
{
v
role
w
in
e
use
[CFdlL03a
wil
is
\c
].
onic
As
al
w
norms"
e
wil
or
se
this
w
eigh
is
ted
the
k
norms
ey
whic
tec
hnical
h
4
R.
to
ol
tha
alo
alterniv
ws
es
W
w
e
e
p
ostp
in
w
eak
en
(3)
reason
the
tro
in
w
the
one
can
fol
to
and
Lla
of
t
consider
for
impro
v
main
e
a
(3)
result
The
ofs
discuon
of
with
til
pro
them.
Section
of
3
the
so
tha
staemn
ts
wil
section.
Not
a
tion
k
e
4.5
can
the
2.
W
e
Remark
one
notai.
subeqn
v
Se
y
t
some
la
.
wh
staemn
duce
de
the
Banc
h
space
+
C
,
k
2
N
,
2
[0
;
Lip
]
endo
w
ed
1
with
the
usal
uniform
norms
as
w
el
as
the
space
C
with
its
usal
!
F
rec
het
top
taking
olgy
and
sup
of
a
C
endo
complex
w
ed
with
a
extnsio.
Banc
h
When
the
top
olgy
size
of
based
the
on
extnsio
is
!
imp
ortan
t,
w
e
wil
denot
it
as
a
subindex,
for
exampl
in
C
.
That
is,
w
e
denot,
when
k
2
N
,
i
k
k
k
=
C
(
U
;X
max
(sup
j
i
=0
;:
r
=
k
+
,
k
2
N
,
j
)
:
;k
x
When
D
)
2
(0
;
Lip
2
U
]
k
r
k
k
=
C
(
U
;X
max
(
k
k
k
;
H
is
the
(
D
)
C
i
wher
H
)
=0
;:
(
U
;X
)
;k
seminor
(5)
H
()
=
sup
j
(
x
)
(
y
)
j
j
x
y
j
:
x;
W
ha
e
v
note
tha,
when
x
~
2
U
;x
=
6=
x
0,
~
for
functios
normalized
to
(0)
=
0
w
e
e
0
k
0
k
H
()
2
k
k
0
C
W
e
les,
adopt
k
sym
b
the
+
Lip
ol
Of
<
Lip
en
ican
tly
k
tak
the
ters.
+
1.
easir
tha
the
v
for
r
the
y
2.
y
ho
ose
to
tha
for
an
y
2
[0
suc
2
index
;
s
r
2
rst
N
ab
b
and
a
=
as
the
result
in
>
ignore
the
Henc,
r
regulait
exprsion
c
note
case
>
h
;
1),
nev
as
erth-
(5),7
,
the
1.
ma
e
Lip
arithmec
alue
read
in
a
tion
In
w
trae
en
en
the
Inde,
cone
Giv
v
es
course,
Lip
con
C
ordelin
case
their
reading
pro
it
wher
ofs
could
are
b
signf-
e
w
orth
to
.
o
v
e,
w
e
wil
denot
(
s
(6)
s
~
;
s
s
(In
the
resp
notai
ectiv
ab
ely
o
and
and
w
v
=
N
e,
w
e
1
note
tha
+
Lip
;
when
s
s
2
N
=
1
;
!
,
then
s
~
=
1
;
!
.
usefl
wn,
mains
kno
Tw
o
2
=
e
elmn
the
omit
existnc
a
tary
inequalts
precis
of
form
pro
are
ducts)
ulation
includg
(since
they
asumption
are
so
w
on
el
do-
In
v
arin
t
manifolds
5
0
H
(
)
H
()
k
0
k
+
k
k
H
()
C
C
(7)
H
(
Æ
)
H
()Lip
W
e
recal
caly
tha
equal
a
to
1
()
:
cuto
in
functio
the
is
bal
of
a
radius
functio
1
and
suc
zero
h
tha
it
outside
a
is
bal
iden
ti-
of
radius
2.
In
is
nite
ob
dimensoal
space,
vious.
whic
Simlary
h
Nev
are
,
smo
oth
erthls,
the
Hilb
ert
{
out
are
of
space
functios
ther
existnc
of
inte
or
the
suc
h
Banc
h
orign
{
dimensoal
a
cut-o
functio
space
do
whose
ha
v
Banc
h
e
norms
cut-o
functios.
space
for
whic
h
no
0
suc
h
in
smo
terv
oth
al
cut-o
do
exist.
es
not
ha
F
v
e
smo
The
main
result
3.1
case.
The
h
diern
t
yp
in
roughly
the
wing
ab
the
b
y
3.1
regulait
but
y
in
of
closed
]).
and
in
ratios
a
[DGZ93
stable
manifolds
map,
limted
of
Theorm
out
on
the
space
ts
fol
conlusi
as
also
is
resul
result
since
regula
is
er
orates
case
C
(se
of
pa
usal
functios
temn
and
oth
as
y
this
the
cut-o
a
incorp
othesi
b
{
regulait
St
of
Theorm
exampl,
oth
3.
tha
or
etc.
the
the
W
e
note
pseudotabl
are
somewhat
stable
case
are
pseudo-tabl
{
case
the
norms.
r
Theorm
3.1
r
2
D
F
N
(0)
+
[0
L
=
;
A
Lip
et
]
X
[
b
f1
;
e
a
!
g
Banch
,
sp
r
1
b
ac
e
e.
L
such
et
F
:
tha
F
X
(0)
!
X
=
0
b
.
e
tha
A)
Ther
e
exist
a
de
c
omp
ositn
X
=
X
X
into
1
such
tha
the
sp
ac
e
X
is
invart
close
d
subp
ac
2
under
A.
That
is,
1
(8)
A
(
X
)
X
:
1
Note
tha
we
do
not
asume
1
tha
X
is
invart
under
A
.
2
Denot
by
,
the
1
pr
oje
ctions
X
,
X
.
1
A
=
A
1
Asume
over
2
(9)
furthemo
;
1
e
tha
A
=
1
we
e
A
:
2
in
one
also
2
2
ar
Denot
of
2
the
fol
lowing
two
c
case
B.1
k
A
k
<
1
:
1
B.12
L
et
s
Asume
=
min
tha
(2
;
r
we
)
,
have
the
fol
lowing
1
(10)
,
Denot
.
Asume
Stable
C
k
A
s
k
A
k
1
2
we
<
1
:
ak
dominac
e
c
ondit
ase:
es
6
R.
In
p
articul,
if
r
de
2
la
Lla
we
v
just
asume
1
(1)
k
e
2
A
k
k
A
k
<
1
:
1
2
Pseudo-tabl
case
B.21
1
k
A
k
<
1
:
2
B.21
The
B.2
F
sp
ac
e
X
admits
smo
oth
cut-o
functios.
1
or
some
r
e
al
numb
er
1
s
,
we
have
1
(12)
k
s
A
k
A
k
<
1
:
1
2
Then,
ther
e
exist
U
a
neighb
orh
o
d
of
0
and
a
map
:
U
X
!
X
1
such
2
tha
i)
i)
is
Lipschtz
(0)
in
=
i)
The
0
gr
the
gr
iv.1)
In
,
D
aph
aph
and
(0)
=
of
of
the
U
0
is
lo
is
stable
c
dier
entiabl
iv.2)
e
In
r
~
is
the
al
ly
we
invart
under
den
d
in
2
(6)
Remark
~
3.2
is
(10)
and
w
its
e
can
ase
radi
W
e
norms
in
up
space,
In
a
yp
the
in
this
~
s
r
the
b
for
whic
norms
ounds
tha
sp
c
the
are
the
the
By
ers,
(12)
of
on
time
v
of
holds.
op
eratos
ectral
ho
b
an
of
close
in
radius
osing
norm
as
[CFdlL03a
]
eratos.
Of
thes
y
adpte
the
as
in
op
erato,
desir
to
e
discuon
wil
in
of
indep
just
of
nite
are
terms
are
w
a
norms
ulation
er
for
course,
adpte
asumption
pa
ase,
the
eratos.
form
the
c
sp
the
dimensoal
elmn
tary
ectral
.
pro
end
use
adpte
t
norm
of
ertis
the
is
form
norm
ulation
used.
of
the
othesi.
Since
w
staemn
e
ts
Banc
ys,
s
norms.
op
since
erthls,
h
Nev
w
~
any
used
in
triangul
trinsc
for
same
of
y
in
e
endix
construi
man
more
ap
er
the
v
the
op
the
r
on
the
and
thes
to
)
adpte
at
of
ref
stable
C
condits
ha
ks
the
have:
2
(6)
using
c
(In
C
we
in
could
arnge
blo
ectral
e
norms
diagonl
sp
d
of
w
the
norm,
den
Instead
(12),
rednig
.
.
s
F
have:
c
e
orign.
invart.
ase
pseudo-tabl
(wher
the
.
c
wher
at
h
space.
wil
w
not
ork
use
just
a
as
sp
w
el
ectral
form
for
real
ulation
Banc
of
h
space
the
h
and
yp
othesi,
for
the
complex
In
Remark
3.
nes
The
staemn
4.18
W
are
v
the
e
an
ery
y
e
the
in
w
in
v
e
b
eha
vior
This
leads
ther
yp
othesi
of
h
yp
othesi
of
er
also
pro
in
the
with
some
in
the
and
uniqes
asumption
orh
o
uniqes
e
case.
the
b
giv
Remark
pseudo-tabl
under
neigh
uniqe-
4.1
tha
and
a
vide
Remark
condits
case
b
of
d
y
In
of
the
imp
regula-
orign.
In
osing
the
condits
on
.
apren
tly
wher
the
wil
them
tha
obtain
y
case
h
ev
e
some
some
of
in
int
to
are
7
uniqes
w
at
manifolds
e
manifold
case,
the
pro
stable
obtain
t
pseudo-tabl
w
the
arin
t
describ
ho
t
arin
of
wil
ticpae,
case,
of
d
W
diern
stable
it
metho
ts.
v
pardo
one
stable
xical
space
part
in
of
situaon.
v
F
arin
t
Theorm
3.1
for
D
for
F
or
exampl,
(0)
the
satie
the
map
F
and
the
1
it
could
the
hap
pseudo-tabl
en
obtained
part
{
inde
aplying
F
urthemo,
w
3.1,
w
e
it
the
only
t
e
c
hoice
c
of
the
manifold
it
claimed
from
en
in
the
w
Section
a
en
the
e
the
manifolds
{
in
p
4.1
obtain
hap
on
erthls,
result
when
e
could
ends
tha
pseudotabl
w
Nev
t.
in
if
.
{
diern
ed
Ev
F
genricaly
the
describ
dep
map
result
hoices.
duce
manifolds
far
for
preation,
pro
the
are
uniqes
{
arbity
ens
result
tha
the
preations
include
hap
o
note
obtain
limnary
w
for
some
for
the
uniqe
se
map
whic
5.4
{
Henc,
of
h
for
Example
case
pre-
manifold
preation.
pseudo-tabl
Theorm
erfom
h
tha
the
Theorm
eac
the
in
v
3.1
arin
t
are
v
ery
uniqe.
Remark
3.4
the
W
result
nes
for
e
result
thes
are
rathe
p
ab
ery
in
v
e,
it
in
write
v
simlar
to
of
is
not
arin
o
in
the
just
the
W
ha
v
obtains
e
dev
elop
also
for
wil
uniqe-
one
e
Note
e
aplying
the
tha
W
result
her.
y
en
w
w.
6.1
of
b
Giv
sho
o
Section
of
used
ws
w.
to
for
pro
tha
o
the
hard
t
direct
for
map
ts
a
Section
Remark
o
t
argumen
to
v
osible
result
time
inde
standr
osible
straegy
some
the
to
tha
p
obtain
to
alude
manifolds
quite
wil
maps
ed
tha
o
it
ws
is
indcate
using
ho
an
w
this
is
6.2
3.5
asumption
Note
on
tha
what
in
is
k
the
A
pseudo-tabl
k
case,
other
than
(12)
w
.
This
e
do
alo
not
ws
mak
to
e
an
y
consider
1
maps
wher
(13)
k
A
k
1
:
1
Of
course,
due
to
(10)
,
w
e
ned
tha
the
p
osible
expansio
of
A
1
1
is
dominate
b
y
the
con
tracion
of
A
requing
2
(notice
tha
s
1).
tha
in
(12)
w
e
are
8
R.
Ev
en
if
stable
it
is
not
stricly
manifolds
the
case
wher
can
(13)
in
the
[HPS7]
b
t
no
v
plemn
elt
y
t,
esn
Remark
pro
impro
v
to
r
e
the
standr
One
p
tion
en
study
of
genralit
pseudo-
y
eÆcien
tly
establihd
b
b
ds
in
y
(13)
since
the
to
metho
ds
her
graph
in
v
the
].
of
in
trans-
W95
tha
of
t
the
[dlL
asumption
presn
y
[Irw80],
simlar
the
e
Inde
[HPS7
],
arince
of
the
the
com-
pseudo-tabl
case
is
of
r
~
and
in
y
s
v
~
in
arin
the
t
conlusi
theory
on
separt
do
manifold
conlusi
a
the
tric
the
[HPS7].
use
requi
in
functor
w
in
regulait
b
quite
of
in
to
e
is
al
of
as
comn
sem
v
er
v
The
quite
in
of
more
metho
pa
same
3.6
is
,
los
study
other
remo
the
the
y
this
the
but
tialy
This
b
in
is
e
ha
and
treamn
only
e
case.
manifolds
in
v
withou
can
stable
Pseudo-tabl
form
Lla
necsary
asume
fails
the
la
logicay
one
discue
de
the
literau.
In
Suc
h
one
b
to
r
1
argumen
ts
often
elong.
erthls,
from
t.
particul,
not
Nev
manifold
argumen
es
.
use
+
Lip
are
v
the
ery
tange
t
k.
osiblt
y
satied
b
for
y
suc
the
h
an
rst
argumen
t
deriv
ativ
e
is
{
to
it
deriv
is
e
a
a
functioal
linear
equa-
equation
on
the
r
ativ
W
deriv
e
e.
ref
Then
to
argumen
h
this
simple
do
fol
also
the
for
impro
v
e
{
of
of
ts
sharp
t
this
a
y
equation
are
simlar
o
situaon.
cur
in
].
W
e
3.1
the
xe
d
and
Consider
aplies
result
such
X
tha
the
=
line
in
and
the
[CFdlL03a
]
do
clasi
aply
the-
{
is
2
R
and
a
arizton
at
C
map
the
F
orign
2
1
C
1
=
2
1
B
C
B
C
1
=
2
=
:
B
C
@
A
2
=
5
1
2
Note
tha
1
=
2
>
2
=
5
le
1
=
B
A
,
is:
0
1
>
1
=
3
>
wil
5
3.7
.
Other
[ElB01
wing.
Example
1
C
her.
Theorm
course
in
regulait
emn
wher
not
solutin
argumen
giv
exampl
orems
tha
]
whic
purse
A
w
[CFdlL03a
ts
not
sho
(1
=
2)
.
=
3
aving
the
orign
the
In
Ther
efor
(as
e,
wel
l
The
as
or
em
others
b
3.1
ase
v
arin
t
manifolds
aplies
d
9
to
on
the
eithr
same
a
=
f
(
x;
0
;
0
;
0
;
0)
j
x
2
R
g
X
=
1
f
b
f
(
x;
y
;
0
;
0
;
0)
j
x;
y
2
R
g
X
=
f
(
x;
0
;
0
;
t;
0)
j
x;
t
2
R
g
X
=
f
(
x;
0
es
not
;
to
0
;
0
;
take
the
ondig
u
)
=
3
j
x;
2
=
5
.
suc
u
h
t
2
R
to
w
et
as
g
(0
=
;
t;
0
;
u
z
;
y
;
;
z
)
j
t;
;
f
(0
;
y
;
y
;
u
0
;
z
)
u
;
z
j
)
t;
;
z
j
t;
;
y
u
t;
;
0)
j
2
u
u
y
R
2
g
R
2
;
Remark
3.8
g
R
g
z
;
t
2
R
g
:
to
note
thels,
the
ar
if
and
the
ear
c
or
e-
naturly
time
one
Jordan
map
blo
resonac
man
adjoin
e
eignvalus
ap
trival
at
c
k.
Un-
sem
y
to
empircaly
kno
v
are
b
wn,
e
but
tical
],
than
a
particul,
w
3.7
er
sytem
e
with
do
not
go
blo
[ElB01
],
gap
ha
=
3
v
e
h
ks.
[CFdlL03a
]
w
e
aplied
replacd
ha
v
the
b
throug
whic
c
condit
could
1
{
Jordan
[dlL97
rathe
Example
master-lv
iden
of
h
pa
literau
has
result
whic
this
e
the
linearzto
the
to
of
sla
the
In
ers
result
no
in
the
tha
in
pa
a
2
ac
The
caled
and
condits.
those
e
=
ck.
3.7
ertis
are
situaon
no-resac
e
blo
Example
of
{
e
v
1
eignsp
phenoma.
elctronis
W
also
of
ha
master
in
of
the
resonac.
explantios
the
ens
ck
or
ap
pro
vidng
3
dan
with
often
blo
=
osiblte
in
sytem
geomtric
pro
When
p
1
Jor
couplings
hap
aply
e
consider
the
ard
to
no-trival
wil
el
the
ondig
mor
a
of
triangul
sytem.
sult
;
under
esp
anlyzed
er
often
(0
f
X
invart
or
those
rigously
Up
e
c
ourse,
sytem
w
y
c
maps
resona
step
ac
e
had
one
derstanig
not
Of
have
time
an
sp
ac
would
System
the
a
eignsp
to
1
a
z
2
suÆc
of
f
=
1
as
;
d
X
,
y
2
d
5
;
=
1
=
(0
c
X
2
splitng
2
c
sp
lowing
b
=
1
or
fol
2
X
It
the
as)
a
X
it
of
ide
y
with
1
suc
=
7.
h
re-
Nev
a
c
e
er-
hange.
Remark
3.9
Note
tha
the
equations
(10)
and
(12)
ha
v
e
the
same
form.
Roughly
sp
min
um
satify
of
the
In
the
the
to
case,
satify
the
the
y
only
e
stronge
w
e
to
certain
giv
the
n
is
of
tha
y
b
of
y
um
b
ers
the
s
tha
is
large
a
to
orde
e
s
,
the
eas-
y
enough
gen
uine
the
.
of
the
Henc,
whic
y
pro
In
.
limtaon
the
s
s
regulait
tha
on
tak
regulait
map.
for
(12)
e
the
the
false
due
in
w
for
is
en
limtaons
manifold
set
large
limtaon
tha,
the
the
limtaons
regulait
y
is
asume
and
tha
(12)
the
considerat
ned
e
the
regulait
for
map
the
for
than
Another
v
case,
for
b
ha
obtained
(12).
Henc
pseudo-tabl
limtaon
ma
the
or
e
.
come
In
of
(10)
w
(10)
conlusi
y
y
in
stable
is
regulait
regulait
condit
the
ier
eaking
the
of
stable
go
of
the
h
es
map.
case,
throug,
the
10
R.
pro
of
w
usal
e
presn
t
go
es
dominat
is
to
the
la
throug
s
fact
Lla
v
with
condit
relatd
de
tha
s
=
1.
the
e
=
2.
The
This
is
reason
wh
noliear
part
w
y
of
eak
w
er
e
the
than
can
the
tak
map
v
e
s
=
2
anishe
to
i
orde
s
This
.
If
w
e
observ
had
D
ation
resonac
N
is
(0)
e
N
wil
b
e
in
a
discu
In
e
ned
this
to
section
the
presn
t
rst
the
e
pro
pseudo-tabl
tion
and
ha
anlysi
v
e
wil
4.1
e
e
.
under
s
extra
k
no-
to
higer
h
orde.
W
e
v
anishg
of
s
N
do
es
not
help
1.
3.1
the
pro
stable
of
of
case
are
Theorm
and
based
3.1
then,
in
the
the
w
same
ork,
W
pro
of
e
for
the
functioal
nev
wil
equa-
erthls,
the
nal
t.
this
section,
problem
w
tha
simplfy
e
can
cary
b
subeqn
t
out
e
p
some
prelimnay
erfomd
withou
los
of
anlysi.
write
D
F
F
Cleary
,
N
With
(0)
=
resp
0,
ect
D
to
N
(0)
(
(0)
the
=
x
)
=
A
=
Ax
+
N
(
x
)
:
0.
decomp
ositn
X
=
X
X
,
1
w
e
can
write
2
A
B
1
(14)
A
=
:
0
A
2
W
e
wil
asume
withou
los
of
genralit
y
tha
k
x
k
=
max
k
x
X
k
;
1
k
x
X
k
2
X
1
2
F
urthemo,
b
y
c
hangi
k
to
k
=
k
with
X
suÆcien
tly
X
2
2
X
2
large,
w
e
can
asume
tha
(15)
k
B
k
X
!
"
n
um
As
b
is
er
a
of
consequ
arbitly
smal.
condits
of
1
Later
tha
"
in
(15)
has
w
the
to
e
"
X
2
wher
pro
ha
satify
v
of
w
e
.
e:
k
A
.
whic
anished
preatoy
the
They
e
transfomi
Theorm
ofs
diern
of
y
W
the
In
genralit
h,
tak
detail.
of
t
comn
rathe
Prelimnas.
preations
v
to
whic
e
condit
pro
some
b
suÆce
]
in
of
for
Both
ould
mak
orde
presn
of
case.
w
it
ts
of
wil
it
[dlL97
clasi
Pr
,
tha
emn
the
use
w
k
can
so
v
case,
just
one
form
impro
4.
In
tha
normal
pseudo-tabl
w
i
in
ws
thes
the
and
for
exploitd
sho
to
not
0
furthe
condits
mak
=
k
max
k
A
k
1
;
k
A
k
2
+
"
wil
imp
ose
a
nite
In
As
standr
in
in
v
arin
v
t
arin
t
manifolds
1
manifold
theory
,
w
e
observ
e
tha
if
w
e
in-
1
tro
duce
a
scaling
F
(
x
)
=
F
(
x
),
w
e
ha
v
e
tha
D
F
(0)
henc
noe
the
of
same
the
pro
can
D
F
(0),
previous
time
=
ertis
arnge
b
y
of
taking
the
linear
big
map
are
enough
alterd.
A
t
tha
r
(16)
k
N
k
C
(
B
"
:
)
1
Henc,
w
and
e
wil
asume,
withou
los
of
genralit
y
tha
w
e
ha
v
e
(15)
(16).
4.1
Pr
case,
ep
w
ar
e
ations
for
ned
a
the
pseudo-tabl
furthe
c
reduction
tha
ase.
In
alo
ws
the
us
pseudo-tabl
to
asume
tha
N
r
is
C
smal
in
X
B
(
1
the
orign
in
X
X
)
1
.
Giv
wher
B
(
2
en
a
X
)
1
cuto
is
the
unit
on
X
,
2
the
bal
cen
terd
at
2
functio
it
suÆces
to
consider
1
maping
~
F
(
x
)
=
Ax
+
(
x
)
N
(
x
)
1
r
Since
w
e
radius
ha
2,
v
e
arnged
the
b
Leibnz
y
scaling
form
tha
ula
for
N
the
is
deriv
C
smal
ativ
es
in
of
pro
the
bal
ducts
of
and
the
~
form
ula
the
for
bal
the
of
pro
ducts
radius
of
H
older
functios
sho
w
tha
F
is
smal
in
2.
~
Note
tha
the
map
F
agres
with
the
map
F
in
a
neigh
b
orh
o
d
of
~
the
orign.
in
v
Henc
arin
t
As
w
for
e
stable
a
F
wil
whic
h
is
in
v
arin
t
for
F
,
wil
b
e
lo
caly
.
se
case,
manifold
later,
wil
the
b
e
uniqes
result
uniqes
establihd
result
for
for
the
manifolds
the
in
v
pseudo-
arin
t
under
~
F
and
whic
h
satify
some
condits
on
the
b
eha
vior
at
1
.
Since
the
~
construi
of
,
it
is
F
quite
p
out
of
osible
F
in
tha
v
olv
es
the
diern
t
c
c
hoice
of
hoices
of
the
cuto
wil
functio
lead
to
diern
t
in
~
v
arin
t
manifolds
for
manifolds
for
4.2
A
F
functioal
v
x
t
equation
standr
If
diern
=
ariton
(
;
and,
for
of
y
F
henc,
diern
t
lo
caly
in
v
arin
t
.
(
y
)
the
is
the
graph
a
p
in
v
arince.
W
transfom
oin
t
in
metho
the
graph
e
fol
w
a
rathe
d.
of
w
e
ha
v
e
F
(
x
)
=
A
y
+
B
(
y
)
+
N
(
1
wher
N
,
N
are
1
The
;
shortand
for
t
y
)
;
N
,
A
(
F
question
(
x
of
)
is
y
)
+
N
(
also
;
(
y
)
resp
ectiv
ely
.
2
in
domains
y
2
N
1
tha
moen
(
2
2
condit
the
y
1
the
graph
on
of
wher
the
reads
comp
{
ignor
for
ositn
is
den{
(17)
A
y
+
B
(
y
)
+
N
(
1
The
ing
equation
(17)
question
y
;
(
y
)
=
A
(
1
of
is,
furthemo
formaly
domains
of
y
)
+
N
(
2
equiv
denitos
of
the
y
;
(
y
)
:
2
alen
t
functios
{
agin
{
ignor-
to
1
(18)
(
y
)
=
A
(
A
y
2
1
+
B
(
y
)
+
N
(
1
y
;
(
y
)
)
N
(
2
y
;
(
y
)
:
12
R.
Remark
4.1
tainly
The
not
equations
h
argued
in
notis
v
e
v
slo
w
la
equiv
se
v
diern
t
b
e
et
dep
and
ha
of
lead
solutin
y
one
v
e
the
to
tha
(17)
are
diern
ob
t
y
diern
t
jects.
Se
formal
the
cer-
diern
solutin.
wh
t
same
on
t
reason
diern
the
endig
(18)
uniqes,
ma
is
manifolds
en
discu
t,
this
exmplify
w
e
alen
]
t
w
e
w
equiv
arin
v
when
[JPdlL95
in
Lla
alenc
wil
formaly
wher
ery
e
W95],
5.4
ha
w
are
[dlL
of
Example
of
As
whic
As
isue
trival.
de
also
equation
precis
ma
requimn
y
ts
on
domains.
W
e
W
e
wil
study
(18)
denot
b
R.HS
of
(18)
as
y
T
.
a
the
xed
op
That
p
oin
erato
t
problem.
whic
h
to
a
functio
aso
ciates
the
is
1
(19)
T
[](
y
)
=
A
(
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
N
(
1
y
;
(
y
)
2
2
This
op
erato
is
transfom,
but
pro
duce
{
it
for
and
T
sligh
The
of
w
on
xed
p
(tha
a
oin
tha
The
the
in
e
v
ofs
t
T
and
iden
wil
on
whic
sho
1
w
a
ts
w
e
wil
erato
consider
in
man
con
y
the
t
W
ation,
duce
is
also
range
the
a
matc
e
stable
the
h
d
the
is
and
pro
of
h
wil
case
clasi
metho
the
pro
el
the
deriv
e
in
w
whic
for
w
tha
and
fol
the
of
is
and
formal
w
diern
tracion.
t
).
functios
pro
T
oin
domains
them
wing
of
erato
p
(18)
(17)
e
op
xed
ers
the
b
of
the
rev
of
wil
is
oin
graph
op
the
a
tha
Both
later
p
the
transfom
is
of
fol
T
and
xed
tha
has
to
ation
space
h
case
wing
it
us
deriv
some
to
graph
tha
solutin
result
tify
ciated
the
the
erato
sho
ws
theorms
e
stable
of
thes
manifold
W
y
e
{
of
op
the
case.
arin
the
alo
the
pseudo-tabl
form.
b
it
ers
of
the
in
tha
w
rev
it
ts
tha
w
aso
[LI83].
consit
tha
since
can
pro
in
h
oin
functios,
sho
(17)
w
is
t
wil
of
so
presn
suc
e
p
of
of
is
w
solutin
e
erato
to
xed
tha
space
t
is,
also
e.g
op
relatd
e
This
ofs,
pro
the
closey
b
simpler.
pro
den
exactly
is
wil
tly
clasi
not
graph
map
presn
t
rst
of
trans-
ed
pseudo-tabl
ofs
in
the
to
itself
pro
case.
of
The
in
case
!
C
and
C
Section
regulait
y
for
the
stable
case
wil
b
e
done
separtly
in
4.5
4.3
F
stable
orm
ulas
and
for
in
Lema
the
deriv
ativ
es.
pseudo-tabl
A
result
case
tha
is
w
the
fol
e
wil
use
wing
b
oth
purely
in
the
formal
4.2
4.2
in
Lema
(19)
.
Asume
tha
for
an
op
en
set
of
y
we
c
an
den
T
[]
as
In
v
arin
t
manifolds
i
If
is
i
C
,
i
r
,
then
i
T
1
D
T
13
[]
=
[]
is
C
and,
mor
e
over,
we
have
i
A
D
(
A
+
B
+
N
(
1
;
)
1
2
(
A
+
B
D
+
D
N
1
(
1
;
)
+
D
N
1
(
;
)
D
i
)
2
1
i
(20)
+
A
D
(
A
+
B
+
N
(
1
;
)
D
N
1
(
2
;
)
D
1
2
1
+
i
A
D
N
(
2
;
)
D
2
2
i
+
R
(
D
;
:
:
:
;
D
1
)
i
wher
e
R
is
a
p
olynmia
in
the
derivats
of
whose
c
o
eÆcients
ar
e
i
p
olynmia
expr
evalut
d
Pr
o
of.
esion
at
The
but
is
involg
form
ula
signca
(20)
tly
The
the
derivats
of
F
{
up
to
or
der
{
main
p
can
b
e
obtained
from
F
a
Di
Bruno
form
ula,
easir.
oin
t
of
(20)
is
tha
w
e
i
can
iden
tify
the
only
term
in
i
D
T
[]
whic
The
h
form
con
tains
ula
as
(20)
is
a
factor
easily
D
.
establihd
b
y
inducto
staring
form
the
i
ob
vious
b
i
case
y
i
taking
=
one
W
e
1.
Asuming
more
note
tha
deriv
tha
ativ
taking
(20)
e
the
on
b
deriv
is
oth
true,
w
e
compute
+1
D
T
[]
side.
ativ
e
of
R
w
e
do
not
obtain
deriv
ativ
es
i
of
of
orde
higer
compute
the
than
deriv
i
ativ
e
of
.
T
o
the
establih
tha
terms
b
y
R
using
is
the
a
pro
j
w
e
tak
e
a
deriv
ativ
e
of
a
factor
j
D
,
w
e
obtain
e
the
are
deriv
of
T
the
o
1
of
desir
F
Æ
claims
observ
when
j
D
w
e
obtain
w
e
w
e
When
+1
and
when
+1
D
F
Æ
D
.
Both
factors
form.
the
to
+
e
establih
suÆces
i
ativ
olynmia
rule.
D
j
tak
p
duct
e
w
e
ab
tha
tak
e
out
the
deriv
the
only
terms
ativ
w
es
of
a
y
with
tha
the
w
higer
e
deriv
get
deriv
exprsion
is
ativ
ativ
tha,
es,
es
of
when
it
orde
w
e
aply
i
the
pro
deriv
duct
rule,
ativ
in
es
deriv
on
of
es
4.
of
of
The
3.1
sp
to
second
on
the
obtain
factor
D
terms
i
+
1
whic
so
tha
h
.
can
If
are
p
w
e
tak
e
olynmias
consider
them
as
when
es
of
r
orde
stable
<
1
case.
.
v
elt
In
w
y
e
wil
with
tha
The
W
case
r
e
no
=
w
1
star
;
!
the
wil
b
e
ect
to
in
w
the
of
literau
w
les
section,
most
the
functios
tly
this
e
in
tro
duce
some
later.
in
the
(sligh
use
resp
theorms
fact
the
functios.
tha
manifold
the
in
4.5
ac
no
t
of
e
tha
3.1
norms
arin
tage
er
Theorm
Section
main
v
w
es
til
and
in
w
lo
Theorm
Some
space
ativ
factors,
orde
of
oned
4.1
deriv
+1
of
ostp
the
.
Pro
pro
e
other
of
R
i
p
the
ativ
part
tak
e
les
the
standr
is
are
seking
regula
pro
tha
case).
v
w
e
anish
tak
Henc
at
ofs
e
of
the
adv
an-
w
orign
e
can
14
R.
use
w
eigh
ted
stronge
norms
con
4.3)
w
siton
eak
the
e
or
pro
This
en
W
(2)
tracion
de
impro
v
Lla
norm
for
ed
v
e
based
ertis
dominace
la
con
on
the
op
deriv
ativ
erato
tracion
es
whic
h
consider
pro
ert
lead
to
(se
y
is
what
Prop
alo
o-
ws
us
to
condit.
consider
r
=
k
+
k
2
N
,
2
[0
;
Lip
]
and
den
the
space
r
=
Æ
;
;Æ
;
:
B
Æ
X
!
1
X
;
1
2
C
;
2
0
k
i
0
k
D
k
Æ
;
i
=
0
;
:
:
:
;
k
;
i
C
(
B
)
1
k
H
(
D
)
Æ
;
(21)
(0)
=
0
;
D
(0)
=
0
;
s
sup
y
2
j
B
f
0
D
(
y
)
j
y
+1
j
<
1
g
1
wher
s
is
Henc,
the
same
when
deriv
r
ativ
as
e
in
2,
(21)
tha
en
the
exp
is
1.
tering
in
one
In
t
case
of
r
j
(10)
=
1
y
,
j
in
+
namely
,
the
,
last
the
s
=
min
(2
condit
exp
;
r
for
one
t
s
+
).
the
1
is
just
.
When
=
0,
the
parmet
in
the
denito
of
0
role
since
Æ
and
0
Æ
w
ould
con
trol
the
do
es
not
pla
y
an
y
k
C
norm
of
D
.
Henc,
w
e
wil
k
0
just
supre
use
Æ
the
Henc,
notai
when
dealing
with
in
tegr
regulait
y
,
w
e
wil
.
Æ
;:
;Æ
0
k
W
e
wil
asume
tha
Æ
1
so
as
to
mak
e
sure
tha
N
(
y
;
(
y
)
is
0
alw
a
ys
w
el
den.
W
e
wil
endo
w
with
Æ
;:
;Æ
;
the
top
olgy
induce
b
y
Æ
0
k
s
(2)
j
j
j
j
j
j
=
sup
y
2
j
B
f
0
D
(
y
)
j
=
j
y
j
g
1
wher,
w
2
(0
;
It
recal
Lip
is
whic
e
]
not
h
s
and
s
hard
0
c
the
tha
2
=
to
satify
Note
=
whenv
er
when
r
hec
k
=
(2)
is
a
(0)
top
olgy
2,
s
=
when
r
=
1
+
,
1.
tha
normalizt
the
r
norm
=
induce
b
in
the
space
of
functios
0.
y
(2)
is
ner
than
the
top
olgy
0
induce
b
An
y
the
imp
C
ortan
norm.
t
result
[LI73
]
Lema
2-5
is
tha
when
r
=
k
+
,
the
0
closure
of
under
C
{
a
forti
under
the
w
eak
er
{
r
top
is
con
tained
in
the
set
of
functios
whic
h
are
=
0
;
D
(0)
=
0
and,
whic
;
i
~
0
k
D
k
Æ
;
0
i
C
(
B
)
1
r
H
(
;B
1
D
1
)
Æ
:
i
w
e
consider
~
C
satify
(0)
olgy
k
;
h,
more
v
er
In
W
e
also
note
tha
v
when
arin
t
r
manifolds
15
2
and
the
space
X
is
separbl,
a
1
v
arin
t
of
the
Ascoli-rze
precomat
a
with
The
fol
op
erations
rst
ery
w
w
under
giv
pro
of
tly
the
comp
tha
study
frequn
y
conludes
the
in
norm
the
b
space
w
con
vior
of
j
transfom
tha
with
eha
graph
(2)
ositn
the
is
en.
ositn
ear
ert
t
olgy
o
ap
pro
el
top
t
tha
The
v
the
wing
argumen
e
j
j
j
j
j
under
aproc
wil
use
is
h.
tha
it
b
eha
v
es
tracions.
1
Prop
ositn
4.3
L
et
:
B
!
B
b
1
e
a
C
functio
such
tha
1
0
(0)
=
0
;
k
D
k
C
(
B
1
:
)
1
Asume
tha
j
j
j
j
j
j
<
1
.
Then,
s
(23)
j
j
j
Æ
j
j
j
j
j
j
j
j
jk
D
k
0
C
(
B
)
1
The
main
in
the
p
b
oin
ound
t
in
of
the
Prop
(23).
In
ositn
the
is
most
t
tha
w
ypical
e
case
obtain
r
the
2,
exp
then
one
the
t
exp
s
one
t
0
s
=
2.
Since
k
D
k
is
C
(
B
smaler
than
1,
this
is
quite
w
orth
while.
)
1
Inde,
this
or
to
is
(10)
the
for
reason
lo
w
wh
y
w
e
can
impro
v
e
(3)
to
(1)
when
r
2
regulaits.
1
Pr
o
of.
W
Cleary
e
,
the
estima
functio
for
y
6=
Æ
is
C
and
it
satie
s
j
D
Æ
(
y
)
j
=
j
y
(
Æ
)(0
=
0.
0
j
s
D
(
D
)
Æ
(
y
)
j
(
y
)
1
j
j
D
(
y
)
j
1
j
s
j
y
1
s
j
j
(
y
)
1
j
"
#
s
s
sup
y
2
(
B
f
0
j
D
y
j
=
j
y
j
)
sup
g
y
2
(
B
f
1
2
0
j
(
y
)
j
=
j
y
j
)
g
1
sup
y
1
1
(
B
f
0
j
D
(
y
)
j
)
g
1
from
whic
the
h
it
cleary
estima
fol
in
Prop
ositn
(23)
ws
tha
j
4.
L
et
N
:
X
N
Æ
2
tha
j
j
j
j
is
nite
and
2
X
!
1
X
satify
2
=
Then,
0
,
D
the
(0)
D
=
0
0
Lip
.
L
(
y
)
=
N
(
y
;
1+
j
(
y
)
b
satie
j
j
j
j
j
k
N
1
k
+
k
N
k
j
C
e,
as
b
efor
satie
j
j
j
j
j
C
e
j
j
j
(
y
(0
)
j
j
j
=
y
sup
2
B
j
f
1
0
g
D
(
y
)
j
=
;
0)
2
et
.
functio
N
1+
,
2
wher
it
2
C
2
(0)
tha
1+
Asume
j
.
j
y
j
:
e
a
C
functio,
=
0
.
16
R.
Pr
o
of.
W
e
ha
v
de
la
Lla
v
e
e
j
D
N
(
y
;
(
y
)
j
=
j
y
j
j
(
D
N
)(
y
;
(
y
)
j
=
j
y
j
1
+
j
(
D
N
)(
y
;
(
y
)
j
D
(
y
)
j
=
j
y
j
2
1+
k
N
1
k
+
k
N
k
j
C
j
j
j
j
j
C
Remark
4.5
is
mainly
W
e
usefl
note
tha
for
the
the
in
case
r
tro
2
duction
1
of
+
[0
;
Lip
the
].
c
In
onic
al
the
norm
case
j
r
j
j
2,
j
w
j
e
j
2
0
could
just
use
the
top
olgy
induce
b
y
k
D
k
.
C
(
B
)
1
2
Since
w
e
are
considerg
space
of
C
functios
whic
h
satify
the
2
0
normalizts
(0)
=
0,
D
(0)
=
0,
w
e
se
tha
k
D
k
is
inde
a
C
norm.
F
or
our
pur
ose,
consider
b
w
e
ha
v
e
the
eha
v
impro
v
es
main
w
ed
el
pro
under
con
ert
y
comp
tha
w
ositn
tracion
pro
e
ned
with
ertis
is
a
con
tha
the
norm
tracion
anlogus
and
to
(23)
tha
.
2
0
F
or
k
D
k
w
C
(
B
e
ha
v
e
)
1
2
2
2
0
k
D
(
Æ
)
0
k
k
D
k
k
D
k
0
C
(
B
)
C
(
B
)
1
1
C
(
B
)
1
2
0
+
k
D
0
k
k
C
(
B
D
k
)
C
(
B
)
1
1
2
2
2
0
k
D
0
k
k
D
k
+
k
D
k
0
C
(
B
)
C
1
C
(
B
(
B
)
)
1
1
whic
h
can
b
e
used
in
a
simlar
w
a
y
as
(23)
pro
vide
tha
w
e
can
mak
e
2
D
smal.
2
0
Henc,
w
e
could
use
k
D
k
in
the
subeqn
t
argumen
ts
rathe
C
than
the
conial
norm.
The
conial
norm
j
j
j
j
j
j
turns
out
to
b
e
some-
2
0
what
simpler
to
estima
and,
since
j
j
j
j
j
j
k
D
k
the
uniqes
C
staemn
ts
in
the
conial
norm
are
sligh
tly
more
genral
(the
space
in
2
whic
h
In
is
the
conial
p
case
norm
tha
w
osible
if
e
N
v
transfomi
is
den
include
consider
functios
functios
anishe
to
whic
h
orde
are
p
k
,
osible
tha
whic
p
h
v
anish
are
to
erhaps
after
under
not
orde
some
nitely
C
k
{
).
whic
h
prelimnay
man
y
no-resac
k
0
condits
{,
it
is
p
osible
to
use
the
norms
k
D
k
,
The
pa
er
[dlL97
]
C
includes
a
thes
genral
discuon
case
are
norms
whic
[ElB01
of
studie
h
with
also
thes
wigh
norms
lead
to
ted
impro
and
norms
v
sho
ed
with
ws.
con
In
higer
tracion
p
[CFdlL03a
pro
o
],
w
ers.
Other
ertis
o
cur
].
in
The
c
4.2
hec
k
tha
the
op
er
RHS
ator
of
T
(18)
is
wel
inde
l
den
dens
d
in
the
an
op
erato
sp
ac
es.
on
W
e
.
wil
rst
In
W
e
rst
note
tha
using
v
arin
t
the
manifolds
con
v
17
en
tions
arnged
in
Section
4.1
w
e
w
e
ha
v
e,
for
j
y
j
1,
2
Æ
;:
;Æ
;
Æ
0
k
j
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
j
k
A
k
1
+
"Æ
+
1
k
N
k
0
1
C
0
(24)
k
A
k
+
2
"
1
If
w
tha
e
imp
ose
the
the
RHS
Once
condit
of
w
e
ha
tha
(24)
v
e
is
tha
"
smaler
is
than
the
smal
enough,
w
e
can
ensur
1.
functio
T
[]
is
w
el
den
in
the
indcate
r
domain,
the
c
hain
rule
tels
us
tha
T
[]
is
C
.
Henc,
the
RHS
of
(18)
can
b
e
den
for
al
the
2
.
Æ
;:
;Æ
;
Æ
0
k
4.3
The
sho
w
r
tha
ange
of
T
diern
t
[
]
er
con
the
on
the
particul,
In
tha
w
get
the
c
ac
es
set
map
ondits
of
sho
In
or
it
is
but
p
Section
with
osible
to
4.1
themslv
The
section,
form
tha
in
to
this
the
w
duce
in
.
of
wil
tro
ed
also
e
in
4.6
sp
another
prenomalizts
domains
Lema
T
in
In
with
nd
ator
tained
parmets.
arnge
can
op
is
tha
one
es.
em
3.1
after
making
the
ad-
s
justmen
in
Se
ction
4.1
so
tha
k
B
k
,
k
N
k
ar
C
(
B
e
smal
l
enough,
)
1
s
=
min
(
r
;
2)
.
Then,
it
is
p
osible
to
nd
Æ
;
;
Æ
;
0
Æ
>
0
as
wel
l
Æ
,
satifyng
Æ
=
k
Æ
=
0
1
,
1
as:
i
T
(
)
Æ
=1
;Æ
=1
0
Æ
;
:
Æ
Æ
=1
1
;Æ
=1
0
;
;Æ
;
Æ
1
k
k
r
Pr
o
of.
First,
it
The
fact
is
clear
tha
b
T
y
the
c
[](0)
=
hain
rule
0
tha
and
D
if
T
2
[](0)
r
C
,
=
0
then
T
are
[]
2
just
C
an
.
easy
calution.
W
e
denot
(25)
[](
y
)
=
A
y
+
B
(
y
)
+
N
(
1
x;
(
y
)
:
1
Therfo:
D
[](
y
)
=
A
+
B
D
(
y
)
+
D
N
1
(
1
x;
(
y
)
1
(26)
+
D
N
(
2
W
e
y
;
(
y
)
D
(
y
)
:
2
estima
0
(27)
Lip
([])
k
D
[]
k
k
A
k
+
"
:
1
C
(
B
)
1
The
fact
tions
tha
4.3
in
and
Prop
F
ertis
of
and
rom
for
of
m
,
ultipcaon,
[]
j
j
jT
j
is
nite
y
m
j
is
erato
T
j
the
in
v
ery
easy
just
to
terms
form
the
left
b
y
estima
those
(27)
Prop
osi-
consider
linear
op
the
deriv
erato.)
ativ
es
of
.
y
ula
case
a
for
of
inequalt
the
of
from
the
obtain
of
triangle
consequ
diers
in
is
j
a
ultipcaon
mater
[]
using
j
op
b
the
j
(20)
jT
(The
4.3
heart
[]
j
4.
ositn
The
T
j
and
,
the
tha
Banc
for
h
functios
algebr
pro-
2
18
R.
de
la
Lla
v
e
,
Æ
;:
;Æ
;
w
e
ha
v
e
b
ounds
on
the
deriv
ativ
es,
w
e
obtain
tha,
when
Æ
0
k
2
2
,
Æ
;:
;Æ
;
w
e
ha
v
e:
Æ
0
k
i
i
i
1
0
k
D
T
[]
0
k
k
D
k
k
A
k
+
3
"
k
A
k
1
C
(
B
)
C
(
B
)
2
1
1
(28)
+
P
(
Æ
;
i
wher
P
is
a
real
p
olynmia
:
:
:
;
Æ
)
0
with
i
p
ositv
1
e
co
eÆcien
ts.
i
The
N
co
.
Ev
eÆcien
en
ts
note
if
w
tha
e
of
P
wil
the
p
are
not
obtained
use
b
it
olynmia
in
P
this
can
y
estimang
pa
the
b
e
er
{
deriv
excpt
asumed
ativ
for
to
b
e
i
=
es
1
;
arbitly
2
of
{
w
e
smal
b
y
i
i
asuming
tha
k
N
k
is
suÆcien
tly
smal.
C
Simlary
,
using
(7)
w
e
estima
the
Holder
seminor
in
the
unit
bal
as:
i
H
(
D
T
[])
;B
1
(29)
i
+
i
H
(
D
1
)
k
A
k
;B
+
3
"
k
A
k
+
P
(
1
Æ
;
i
:
:
:
;
Æ
)
1
i
:
1
1
2
The
rest
of
whetr
r
F
or
r
the
pro
2,
r
2,
of
2
1
asumption
whic
+
,
w
;
is
e
Lema
(0
h
(1)
of
4.6
Lip
]
the
or
r
w
as
in
1
b
e
diern
t
acording
to
1.
case,
arnge
k
=
main
can
wil
e
note
tha,
Section
b
4.1,
ecaus
of
the
tha
2
A
k
D
k
<
1
0
2
C
(
B
)
1
0
k
D
k
<
C
(
B
1
:
)
1
W
e
c
ho
ose
Æ
=
Æ
=
0
1
;
Æ
=
1
1.
Becaus
of
(28)
w
e
get
2
2
0
k
D
T
[]
k
Æ
+
P
(1
2
;
1)
2
C
Recaling
tha
w
e
can
mak
e
the
co
eÆcien
ts
or
P
as
smal
as
desir
2
2
b
y
arngi
tha
k
N
k
,
k
B
k
are
suÆcien
tly
smal,
w
e
can
arnge
C
tha
2
0
(30)
k
D
T
[]
k
1
:
C
Using
tha
can
use
w
the
e
ha
mean
v
v
e
the
normalizts
alue
T
theorm
to
obtain
[]
=
from
0,
D
T
[]
=
0,
w
e
(30)
0
k
D
k
1
;
k
k
1
=
2
:
C
C
0
This
establih
the
desir
result
for
r
=
2.
In
case
tha
r
=
k
+
>
2,
w
e
pro
ced
to
c
ho
ose
the
Æ
,
Æ
so
tha
i
the
desir
conlusi
hold.
It
is
imp
ortan
t
to
emphasiz
tha
the
2
smalne
condits
tha
w
e
wil
b
e
imp
osing
on
k
N
k
,
k
B
k
wil
b
C
1
indep
W
end
e
observ
t
e
of
tha,
k
.
This
wil
w
e
b
ha
v
e
e
the
for
basi
k
1
k
A
of
i
>
the
study
2
of
k
i
D
k
<
0
k
2
C
(
B
)
1
1
:
the
C
case.
e
In
Using
(28)
w
e
ha
v
v
arin
t
manifolds
19
e
i
0
k
D
T
[]
k
Æ
+
i
P
(1
i
;
Æ
;
i
:
:
:
;
Æ
)
2
i
1
C
i
0
Henc,
w
e
can
c
ho
ose
recusiv
ely
Æ
so
tha
k
D
T
[]
k
Æ
.
i
i
C
Simlary
,
taking
in
to
acoun
t
(29)
w
k
H
[
T
[]
Æ
k
1
k
k
A
obtain
D
wher
e
k
D
+
+
P
(1
;
k
;
Æ
;
;
:
:
:
;
Æ
)
2
k
k
<
1.
Henc,
it
is
p
osible
to
c
ho
ose
0
k
;
2
C
(
B
)
1
also
the
Æ
This
.
nishe
the
Remark
4.7
pro
W
asumption
e
(1)
condit
As
it
lo
w
w
in
somewhat
the
case
this
w
r
case,
eak
w
er
considerg
higer
e
e
than
2.
ha
v
the
the
orde
deuc
ha
v
e
estima
deriv
second
deriv
tha
when
of
higer
a
ativ
orde
ativ
e
This
b
e
only
used
usal
comp
the
dominace
ositn
etr
ativ
es
ecaus
functio
with
estima
w
y
relatd
part
e
con
than
the
lo
ha
so
v
e
deriv
the
trac-
deriv
ativ
ativ
e
es
from
those
tha
could
estima
ativ
tha,
y
of
the
rst
e.
b
tangecy
es
normalizt
w
fact
a
er
deriv
the
has
w
tha
second
to
noliear
the
zero,
b
course
the
b
at
the
of
e,
for
tangecy
and
is,
riv
es
case,
the
has
is
in
orde.
our
for
4.6
tion
ens,
ativ
er
In
Lema
aten
This
hap
deriv
of
of
.
often
tions,
cal
.
(3)
of
the
denito
orde
of
2
with
the
de-
linear
part.
The
same
orde
t
yp
higer
e
of
argumen
than
enough
t
2
orde.
if
w
Inde
no-resac
e
in
reduc
it
is
obtain
p
so
tha
to
condits
p
out
is
sho
than
a
tha
e
are
(1)
b
certain
of
v
ed.
higer
ecaus
exp
aribles
In
deriv
the
or
hig
satie
hanges
presv
with
es
of
A
c
ativ
tangecy
if
mak
ts
deriv
has
wn
to
argumen
er
using
N
tangecis
simlar
eak
caried
osible
orde
use
w
it
is
hig
osible
e
tha
]
it
N
case,
b
ensur
[dlL97
condits,
tha
can
can
suc
h
ativ
one
t
es
of
k
a
and
A
k
is
1
biger
than
2.
Argumen
ts
op
eratos
with
a
hap
W
e
are
en
also
ref
v
or
but
aplied
to
somewhat
diern
t
].
Example
5.1
to
sho
w
tha
some
of
thes
condits
.
No
w,
w
case
e
for
F
a
[CFdlL03a
to
necsary
simlar
in
or
consider
the
whic
h
the
the
range
use
case
r
of
of
=
the
1,
w
the
op
second
e
erato
T
deriv
observ
e
ativ
tha
in
e
the
is
the
lo
not
p
w
regulait
y
osible.
estima
(28)
w
e
ha
v
1
0
k
D
T
[]
k
Æ
1
+
"
wher
=
1
k
A
k
A
k
1
.
In
this
case,
the
as-
1
C
2
sumption
in
Theorm
3.1
imply
tha
<
1.
Henc,
w
e
can
c
ho
ose
1
0
the
Æ
=
1
so
tha
k
D
T
[]
k
1
Æ
.
1
C
Once
w
e
ha
v
e
tha,
w
e
can
e
R.
c
20
ho
ose
Æ
>
Æ
.
0
T
[](0)
Using
the
de
la
mean
v
Lla
v
alue
e
theorm
and
the
normalizt
1
=
0,
w
e
obtain
tha
T
(
)
Æ
;:
;Æ
;
:
Æ
Æ
;:
0
;Æ
;
Æ
0
k
k
In
the
case
r
=
1
+
,
0
<
Lip
,
w
e
c
ho
ose
Æ
=
1,
Æ
=
0
By
the
c
hain
rule,
w
e
ha
v
e
D
T
[]
=
(
D
)
Æ
D
1,
Æ
=
1.
1
.
Henc,
using
(7)
,
1
0
H
[
D
T
[]
k
A
k
H
[
D
]
Lip
()
k
D
0
k
+
k
D
k
H
D
:
C
C
2
Note
tha
H
[
D
]
H
[
arbitly
Henc,
w
b
e
D
N
]
and,
since
>
0,
this
can
b
e
made
smal
ha
v
y
e
rescaling.
for
functios
in
1
H
[
D
T
[]
;
1;
+
"
1
wher
k
1+
A
k
D
k
<
1
b
y
asumption.
Adjusting
tha
"
is
0
2
C
smal
enough,
w
This
e
nishe
W
e
obtain
the
the
pro
emphasiz
desir
of
tha
of
the
result.
Lema
4.6
condits
of
smalne
tha
w
e
ha
v
e
imp
osed
2
on
k
B
k
,
k
N
k
,
are
indep
end
t
of
r
.
The
w
a
y
to
ensur
tha
the
hig
C
orde
deriv
ativ
es
get
trap
ed
is
b
y
c
ho
osing
the
Æ
i
2
and
Æ
en
tering
i
in
the
denito
of
the
to
Æ
;:
;Æ
;
b
e
large
enough.
Æ
0
k
Note
ha
also
v
e
tha
al
the
the
argumen
t
smaler
than
relid
hea
one
vily
with
on
the
condits
fact
tha
tha
w
are
e
indep
could
end
t
i
of
i
.
This
is
certainly
true
pseudo-tabl
in
the
stable
case,
but
wil
b
e
false
in
the
case.
Remark
4.8
to
the
In
space
case
{
tha
e.g
w
e
when
can
aply
X
is
Ascoli-rze
a
a
separbl
space,
theorm
w
aplied
e
obtain
a
pro
of
1
of
the
T
yc
existnc
of
hon
v
The
op
theorm
This
T
giv
es
since
the
e
the
a
h
a
p
the
y
geomtric
of
of
not
obtain
case
osible
t
to
b
argumen
in
the
v
Sc
Then,
as
a
w
ex.
arin
t
mani-
On
Nev
e
are
erth-
ts
[Sh
establihng
u87
],
[LI83
]
uniqe.
the
stable
hauder
one
refncs
v
tracion.
ertis.
manifold
e
con
v
con
pro
pro
the
ounds.
ts
in
is
argumen
to
aply
the
T
uniqes
is
hauder-
graph.
of
the
Sc
cleary
closed
tha
manifold
argumen
is
and,
a
geomtric
stable
the
existnc
tha
are
the
aplying
has
obtain
the
the
ther
proagted
the
y
compat
it
to
in
tha
space
b
do
tha,
is
ecaus
estima
manifold,
dimensoal
lishe
pro
b
b
short
just
space
uos
the
e
(18)
the
tin
oid
e
establih
Henc,
ecaus
con
ery
v
w
observ
stable
whic
v
e
hand,
w
is
is
a
w
other
les,
of
b
erato
folds
solutin
manifold
theorm
can
establih
ab
for
once
o
the
one
v
e.
nite
estab-
uniqes
In
4.
Contr
action
Lema
pr
4.9
In
with
Æ
op
arin
ertis
the
the
v
c
norm
t
of
manifolds
21
the
ondit
op
of
(2)
er
L
ator
T
ema
on
4.6
T
.
is
a
c
ontr
action
on
.
Æ
r
0
Recal
tha
1
D
T
[](
y
)
=
A
D
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
1
2
A
+
B
(
y
)
+
D
N
(
y
;
(
y
)
(31)
1
1
1
1
+
A
D
N
(
1
y
;
(
y
)
+
D
N
2
(
2
y
;
(
y
)
D
(
y
)
2
2
~
b
F
rom
y
(31)
it
ading
wil
b
and
W
e
wil
use
tha
a
al
ula
j
As
ignor
form
straigh
(31)
tforw
terms
4.10
tha,
rathe
subtracing
also
Remark
e
j
the
1,
k
jects
k
1,
ativ
k
b
amoun
j
k
the
can
e
D
for
tha
deriv
estima
j
jT
[]
T
[
]
j
j
j
.
guide
ob
to
aproitely
heuristc
the
for
y
ard
1,
k
D
subeqn
e
ts
k
t
made
estima
is
arbitly
to
1.
smal,
the
just
1
(32)
A
D
(
A
(
y
)
A
:
1
1
2
As
w
e
wil
wil
b
con
se,
e
since
al
obtained
b
tracion
in
The
con
y
(32)
the
of
W
are
and
smal,
the
subtracing
pro
of
terms
of
con
from
tracion
the
pro
of
of
.
tracion
estima.
terms
ading
e
ha
v
(32)
is
pro
v
ed
using
the
impro
v
ed
con
tracion
e:
1
1
~
k
A
D
(
A
y
)
A
A
1
D
(
A
y
1
)
A
k
1
2
1
2
1
~
k
A
k
A
kj
D
(
A
y
1
)
D
(
A
y
1
)
j
1
2
1
~
k
A
k
A
kj
j
j
j
j
j
A
y
1
j
1
2
1
2
~
k
A
k
A
k
j
j
j
j
j
j
y
j
:
1
2
F
rom
(32)
the
is
No
a
ab
w
con
w
e
estima,
to
j
wil
j
j
ws
tha
e
other
main
part
of
T
(as
in
for
to
wil
pro
ha
v
terms
adjustmen
the
.
try
w
the
the
j
fol
estima
e
but
al
with
j
ving
w
4.10,
temaicly
j
pro
eaking
Remark
smal
it
in
turn
sp
the
v
tracion
e
Roughly
o
in
ful
ced
e
T
along
to
pa
whic
ts
the
y
h
not
the
aten
just
lines
tion
wil
Section
and
.
indcate
to
turn
(32)
in
estimang
out
sy-
to
b
e
arbitly
4.1
~
~
0
j
D
N
(
y
;
(
y
)
D
1
N
(
x;
(
y
)
j
k
D
D
1
N
1
k
j
(
y
)
2
C
(
B
)
1
(3)
2
~
A
forti,
simlar
b
ounds
are
true
for
"
N
j
j
1
,
j
N
in
2
j
j
j
place
y
j
:
of
N
.
(
y
)
j
2
R.
More
v
de
la
Lla
v
e
er,
~
j
D
N
(
y
;
(
y
)
D
(
y
)
D
N
2
(
y
;
~
(
y
)
D
(
y
)
j
2
~
j
D
N
(
y
;
(
y
)
j
D
(
y
)
D
(
y
)
j
2
~
+
j
D
N
(
y
;
~
(
y
)
D
N
2
(
y
;
~
(
y
)
j
D
(
y
)
j
2
~
wher
w
e
ha
Again,
v
w
e
e
"
j
j
j
used
j
agin
note
j
j
y
j
(3)
tha,
to
the
estima
same
the
second
estima
factor.
remain
v
alid
for
N
,
N
in
1
place
N
W
.
e
denot
b
[](
y
y
)
=
A
y
+
B
(
y
)
+
N
(
1
D
[](
y
)
=
e
y
;
(
y
)
1
A
+
B
D
(
y
)
+
D
N
1
W
2
(
1
y
;
(
y
)
+
D
N
1
(
y
;
(
y
)
2
estima
0
k
D
[]
k
k
A
k
+
"
:
1
C
(
B
)
1
Henc
j
[](
y
)
j
j
A
j
+
"
k
y
k
:
1
More
v
er,
w
e
ha
v
e
~
j
D
[](
y
)
D
[](
y
)
[
[
~
](
y
)
j
"
j
j
j
~
j
j
j
j
y
j
~
](
y
)
j
"
j
j
j
j
j
j
y
j
~
The
only
terms
left
to
estima
in
D
T
[]
D
T
[
]
can
b
e
exprsd
as
1
~
(34)
A
(
D
Æ
[](
y
)
D
[](
y
)
D
~
Æ
[
~
](
y
)
D
[
](
y
)
:
2
The
in
norm
the
of
(34)
is
denito
of
j
j
b
j
ounde
j
j
j
b
y
(w
e
recal
tha
s
=
min
(2
;
r
)
en
terd
)
1
~
k
A
k
j
D
Æ
[](
y
)
D
~
Æ
[
](
y
)
j
D
[](
y
)
j
2
~
+
j
D
~
Æ
[](
y
)
D
~
Æ
[
](
y
)
j
D
[](
y
)
j
~
+
j
D
~
Æ
[
~
](
y
)
j
D
[](
y
)
D
[
](
y
)
j
1
s
1
~
k
A
k
j
j
j
j
j
j
k
A
k
+
"
j
y
j
k
A
k
1
+
"
1
2
s
~
+
j
[](
y
)
[
1
~
](
y
)
j
k
A
k
+
"
+
"
j
j
j
j
j
j
y
j
1
1
s
s
1
~
k
A
k
(
k
A
k
+
"
)
+
"
j
j
j
j
j
j
y
j
:
1
2
Colecting
the
Remark
4.10,
previous
estima,
the
op
erato
w
T
has
a
Lipsc
e
obtain
tha
hitz
consta
as
t
indcate
in
in
j
j
j
j
j
j
whic
h
1
s
is
k
A
k
A
k
+
"
.
By
the
asumption
in
Theorm
3.1,
this
is
a
1
2
con
tracion
on
.
Æ
;:
;Æ
;
0
k
Æ
Henc,
w
e
obtain
tha
ther
a
xed
p
oin
t
In
of
T
{
therfo
a
v
solutin
arin
t
of
manifolds
23
(18)
{
in
the
closure
of
for
the
j
j
j
j
j
j
r
Aplying
Lema
Henc,
2.5
w
case
k
A
e
k
ha
<
v
1,
in
e
[LI73],
w
establihd
r
e
conlude
tha
the
2
N
+
[0
;
Lip
this
conlusi
xed
of
p
oin
Theorm
t
is
3.1
.
~
C
for
.
the
].
1
Remark
4.1
W
e
note
tha,
since
w
e
ha
v
e
used
a
con
tracion
argu-
men
t,
w
e
ha
v
e
sho
wn
tha
ther
is
exactly
one
xed
p
oin
t
in
,
Æ
;:
;Æ
;
Æ
0
k
wher
denots
Note
the
tha
if
`
closure
>
in
the
top
olgy
w
e
consider.
k
(35)
:
Æ
Æ
;:
;Æ
;Æ
;:
Æ
;
;:
;Æ
;
Æ
Æ
0
k
0
k
It
is
imp
ortan
ularties
is
hig
w
to
b
are
when
e
e
r
w
e
tak
the
the
not
e
for
In
diern
this
lo
w
t
w
a
y
orde.
,
reg-
the
In
tha
con
a
ed
is
on
e
's
space
in
map
tak
.
of
obtain
tracion
end
w
Æ
Æ
this
of
w
get
a
y
map
,
ed
(35).
This
dep
when
Æ
the
in
space
2.
the
y
the
to
of
regulait
condits
r
hoice
regulaits,
as
get
do
tha
c
hangi
con
for
e
Notice
t
establih
to
tha
in
the
only
higer
our
c
nestd
tha
w
condits
with
tha
diern
are
also
{
`
inducto
consider
Notice
the
y
+1
withou
es
r
realiz
ade
themslv
with
to
done
index
in
t
k
b
do
w
to
ativ
y
do
the
es
not
c
hange
{
hange
and
eithr
asumption
than
(1)
c
norm
smalne
higer
asumption
not
regulait
itself
ecaus
deriv
the
in
tracion
lo
,
the
w
e
second.
obtain
con
tracion
2
only
on
C
space
Example
or
5.3,
this
2
uniqe
W
are
e
it
2
o
furthemo
in
is
observ
v
t
osible
e
to
y
Ev
en
in
obtain
y
b
enough
whic
if
.
As
w
case
e
wil
tha
w
intely
b
y
e
se
e
man
y
previous
uniqes
in
obtain
a
manifolds
in
atrcion
in
set
of
the
wil
basin
b
Their
the
obtain
of
tained
ounde
iteraons.
h,
w
basin
con
an
taking
tha
the
manifolds
with
ifolds,
p
to
arin
terscion
bal
.
.
proagtes
w
regulait
sharp
Æ
C
uniqes
t
higer
is
manifold,
2
h
with
Æ
C
whic
space
conlusi
of
e
ev
en
images
a
bal,
Giv
atrcion,
en
their
map
also
ed
b
staemn
the
orign.
tualy
wil
uniqes
the
t
e
in
v
wil
in-
in
to
arin
ha
the
t
v
e
to
man-
agre.
4.5
Pr
o
of
of
The
or
em
3.1
in
the
stable
c
ase
when
r
=
1
;
!
.
The
case
1
C
is
a
v
ery
simple
consequ
of
the
observ
ations
made
in
Remark
4.1
W
e
note
tha,
w
e
can
nd
a
sequnc
f
Æ
g
suc
i
k
2
N
,
is
Æ
;Æ
;:
0
map
ed
in
to
itself
to
T
i
2
h
tha,
for
ev
ery
N
and
T
is
a
con
tracion
in
;Æ
1
k
the
distance
Becaus
j
of
j
j
j
j
j
.
the
nestig
(35),
the
xed
p
oin
ts
in
those
space
T
~
to
k
coinde.
The
xed
p
oin
t,
therfo
is
in
C
k
=
1
C
.
ha
v
e
24
R.
The
anlytic
space
case
and
space
w
are
e
is
recal
v
ery
tha
anlytic
de
la
v
simple.
e
W
diern
(se
Lla
e
tiable
[Kat95
]
w
ork
in
a
functios
complex
in
a
complex
Banc
h
Banc
h
).
2
If
w
e
cary
obtain
out
tha
the
the
C
pro
functio
is
of
as
ab
diern
o
v
e,
tiable
but
in
in
the
a
complex
unit
bal,
bal
and,
w
e
therfo
anlytic.
4.5
Pro
fol
of
ws
w
e
v
ha
pro
v
er
tha
w
X
this
Æ
e
in
v
e
stable
case.
and
in
man
of
y
tha
of
wel
no
w
w
e
a
e
ys
erthls
the
somewhat
simpler
to
wil
of
Inde,
in
the
con
manifolds,
w
pro
Nev
norms.
used
pseudo-tabl
and
l
den
not
d.
ha
den
to
turns,
is
e
is
ha
the
The
traciv
so
not
ha
v
e
to
e
tha
the
distnguh
regulaits.
w
Henc,
is
norms
case
T
tha
in
space
ard
ator
is
case.
of
of
straighfow
t
op
case
pro
hoices
ted
the
more
diern
The
eigh
for
e
of
ensur
w
usefl
b
case
stable
c
Pseudo-tabl
case
of
not
the
the
t
pseudo-tabl
wil
4.51
in
as
diern
devics
ofs
the
e
the
are
pro
lines
mak
the
case
3.1
simlar
to
of
since
Theorm
ery
e
of
of
v
in
e
a
consider
us
bal
if
D
diernc
j
of
to
main
j
space
forces
The
tha
<
Henc,
w
the
e
canot
is.
functios
consider
with
1.
den
in
no-liearts
N
the
whole
tha
are
uni-
1
formly
smal
on
X
B
(
1
in
Section
X
).
1
This
is
precisly
what
w
as
acomplished
2
4.1
W
e
wil,
smal
therfo
in
X
asume
B
(
1
X
in
).
1
In
the
this
fol
wing
tha
N
cirumstane,
the
is
op
den
and
erato
T
is
w
el
2
den.
Some
v
4.52
ery
sp
simlar
W
ac
es
to
e
of
the
consider
functios.
The
space
r
=
k
+
space
tha
as
w
e
usal
tha
w
consider
e
in
and
consider
wil
consider
are
Section
4.1
the
space
(36)
r
i
0
~
=
Æ
Æ
;
:
X
!
Æ
X
;
1
2
C
;
k
D
k
Æ
;
2
i
=
0
;
:
:
:
;
r
;
i
C
(
X
)
0
1
k
k
H
(
D
)
Æ
(0)
The
space
den
in
~
the
dier
from
whole
the
space
X
=
space
b
rathe
0
;
D
(0)
ecaus
=
than
the
just
0
g
functios
on
the
in
unit
~
are
bal.
1
The
main
tec
hnical
diernc
with
the
space
~
is
tha
w
e
consider
0
the
~
space
endo
w
ed
with
the
C
(
X
)
top
olgy
.
1
In
the
pseudo-tabl
case,
w
e
wil
not
use
w
eigh
ted
norms,
since
in
0
this
case,
the
extra
factors
k
D
k
than
C
do
1.
not
help
since
they
are
biger
In
The
anlogues
whic
h
of
are
rathe
Prop
ob
v
arin
t
ositn
manifolds
25
4.3
and
4.
are
the
fol
wing
result
vious.
1
Prop
ositn
4.12
L
et
:
X
!
X
b
1
e
a
C
functio
such
tha
(0)
=
1
0
0
,
k
D
k
C
(
X
1
.
)
1
Then,
0
(37)
k
Æ
0
k
C
(
X
k
k
:
)
C
(
X
)
1
Prop
ositn
4.13
L
et
N
:
X
2
1+
Asume
tha
N
2
1
B
(
1
X
)
1
!
X
satify
2
D
N
(0
2
1+
,
0
Lip
.
L
;
0)
=
0
.
2
C
et
b
e
a
C
functio,
2
(0)
=
0
,
Then,
D
(0)
=
the
0
.
functio
(
y
)
=
N
(
y
;
(
y
)
satie
2
0
k
0
k
k
N
k
:
2
C
(
X
)
C
(
X
1
4.53
The
r
anlogue
ange
of
in
w
e
wil
the
se
diern
op
er
ator
T
pseudo-tabl
the
t
the
due
to
on
ds
the
are
fact
(
v
of
tha,
ery
p
X
)
1
the
case
metho
B
1
sp
2
ac
the
es
~
.
result
ev
k
section
in
simlar
osibly
This
A
en
k
Section
if
an
4.3
the
1,
is
As
conlusi
are
henc
hig
p
o
w
ers
of
1
1
k
of
A
are
not
con
tracions
and
k
A
k
A
k
1
wil
not
b
e
a
con
tracion
1
2
for
hig
k
.
Lema
4.1
In
the
c
ondits
of
The
or
1
k
em
3.1,
asume
tha
r
A
k
A
k
<
1
:
1
2
Denot
r
=
Then,
k
+
after
as
usal.
making
the
adjustmen
in
Se
ction
4.1
so
tha
k
B
k
,
1
k
N
k
is
smal
l
enough,
it
is
p
osible
to
nd
Æ
;
:
:
:
;
Æ
;
0
C
(
X
B
(
1
Æ
=
Æ
0
X
1
Æ
0
,
is
such
a
way
tha
i
T
(
~
)
Æ
;Æ
0
Æ
;
~
:
Æ
Æ
;:
1
;Æ
;
of.
The
pro
Lema
of
is
v
ery
Æ
0
k
o
,
2
,
1
Pr
Æ
k
)
1
=
k
simlar
to
{
but
simpler
than
{
the
pro
of
of
4.6
W
(w
e
e
just
note
just
tha,
ned
b
to
c
ecaus
of
hange
the
(20)
w
domains
e
ha
v
wher
e
w
a
e
direct
anlogue
cary
of
out
the
(28)
estima).
(38)
i
i
i
1
0
k
D
T
[]
0
k
k
D
k
k
A
k
+
3
"
k
A
k
+
P
(
1
C
(
X
)
C
(
X
Æ
;
i
:
:
:
;
Æ
)
0
i
1
)
2
1
wher
P
is
a
1
real
p
olynmia
with
p
ositv
e
co
eÆcien
ts.
i
W
e
also
ha
v
e
an
anlogue
of
(29)
just
c
hangi
the
domains.
(39)
i
i
H
(
+
i
D
T
[])
H
;N
(
D
1
)
k
A
k
;N
+
3
"
k
A
k
+
P
(
1
1
Æ
;
i
1
1
Henc,
2
w
e
pro
ced
b
y
inducto
asuming
Æ
=
0
1
;
tha
Æ
=
1
1
:
w
e
ha
v
e
set
:
:
:
;
Æ
)
i
1
26
R.
As
in
the
con
traciv
e
case,
w
de
e
la
just
Lla
v
e
note
tha
this
can
b
e
arnged
if
w
e
1
arnge
tha
k
N
k
;
k
B
k
are
suÆcien
tly
smal.
C
Asume
inductv
ely
tha
w
e
ha
v
e
detrmin
Æ
=
1
;
Æ
=
0
Then,
c
ho
ose
Æ
in
i
suc
h
a
Æ
i
=
k
y
1
;
:
:
:
;
Æ
,
1
i
tha
Æ
i
i
A
+1
1
a
+1
(40)
wher
w
k
+
+1
P
(
i
i
Æ
;
+1
:
:
:
;
Æ
)
1
1
+1
k
.
1
i
+1
2
C
This
c
hoice
of
Æ
satifyng
i
By
the
<
i
(40)
for
i
k
p
osible
pro
vide
tha
<
i
in
1
is
+1
asumption
Theorm
3.1,
w
e
se
tha
w
e
can
1.
+1
adjust
tha
.
+1
It
is
clear
tha
if
Æ
satie
i
(40)
and
T
(
)
+1
Æ
;:
,
;Æ
Æ
;:
0
T
(
)
Æ
;:
;Æ
.
;Æ
Æ
;:
0
;Æ
Henc
w
e
can
then
;Æ
0
i
recusiv
i
ely
,
nd
the
Æ
.
;Æ
i
0
i
i
+1
i
i
+1
Simlary
,
using
(39)
1
tha
k
k
A
k
+
,
w
e
can
ensur
tha
w
e
can
nd
Æ
pro
vide
k
<
1.
1
2
C
Remark
4.15
in
The
Lema
4.6,
main
the
con
w
tras
er
with
the
decrasing
pro
as
of
i
of
Lema
4.6
increasd.
In
is
the
tha
pseudo-
i
stable
case
consider
her,
the
are
increasg
with
i
and
for
large
i
enough
i
the
condit
<
1
is
violated.
This
is
what
mak
es
the
i
inducto
ndig
regulait
y
ther
the
.
As
are
y
action
go
es
pr
Lema
v
4.16
with
Æ
the
of
the
norm
op
er
stop
for
an
hig
artifc
and
regulait
ator
T
y
the
of
L
on
result
as
ondit
(2)
more
main
lines
c
of
not
than
the
The
simlar
the
pro
is
t.
4.
In
the
this
get
argumen
ertis
ery
es
5.4,
canot
this
op
mak
Example
one
y
Section
along
henc,
in
wher
to
of
and,
se
b
anlogus
pro
stop
wil
predict
Contr
quite
Æ
e
exampls
regulait
4.5
w
is
pro
of
ema
of
4.6
~
.
This
section
Lema
is
4.16
Lema
whose
4.9
T
is
a
c
ontr
action
on
.
Æ
r
0
Again
w
w
e
ha
v
main
e
e
p
remak
tha
erfomd
a
part
al
of
the
op
usefl
the
heuristc
idea
prelimnay
erato
is
to
adjustmen
T
is
ts
note
tha,
in
after
Section
4.1,
the
just
1
(41)
R
[]
A
Æ
A
:
1
2
Henc,
w
of
the
v
e
ery
exp
ect
tha
simple
the
op
estima
for
T
are
simlar
to
the
estima
erato
1
~
~
0
kR
[]
R
[
]
0
k
k
A
k
k
k
2
C
(
X
B
(
1
No
w,
w
e
pro
ced
diÆcult
diern
most
to
t
terms
p
oin
ts
whic
C
dep
(
X
e
terms
those
end
B
(
1
the
b
h
)
2
estima
wil
{
X
1
in
on
whic
.
h
F
the
ful
or
include
op
this
terms
1
erato
ev
X
)
2
T
.
w
aluted
e
at
wil
ned
The
t
w
to
o
In
tha
asume
~
in
suc
h
a
is
w
a
at
y
least
v
arin
t
Lipsc
tha
the
manifolds
27
hitz.
Lipsc
This
hitz
is
arnged
consta
t
b
is
b
y
c
ho
osing
the
ounde.
~
W
e
note
tha
if
;
2
~
,
w
e
ha
v
e
~
~
0
j
N
(
y
;
(
y
)
N
(
y
;
(
y
)
j
"
k
k
C
wher,
as
b
desir
b
simlar
efor,
y
b
w
c
ho
e
denot
osing
b
the
ounds
for
y
"
terms
tha
adjustmen
the
ts
N
;
N
in
can
b
Section
e
made
4.1
as
A
smal
as
forti,
w
e
ha
v
e
.
1
2
Simlary
j
[
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
N
(
1
~
[
A
y
+
B
y
;
(
y
)]
2
~
(
y
)
+
N
(
1
y
;
~
(
y
)
N
(
1
y
;
(
y
)]
j
2
~
0
"
k
k
C
Finaly
,
~
j
(
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
~
(
A
y
1
+
B
~
(
y
)
+
N
(
1
y
;
(
y
)
j
1
(
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
1
~
(
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
y
)
1
~
+
(
A
y
+
B
(
y
)
+
N
(
1
y
;
(
y
)
1
~
~
(
A
y
+
B
~
(
y
)
+
N
(
1
y
;
(
1
~
~
~
0
"
k
0
k
+
Lip(
)
"
k
k
:
C
Remark
4.17
ties
(12)
In
to
C
the
obtain
pro
of
tha
w
the
e
ha
v
space
e
presn
~
ted,
get
w
map
ed
e
only
in
use
to
eac
the
h
pro
er-
other.
Nev
er-
1
thels,
the
con
tracion
part
of
the
argumen
t
only
use
tha
k
A
k
<
1.
2
It
is
p
osible
{
but
w
e
wil
not
cary
out
the
details
her
{
to
sho
w
s
tha
the
op
usefl
if
tions
erato
T
one
or
w
to
Remark
pro
an
is
ts
v
a
e
to
smo
4.18
con
v
tracion
on
alidte
the
oth
Notice
dep
C
on
w
e
ha
v
This
pro
of
endc
tha
.
result
e
ert
some
n
y
is
someti
umerical
computa-
parmets.
establihd
uniqes
of
the
xed
1
p
oin
t
in
the
space
~
under
the
asumption
tha
k
N
k
;
k
B
k
are
smal.
C
W
in
e
note
tha
Section
ma
this
4.1
y
aect
Nev
the
Note
can
if
e
arnged
b
erthls,
in
tha
b
v
k
it
arin
t
A
k
is
y
imp
scaling
manifold
and
ortan
t
arbitly
1,
the
in
v
cut-o
to
note
tha
close
arince
as
to
the
the
equation
indcate
cut-o
orign.
(18)
can
proagte
1
an
smal
W
disturbance.
e
wil
ilustrae
The
incorp
orate
This
this
uniqes
condits
phenom
in
claime
some
her
are
condits
v
of
is
ery
diern
Example
5.3
obtained
gro
only
t
wth
from
the
at
int
in
the
condits
y
of
space
the
~
whic
h
w
functios
e
obtained
.
for
28
R.
the
stable
manifolds
whic
h
w
de
la
er
Lla
v
just
e
regulait
y
of
the
manifolds
at
the
orign.
Remark
4.19
A
uniqes
is
v
ery
w
obtained
el
kno
only
wn
consequ
after
w
of
e
imp
ose
a
the
fact
cut-o
tha
is
the
tha
cen
ter
1
manifolds
ma
y
fail
to
b
e
C
.
This
comes
ab
out
b
ecaus
the
xed
p
oin
ts
r
pro
duce
in
a
r
those
of
+
C
space
b
y
carying
out
In
,
this
whic
h
section,
requis
w
claimed
in
Some
of
a
In
the
t
from
cut-o.
exampls
tha
canot
are
b
relatd
w
impro
e
impro
to
sho
v
ed
w
in
exampls
tha
the
certain
in
v
re-
directons.
[dlL
W95],
[dlL97
],
e
sho
w
tha
the
sp
ectral
gap
5.1
condits
(1)
ed.
2
Example
diern
].
exampl,
e
t
some
3.1
[CFdlL03c
rst
b
are
exampls
colet
exampls
],
canot
e
diern
Some
Theorm
thes
[CFdlL03a
cut-os
k
C
5.
sult
some
Consider
the
map
F
:
2
R
!
R
given
by:
1
1
2
(42)
F
(
x
;
x
)
1
=
x
;
2
x
+
1
x
:
2
1
2
4
2
Then,
the
the
sp
ac
map
e
X
do
=
f
(
es
x;
not
0)
j
have
x
2
any
R
g
C
invart
manifold
tange
to
.
1
Note
tha
equalit
y
the
.
Al
exampl
the
satie
other
h
yp
(1)
othesi
with
of
the
inequalt
y
theorm
are
replacd
b
y
satied.
2
Pr
o
of.
An
y
C
in
v
arin
t
manifold
can
b
e
tange
t
to
X
at
the
orign
1
can
b
e
writen
lo
caly
as
the
graph
of
an
functio
:
X
!
X
.
1
The
functio
should
satify
the
equation
(17),
whic
2
h
in
our
case
reads
1
1
2
(43)
x
=
(
x
)
1
+
x
1
1
2
T
aking
deriv
ativ
es
of
4
(43)
t
wice
and
ev
aluting
at
the
orign,
w
e
obtain:
2
1
1
2
D
2
(0)
=
D
2
Cleary
,
this
sho
ws
(0)
+
2
4
tha
ther
is
no
functio
satifyng
(43)
and
2
h
functio.
whic
has
t
w
o
deriv
ativ
es
at
the
orign.
A
forti,
ther
is
no
C
In
Remark
5.2
W
e
note
v
tha
arin
t
the
manifolds
29
main
reason
wh
y
Example
5.1
w
orks
is
2
b
ecaus
(1
[dlL97
],
=
2)
=
[ElB01
1
],
condits
=
4.
Henc,
ther
[CFdlL03a
]
rathe
than
sp
is
study
a
resonac.
Inde,
sytem
ectral
tha
gap
satify
5.3
Consider
the
Given
ar
>
e
0
,
it
map
admits
tange
F
:
intely
to
the
sp
e
no-resac
2
R
!
R
many
ac
ers
in
2
whic
pa
condits.
2
Example
the
X
=
exampl
C
f
(
x;
.
0)
j
5.1
x
2
invart
R
manifolds
g
.
1
Pr
o
of.
Again,
W
e
it
wil
suÆces
to
recusiv
ely
pro
duce
solutin
detrmin
for
the
(43)
functios
.
on
the
in
terv
als
I
=
i
i
1
i
[2
;
2
]
staring
with
an
arbity
c
hoice
on
the
in
terv
al
I
whic
0
has
sup
ort
in
the
in
terio
of
the
in
terv
h
0
al.
P
2
W
e
wil
write
(
x
)
=
(
1
x
)
i
+
x
wher
the
ha
1
v
e
sup
ort
in
i
1
i
the
in
terio
of
I
.
i
Notice
tha
functio
the
equation
of
detrmin,
w
e
in
v
can
arince
is
nd
.
i
is
equiv
i
alen
t
suc
h
tha
Inde,
if
w
e
w
e
ha
obtain
v
e
the
tha
(43)
+1
to
1
2
(4)
(
i
x
)
+1
=
(2
1
x
)
i
+
x
1
1
4
Henc,
w
e
can
den
b
y
recusion
the
.
By
standr
estima,
w
e
i
ha
v
e
from
(4)
1
2
2
k
2
2
k
i
2
k
k
+1
+
2
:
i
C
(
I
)
i
C
(
I
)
+1
i
4
2
This
sho
ws
Example
tha
the
5.4
seri
We
c
givn
onsider
the
con
v
ergs
uniformly
in
(
x
;
x
)
1
1
F
.
functio
(45)
C
1
=
x
;
2
x
+
1
(
x
)
2
2
1
3
1
wher
e
identc
is
al
a
ly
Then,
C
zer
functio
with
sup
ort
c
ontaie
d
in
(1
;
2)
whic
is
o.
ther
e
is
one
and
only
one
invart
manifold
whic
is
a
log
of
a
b
ounde
d
functio.
log
but
not
Pr
o
of.
not
3
=
F
log
2+
or
r
evry
>
0
,
this
functio
is
3
gr
=
aph
log
C
2
C
.
The
equation
for
in
v
arince
is
1
(46)
(1
=
2
x
)
=
(
x
)
1
+
(
x
1
)
1
3
P
i
Again,
w
e
write
=
wher
has
i
i
The
equations
for
2
sup
i
Z
are
(47)
i
(
i
+1
x
)
1
=
3
(2
i
x
)
1
ort
on
.
(2
i
;
2
+1
).
30
R.
when
i
6=
1
de
la
Lla
v
e
and
(48)
(
x
)
1
Aplying
(47)
rep
=
3
(2
1
x
)
0
eatdly
,
w
e
+
(
(
x
ha
)
n
Henc,
the
0.
only
This
p
can
b
v
e
=
n
3
(2
1
osiblt
e
x
)
0
y
to
arnged
w
e
e
c
:
1
mak
if
)
1
n
x
1
ho
b
ose
ounde
at
=
1
1
is
to
ha
v
e
.
0
Then,
w
e
are
forced,
aplying
(47)
,
n
(
x
)
n
=
to
ha
v
1
3
n
+1
n
+1
e
(2
1
x
)
1
1
(49)
n
=
1
3
(2
x
)
1
:
1
log
Unles
the
is
iden
ticaly
zero,
w
e
se
tha
(49)
is
not
3
=
log
2+
C
.
Note
tha
w
Example
e
5.4
can
run
run
in
the
the
argumen
op
t
osite
in
the
pro
directon.
of
of
That
the
is,
staemn
w
e
ts
can
tha
log
the
only
in
a
w
a
neigh
y
b
to
obtain
orh
o
a
d
of
functio
the
satifyng
orign
is
(46)
to
ha
v
e
whic
0
h
is
whic
in
argue
3
=
log
2+
C
h
.
in
turns
forces
1
exp
one
tial
gro
Henc,
wth
at
Example
ditons
5.4
in
Unles
Remark
W
e
4.1,
zero,
ma
linear
cut
manifolds
pro
Note
o
map
the
are
t
so
the
map
the
o
to
uniqes
sp
eak
fail.
Section
ounde
in
space.
in
to
in
b
linear
orthgnal.
goin
indcate
course,
preations
con-
is
as
Of
just
w
other
(45)
map.
the
the
are,
satied
linear
case,
h
4.18
is
the
pseudo-tabl
This
Section
v
sho
4.1
arin
t
ws
do
tha,
aect
the
duce.
also
the
e
whic
Remark
them
w
just
the
in
in
of
if
.
exampl
and
obtain
for
the
an
one
tha
y
manifolds
in
is
if
note
e
y
4.1
is
also
w
int
tha
manifold
in
is
Example
more
5.4,
regula
w
than
e
ha
the
v
e
the
phenom
crital
tha
regulait
y
log
3
=
once
log
2
then
1
it
is
C
W
.
e
also
note
tha
in
the
case
of
the
linear
map,
log
itely
man
y
in
v
arin
t
manifolds
x
=
A
j
x
2
Henc
uniqes
do
es
not
W
m
e
do
not
ulation
form
for
[CFdlL03a
ev
ulate
Resul
en
in
ts
f
precisly
diern
v
log
w
e
ha
v
e
2
in-
log
whic
the
or
h
crital
flo
the
tiable
=
are
3
=
log
.
regulait
for
elds
.
ws
result
ector
y
is
o
quite
ws
since
the
standr.
for-
Se
e.g
].
In
6.1,
w
3.1
when
Section
for
maps
e
ha
v
w
e
imply
some
e
wil
presn
t
uniqes
a
result
result
for
v
rigous
argumen
for
ector
2
C
1
hold
6.
3
j
elds.
the
t
maps,
the
tha
sho
result
ws
Theorm
tha,
In
The
v
main
result
eris
the
under
tha,
t
map
manifolds
31
vide
of
time-1
arin
pro
asumption
the
o
is
v
tha
the
Theorm
pro
3.1,
duce
b
y
time-1
map
the,
the
Theorm
of
the
manifold
3.1
is
o
in
in
v
arin
t
v
w
arin
t
under
the
w.
Ev
en
for
if
o
to
the
ws
p
b
oin
6.2
t
w
the
e
argumen
y
in
Section
them
tha
it
presn
is
t
result
p
an
for
easily
t
reducing
ws.
h
W
{
e
e
rigous
a
it
for
of
read
of
think
of
pro
the
e
pro
rigous
tha
pro
w
out
a
a
maps,
cary
not
hop
es
for
to
etc
giv
result
osible
sk
o
6.1
to
o
{
e
tersing
In
a
b
result
in
ws.
of
wil
of
is
Section
direct
pro
able
to
of
l
of
details
.
6.1
Deducing
Giv
result
en
a
o
w
f
S
for
g
ws
genrat
t
the
o
t
asumption
2
b
from
y
a
Theorm
smo
oth
3.1
v
ector
eld
for
Y
,
w
maps.
e
note
tha
R
of
Theorm
3.1
can
b
e
satied
for
al
the
maps
S
.
t
W
e
furthemo
ha
v
e
tha
D
S
(0)
=
exp
(
tD
Y
(0).
Henc,
if
D
Y
(0)
has
t
an
in
v
arin
t
decomp
ositn
so
do
es
D
S
(0).
W
e
wil
denot
D
Y
(0)
=
A
t
and
denot
the
W
e
also
decomp
note
ositn
tha
as
the
result
in
(14)
in
.
Section
4.1
can
b
e
easily
adpte
for
r
o
ws.
Namely
,
b
y
scaling
w
e
can
ensur
tha
j
D
S
(
)
exp
(
tA
)
j
t
C
(
X
)
1
is
as
smal
as
In
the
desir
v
B
(
1
a
If
e
[0
;
eld
)
and
w
to
tha
1].
e
can
pro
ensur
ced
tha
therfo
as
the
w
e
v
can
in
ector
a
Section
4.1
eld
den
S
is
in
2
w
w
ector
X
2
h
2
case,
the
suc
t
pseudo-tabl
cut-o
X
for
X
tha
it
aply
is
close
to
Theorm
3.1
to
ounde
t
y
b
in
B
(
1
X
)
1
in
2
linear.
to
eac
h
of
the
maps
S
w
e
obtain
an
in
v
arin
t
t
manifold.
The
manifolds
only
are
W
e
thing
the
note
tha
same
tha
w
for
since
al
S
Æ
S
,
then
so
is
S
w
can
v
e
to
w
diern
=
ory
ab
t
S
Æ
s
M
t
If
ha
the
S
t
under
e
S
,
s
v
if
out
alues
a
is
of
t
whetr
thes
.
manifold
M
is
in
v
arin
t
t
.
s
e
ha
v
e
some
conlude
uniqes
tha
staemn
S
M
=
M
t
.
That
is
for
in
M
v
is
in
arin
v
t
manifolds,
arin
t
w
for
the
whole
s
o
w.
In
if
the
stable
M
is
case,
tange
the
t
at
uniqes
the
staemn
orign
to
X
t
,
then,
tha
w
cleary
e
can
so
use
is
S
1
S
M
is
also
as
regula
as
M
,
is
M
.
tha
Since,
s
then,
the
conlusi
of
Remark
4.1
s
alo
w
In
us
to
conlude
the
the
desir
pseudo-tabl
graph
of
result.
case,
a
functio
:
the
X
!
observ
ation
X
2,
whic
is
h
is
tha
since
M
uniformly
b
is
the
ounde
and
1
S
diers
from
exp
sA
b
y
a
map
of
smal
Lipsc
hitz
consta
t,
w
e
obtain,
s
therfo
tha
S
M
is
also
a
graph.
s
Using
S
M
s
=
the
M
.
uniqes
staemn
ts
in
Remark
4.18,
w
e
obtain
tha
e
32
R.
Of
course,
used
for
diern
Nev
eac
t
w
y
b
e
e
p
oin
t
b
diern
6.1
es
not
only
An
use
use
v
ery
etc
w
e
erfctly
w
sho
logica,
wn
in
t
e
of
ha
v
Example
5.4,
uniqes
e
course.
the
argumen
ts
ation
y
the
ector
v
argumen
v
t
W
e
tiable.
tial
tin
course
e
tiable.
Diern
discon
of
o
diern
diern
artil
henc
is,
is
is
P
ab
X
eld
in
{
This
the
eld
en
ounde
ws.
tha
ector
v
hap
b
is
v
equations
uos
ery
{
w
el
op
eratos
kno
wn
for
a
direct
a
pro
of.
rigous
pro
Ev
tiong
tha
of
one
en
for
if
o
can
the
ws
also
argumen
from
t
giv
the
e
a
presn
ted
result
direct
for
pro
of
in
maps,
of
it
the
result
ws.
The
metho
metho
d
d.
pro
b
p
diern
the
un
o
a
es
men
o
case
is
[Sho97].
of
giv
orth
for
pseudo-tabl
the
observ
this
has
oth
e.g
h
t
b
one
se
6.1
is
the
wil
of
tha
genratd
smo
{
Section
e
This
as
h
fact
of
ery
time
Sk
tha,
exampls
often
v
6.2
or
eac
ortan
the
w
t
genrat
long
al
o
ortan
wher
v
ts.
the
imp
at
the
Imp
stable
out
y
Lla
t.
Remark
do
the
la
staemn
selctd
ma
of
uniqes
erthls
manifolds
h
de
F
of.
or
Ev
presn
ofs
W
e
the
t
of
y
b
the
usal
P
w
not
e
b
usefl
ws
e
for
than
the
a
in
presn
t
pro
reads
the
tial
e
eron's
complet
the
with
diern
one
of
completns,
wil
o
write
arin
e
this
for
comp
v
sak
if
ma
pro
a
the
en
taion
the
is
ofs
equations
w
are
for
equation
sk
of,
who
pro
an
tegral
etc
h
e
of
hop
a
e
more
direct
tha
this
familr
with
dieomrphs.
genrati
the
o
w
separting
ts.
x
_
=
A
x
1
+
1
B
x
+
1
N
(
2
x
;
1
x
);
1
2
(50)
x
_
=
A
x
2
As
it
is
graph
standr,
is
F
or
in
w
v
the
arin
diern
t,
sak
e
of
tiable,
with
smo
oth
if
the
;
w
);
2
an
tha
,
x
1
e
w
sem
only
x
deriv
sho
it
(
2
rst
wil
y
en
equation
for
the
e
asume
the
some
of
the
aproite
v
gro
whose
a
tha
tha
functio
has
tha
(satifyng
a
equation
wil
clear
asumption
semigroup
detail
wil
e
N
2
simplct
ev
throug
e
w
+
2
the
v
solutin.
ector
eld
result
ector
w
eld
wth
is
ould
go
genrats
a
condits,
whic
h
w
later).
e
W
e
note
tha
if
w
e
ha
v
e
x
=
(
x
)
2
x
_
=
A
x
1
+
1
w
e
ha
v
e
1
B
(
x
)
1
+
N
(
1
x
1
;
(
x
);
1
1
(51)
_
(
x
)
=
A
1
W
x
.
e
If
note
tha,
is
if
Lipsc
(
x
w
2
e
hitz,
x
w
e
,
se
)
+
N
(
1
x
2
the
rst
tha
the
;
(
x
1
);
1
equation
b
rst
ecoms
equation
an
of
ODE
(51)
wil
for
ha
v
uniqe
1
solutin.
W
e
denot
the
condits
x
b
y
(
1
x
).
1
t
solutin
of
this
equation
with
intal
e
a
In
W
e
ha
v
e
v
arin
t
manifolds
3
tha
A
t
"t
1
j
(
x
)
j
k
e
k
e
t
0
1
t
wher
"
men
is
ts
a
in
The
n
of
b
W
e
b
er
h
to
the
y
use
P
pro
arbitly
e
of
smal
smal
is
the
same
d
ariton
of
if
w
e
c
ho
ose
the
arnge-
enough.
metho
v
and
is
b
eron's
the
(51)
whic
4.1
idea
manifold
of
um
Section
as
(se
the
standr
e.g
pro
[Hal80
parmets
]
form
Ch.
of
VI
ula
in
of
I
the
the
p.
stable
25
.
second
).
equation
obtain
Z
t
A
t
A
(
2
(52)
(
(
x
)
=
t
s
)
2
e
(
x
)
1
+
ds
e
N
(
1
(
x
)
2
;
(
(
x
)
1
t
1
s
s
0
Equiv
alen
tly
,
Z
t
A
t
sA
2
(53)
2
e
(
(
x
)
=
(
x
)
1
+
ds
e
N
(
1
(
x
)
2
;
(
(
x
)
1
t
1
s
s
0
In
the
stable
case,
w
e
note
tha,
if
w
e
consider
functios
whic
h
are
2
C
and
satify
(0)
=
A
0,
D
t
(0)
=
0,
w
e
ha
v
A
(54)
j
e
t
2
A
t
2
e
(
(
x
)
j
C
k
"t
2
1
e
k
(
k
e
k
e
)
1
t
The
condit
A
t
A
t
2
(5)
k
for
some
cleary
>
an
0,
Under
es
of
to
as
t
k
e
RHS
k
of
(5),
!
graph
(
1
of
.
2
e
)
(5)
C
t
go
e
es
to
zero
exp
one
tialy
,
is,
(1).
asumption
0
the
k
the
of
this
go
tha
anlogue
"t
1
e
w
Henc,
w
e
e
obtain
tha
obtain
the
tha
a
rst
term
condit
in
for
in
v
(53)
arince
is
Z
1
sA
2
(56)
(
x
)
=
ds
e
N
(
1
(
x
)
2
;
(
(
x
)
1
1
s
s
0
Note
also
tha
(5)
and
the
quadrtic
v
anishg
of
N
,
at
the
orign
2
also
imply
tha
Henc,
the
w
den
b
A
y
e
tegral
in
consider
the
usefl
in
wil
the
(56)
RHS
as
a
of
xed
(56)
p
con
oin
t
v
ergs
uniformly
equation
for
.
the
op
erato
RHS.
heuristc
guide
is
tha
this
op
erato
is
v
ery
simlar
to
Z
1
sA
A
s
2
ds
A
s
1
e
N
(
1
e
(
x
)
2
;
(
e
x
)
1
1
0
Again,
if
ator
w
e
tak
obtained
has
a
b
the
T
o
norm
sho
conial
e
w
whic
tha
space
y
with
comp
h
osing
is
the
norm
w
b
y
T
more
norms,
the
b
erato
ws
ted
on
ounde
op
fol
eigh
righ
the
t
les
a
obtain
con
the
e
line
op
er-
functio
tracion.
when
same
the
traciv
con
tracion
the
tha
a
of
con
e
with
square
is
or
w
of
top
argumen
olgized
t
b
than
y
34
R.
in
the
case
til
w
of
e
get
dieomrphs,
to
The
pro
de
Lla
v
e
namely
the
of
la
heuristc
of
ading
and
subtracing
terms
princle.
o
ws
is
sligh
tly
more
compliated
than
in
the
case
of
A
s
1
dieomrphs
since
w
e
ha
v
e
to
obtain
estima
for
(
x
)
e
x
.
1
1
s
Of
course,
thes
tions
of
can
ODE's
anlogues
for
W
in
to
m
ulas
e,
of
obtained
from
the
but
they
dep
tak
endc
e
on
longer
to
the
solu-
write
than
the
dieomrphs.
also
es
for
ha
but
the
v
e
the
hig
deriv
to
sho
pro
w
of
deriv
ativ
The
e
parmets,
course,
themslv
higer
b
on
tha
is
ativ
the
op
extrmly
es
erato
maps
simlar
of
the
op
the
once
erato
w
T
whic
e
h
space
obtain
for-
separt
the
es.
pseudo-tabl
case
is
in
man
y
w
a
ys
easir.
In
this
case,
w
e
canot
asume
tha
is
a
con
tracion,
so
w
e
ned
to
mak
e
sure
tha
s
the
functios
ha
v
e
domain
in
al
of
X
and,
therfo
tha
the
N
are
1
smo
oth
in
al
of
X
B
(
1
X
).
1
This
is
done
in
Section
4.1
2
sA
2
Since
w
e
are
asuming
tha
e
is
a
con
tracion
for
large
s
,
ther
is
0
no
problem
sho
Also,
using
sho
w
wing
tha
the
tha
the
same
the
a
w
ork
space
Dean's
of
m
felo
y
at
Seminar,
out
v
ab
ery
the
in
en
to
con
tracion
in
ativ
es,
themslv
it
C
is
p
norm.
osible
to
es.
sup
of
T
orted
exas
F
al
20,
the
made
a
gran
W
v
v
b
studen
ts
alube
y
of
comen
to
reading
and
Dynamicl
also
careful
ts
orking
ery
Thanks
and
NSF
The
in
materil.
y
Austin.
ts
this
discuon
b
at
particn
exas
of
tenig
b
the
taion
a
deriv
wledgmnts
Univ.
U.T
presn
enligh
ckno
and
is
hig
ed
has
course
System
A
(56)
the
map
author
wship
gradute
for
get
the
of
ulas
7.
The
RHS
form
of
E.
F
the
ts
on
pa
tic
h
for
er.
Refrncs
[CFdlL03a]
Xa
vier
Cabr
terizaon
e,
Ernest
F
metho
no-resa
d
t
MP
AR
[CFdlL03b]
Xa
C
vier
tic
in
v
subpace.
h,
and
arin
Rafel
t
Ind.
de
manifolds
la
I:
Univ.
Math.
Lla
v
e.
The
manifolds
parme-
aso
Jour
,
ciated
to
52():83{,
203.
#02-51.
Cabr
terizaon
e,
Ernest
F
metho
to
on
for
d
parmets.
for
on
Ind.
in
tic
v
h,
and
arin
Rafel
Univ.
t
de
manifolds
Math.
I
Jour
,
la
I:
Lla
v
e.
The
regulait
parme-
y
52():39{60,
with
resp
203.
MP
ect
AR
C
#02-51.
[CFdlL03c]
Xa
vier
Cabr
izaton
metho
In
pr
[DGZ93]
ep
ert
r
Rob
enormigs
e,
Ernest
F
ar
d
ation
for
,
in
on
tic
h,
arin
t
and
Rafel
de
manifolds
la
I:
o
Lla
v
erviw
v
e.
The
and
parmet-
aplictons.
203.
Devil,
Giles
in
v
Banch
Go
defro
sp
ac
y
es
,
v
,
and
olume
V
acl
64
v
of
Zizler.
Pitman
Smo
Mon
othnes
and
gr
aphs
and
In
Surveys
in
nical,
[dlL97]
e
w,
Rafel
de
tral
[dlL
Pur
Harlo
la
Lla
de
la
Mohamed
7,
Mathemics
arin
.
t
v
manifolds
Longma
Scien
aso
Phys.
e
and
,
C.
.
ciated
tic
&
T
ec
h-
W
a
Math.
yne.
On
Z.
Sub-stale
and
sp
noresa
t
sp
ec-
197.
Eugen
t
to
87(1-2):{49,
theorm.
no-resa
k
K.
Hale.
lishng
Or
Co.
[HP70]
W.
b
olic
dinary
Inc.,
Moris
w
ectral
,
Irwin's
pro
of
219():30{,
of
the
195.
eak-stbl
manifolds
subpace.
h
In
ential
tingo,
e
N.Y,
and
Glob
aso
Math.
A
C.
nalysi
1968)
Z.
.
pages
Rob
ert
,
ciated
236(4):71{
c.
Pub-
manifolds
Symp
os.
13{6.
Krieg
1980.
Stable
o
E.
editon,
Pugh.
(Pr
,
quations
second
Charles
al
Calif.,
R.I,
dier
Hun
Hirsc
set.
Berkly,
and
Pur
Amer.
e
h
Math.,
V
Math.
So
c.,
yp
ol.
er-
Pro
XIV,
videnc,
1970.
[HPS7]
M.W
Hirsc
V
h,
erlag,
[Irw80]
C.
[JPdlL95]
new
Math.
M.
Y
c.
a.
B.
termdia
P
osi
,
Kato.
Oscar
and
Da
ork
of
vid
159{2,
E.
theorm.
Lla
v
e.
On
the
in
dieomrphs.
for
J.
tegrabil
y
Er
line
T
ak
c
op
e
p
ens.
erio
In
dings
a
ar
go
dic
of
The
Clasic
ory
in
the
1980
in
to
in
Daniel
pr
in
t
Physic
al
instue
Notes
arin
Joseph,
the
sumer
Lectur
v
D.
oblems
le
Math-
editon.
solutin
Stakgold,
Batel
erlag.
.
of
dic
Iv
ar
of
Springe-V
ators
t
Nonlie
e
er
Reprin
editors,
o
1973.
ar
195.
of
Pr
I.
In
sytem.
Houches,
In
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Da
vid
.
R.
,
Ruel.
ory
Sho
w
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in
pages
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Sh
Y
ub.
1987.
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T
a
es
:
lave@mth.ux
dier
entiabl
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dy-
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es
1983.
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of
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op
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ork,
Texas
theory
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and
bifur
c
ation
the-
198.
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in
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sp
ac
e
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nolie
So
ciet
ar
y
,
Pro
vi-
197.
hael
New
matheicl
of
Bostn,
Monte
e
the
ehavior
North-Hland,
of
ential
RI,
Mic
to
3{51.
Inc.,
alter.
dier
denc,
duction
b
pages
Pres
E.
tro
Chaotic
Elemnts
Academi
artil
u87]
la
v
Bifurcaton
gy:
Lanford
namicl
p
Springe-
583.
32.
Oscar
[Sho97]
ol.
manifold
de
ory
and
biol
Berlin,
ol.
[Rue89]
R.
Berlin,
Satinger,
and
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V
195.
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H.
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the
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ation
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E.
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the
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V
of
esin,
Pertub
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[LI73]
of
for
System
T
ub.
in
21:57{6,
distrbuon
Dynam.
[Kat95]
,
Sh
Notes
pro
So
Jiang,
in
M.
Lectur
A
ond
and
197.
Irwin.
L
Pugh,
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M.C
adr
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201.
Jac
of
v
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[Hal80]
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In
Stais.
Lla
S.
with
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d
manifold
[ElB01]
Univ.
e.
J.
Rafel
t
Aplie
v
pseudotabl
[Sh
and
arin
193.
subpace.
W95]
v
t
al
stabily
of
With
ranslted
the
from
A
ustin
A
dynamic
al
colab
the
oratin
ustin,
F
TX
s.
edu
renc
h
7812-0,
sytem
of
b
y
.
Alb
Joseph
ert
U.SA
Christ
Springe-V
F
erlag,
athi
y
and
.
R
emi