Publicacions
Ma emá iques,
Vol
37
(1993),
255-269
.
DIFFEOMORPHISMS
OF
Rn
WITH
OSCILLATORY
JACOBIANS
WALDYR
M
.
OLIVA
*
,
NELSON
M
.
KUHL
AND
LUIz
T
.
MAGALHAES
A
bs ac
The
pape
p esen s,
mainly,
wo
esul s
:
a
new
p oo
o
he
spec-
al
p ope ies
o
oscilla o y
ma ices
and
a
ans e sali y
heo em
o
di eomo phisms
o
R'
wi h
oscilla o y
jacobian
a
e e y
poin
and
such
ha
Nm( (x)
--
(y))
<
NM(x-y)
o
all
x, y
E
R',
whe e
Nm(x)
-
1
deno es he
maximum
numbe
o sign
changes
in
he
componen e
zi
o
zE
R",
whe e
all
zi
a e
non
ze o
and
z
a ies
in
a
small
neighbo hood
o
x
.
An
applica ion
o
a
semi-
implici
disc e iza ion o
he
scala
hea
equa ion wi h
Di ichle
bounda y
condi ions
is
aleo
made
.
I
.
In oduc ion
The
p esen
pape
deals
wi h
di eomo phisms
:
Rn
->
R'z
such ha
(x)
is
an
oscilla o y
ma ix
o
all
x E
R'
.
Oscilla o y
ma ices
we e
s udied
ex ensi ely
by
Gan mache
and
K ein
(see
[G
K ])
and
p esen
e y
in e es ing
spec al
p ope ies
.
They
belong
o
he
class
o
ma ices
ha
a e
a ia ion-diminishing,
Le
.,
ans o ma ions ha dec ease
he
numbe
o
sign
changes
in
he
componen s
o a
ec o
.
These
ma ices
ha e
applica ions
in
mechanical
sys ems,
in
app oxima ion
heo y
and
in
p obabili y
.
Many
esul s
abou
oscilla o y
ma ices
as well as
his o ical
no es
and
a
long
lis
o
e e en es
appea
in
he
book
"To al
Posi i i y",
by
S
.
Ka lin
(see
[Ka])
.
In
Sec ion
II
we
decided
o
p esen
ano he
p oo
o
he
spec al
p op-
e ies
o oscilla o y
ma ices,
summa ized
in
Theo em
2
.12
;
ou
app oach
is
based
on
some
ideas
appea ing
in
[FO-1]
and
[FO-2],
whe e
o he
kind
o
ma ices
we e
also
s udied
.
Theo em
2 .13
shows
ha
di eomo phisms
wi h
oscilla o y
jacobians
sa is ying
an
addi ional
hypo hesis
ha e
he
*This
esea ch
was
suppo ed
in
pa
by
he
"P oje o
BID-USP"
and
by
FAPESP,
p os
.
n°
90/3918-5
.
256

W
.
M
.
OLIVA,
N
.
M
.
KUHL,
L
.
T
.
MAGALHAES
p ope y
ha
gi en
wo
hype bolic
ixed
poin s,
he
co esponding
un-
s able
and
s able
mani olds
a e
ans e sal
.
The
p oo
ollows
some
echniques and, as
a
ma e
o
ac ,
was
mo i a ed
by
he
main
esul
o
[FO-1]
whe e
i is
s udied
a
class
o o dina y
di e en ial
equa ions
whose
low
map
a
a
ixed
ime
is
a
di eomo phism
wi h
oscilla o y
jacobian
.
In Sec ion
III,
a e
a
semi-implici
double
disc e iza ion
o
he
hea
equa ion
wi h
Di ichle
bounda y
condi ion,
we
cons uc ed
a
class
o
Mo se-Smale
di eomo phisms
o
R'
wi h
oscilla o y
jacobians
.
The
i s
au ho
wan s
o
hank
Jaume
Llib e
and
Ca les
Simó
by
he
in i a ion
o
isi
he
"Cen e
de
Rece ca
Ma emá ica"
o he
"Ins i u
d'Es udis Ca alans"
whe e
in
Oc obe
1990
he
de eloped
his
con ibu ion
o
he p esen
wo k
.
II
.
De ini ions
.
Basic
esul s
.
Main
Theo em
All
he ma ices
we
will
use a e
eal
.
2
.1
.
No a ion
.
Fo
a
m
x
n
ma ix
A,
le 's
use he
no a ions
1
<
<
min(m,
n)
;
2
.2
.
De ini ion
.
o
all
1
<
<
min(m,
n),
a22
,
,
aj
.
J
1<21<i2<
.
.
.<i

<m
-
,1< ,2<
. .
.<
<n
.
A
m
xn
ma ix
A
is
called
o ally
posi i e
(s ic ly
o ally posi i e)
i
A
(
il

i2

.
. .

a
)
>
0

( esp
.
>
0)
,1
ice
Le
.,
i
all i s
mino s
a e
non-nega i e
( esp
.
posi i e)
.
A
squa e
ma ix
A
is
called
oscilla o y
i i is
o ally
posi i e
and
i
some
o
i s
powe
AX
(X
a
posi i e
in ege )
is
a
s ic ly
o ally
posi i e
ma ix
.
2
.3
.
The
p oduc
C
o
wo
o ally
posi i e
ma ices
A
and
B
is
a
o ally
posi i e
ma ix
.
The
p oduc
C
o
wo
squa e
ma ices
A
and
B,
whe e
one
is
s ic ly
o ally
posi i e
and
he
o he
is
o ally
posi i e
and
non-singula
is
a
s ic ly
o ally
posi i e
ma ix
.
i2
. . .
i
aZaK1
ai2 -2
. . .
)
-
de
K2
. . .
K
La
¡
,
.
.,
ai

w2
. .
.
DIFFEOMORPHISMS
WITH
OSCILLATORY
JACOBIANS

257
2
.4
.
The
p oduc
C
=
AB
o
wo
oscilla o y
ma ices
is
an
oscilla o y
ma ix
.
The
p oduc
A
=
A
j
A
2
. . .
A

,,
o
m
>_
n
-
1
n
x
n
oscilla o y
ma ices
is
a
s ic ly
o ally
posi i e
ma ix
.
Le
A
be
a
o ally
posi i e
squa e
ma ix
.
Then,
A
is
oscilla o y
i ,
and
only
i ,
A
is
non-singula
and
aij
>
0
o
1
i
-
j1
=
1
.
2
.5
.
Fo
a
ma ix
A,
le
A*
deno e
he
ma ix
de ined
by
a*,
=
Then,
i
A
is
an
oscilla o y
ma ix,
he
same
is
ue
o
(_1)i+ia
ij
.
(A*)
-1
.
2
.6
.
Le
J=
al
b1

0
cl a2
b2
bn-1
0
C
n
-1
an
be annxn
Jacobi
ma ix
.
Then, J
is
ocilla o y
i ,
and
only
i ,
he
mino s
j(1
2
. . .
)
,
1
<
<
n, aj
,
1
<
j
<
n,
and
he
coe icien s
bi
and
ci, 1
<
i
<n-
1,
a e
posi i e
(>
0)
.
Following
Fusco
and
Oli a,
le
us
de ine
now
he
disc e e
unc ional
al eady
conside ed
in
[FO-1]
.
(See
also
[K-L-S])
.
Fo
x
=
(x
1
,
.
. . ,
xn)
E
Rn
such
ha
x
i =,A
0,
1
_< i _<
n,
le
N(x)
-
1
be
he
numbe
o
sign
changes
in
he
sequence
x1,x2'
. .
.'
x
n
.
Fo
an
a bi a y
x E Rn,
de ine
.
,,(x)
and
Nm(x)
as
he
minimum
and
maximum
alue
o N(x'),
o
x'
a ying
in
a
small
neighbou hood
o
x
wi h
(x')
i
0
0,
1
_<
i_<
n
.
Ex end
N
o
he
se
N
=
{x
E
Rn
N,,(x)
=
Nm(x)}
making
N(x)
=
N,,(x)
=
Nm(x)
o
all
x E
N
.
N
is
he unc ional
used
in
[FO-1]
.
The
nex
wo
esul s
can
be
ound
in
[Ka],
and
o
comple eness
we
ep oduce
he p oo s
.
2 .7
.
P oposi ion
.
Le
A
be
an
n
x
m
ma ix
(m
<
n)
such
ha
all
he
mino s
A
(i1
2
.
.
.
m
a e
nonze o
and
Na e
he
same
sign,
.
.
.
)
and
le
xE
R
m
,
x
=,,~
0
.
Then
y
=Ax
sa is ies
NM
(y)
<
m
.
P oo
.
.
Suppose,
a guing
by
con adic ion,
ha
he e
is
a
nonze o
ec o
x
E
Rm
such
ha
Nm(y)
>_
m
+
1
(y
=
Ax)
.
Le
y
=
(
y
1
,
y
2
. . . .
,
yn)
.
Then
he e
exis
indices
il
<
i2
<
.
.
.
<
in,
.+1
wi h
he
p ope y
ha
o
=
1, 2,
.
.
.,m+
1
he
quan i y
(-
1)'y'-
is
o
cons an
sign,
e en
in
258

W
.
M
.
OLIVA,
N
.
M
.
KUHL,
L
.
T
.
MAGALHE E
S
he
possible
case
whe e
se e al
o
he e
numbe s
a e
ze o
.
Now
sople
o
he
y'-'s
a e
nonze o
;
o he wise,
i
y
i1
,
y'2,
. . . ,
y'- a e
all
ze o,
hen
he
sys em
o
homogeneous
equa ions
has
a
non i ial
solu ion,
which
implies
in
con adic ion
wi h
he
hypo hesis
.
Conside
now
he
de e minan
P oo
.
ai,7x7
=0

=1,2,
. .
.,m
A
21
i2
. . .
i
n
,
__
(

0
1
2
. . .
m)
This
de e minan
is
ze o,
since
yi

=
elemen s
o
he
las
colunln,
we
ob ain
1
.
(
-
1)m+1+
y
i

A
21
i2
. . .
1
2
. . .
m1
a?
.jx3
.
Expanding
by
he
-1
i +1
. . .
%m+1
.
Now,
since
he
y's
al e na e
in
sign,
some
o
hem
being
nonze o,
and
since
he A's
all
Na e
he
same
p ope
sign,
he
igh hand
side
canno
be
ze o
.
This
con adic ion
comple es
he
p oo
.
2
.8
.
Lemma
.
Le
A
be
an
n
x
m
s ic ly
o ally
posi i e
ma iz
and
le
x
E
R"1, x
:7~
0
.
Then
y
=
Ax
sa is ies
NM(y)
<
Nm(x)
.
Le
p
+
1
=
Nm(x)
.
Then
he
componen s
o
x
can
be
di ided
in o
p
+
1
g oups,
1
2
l
+1
+2
p

y
+1
p
+
2
m
(x
,x
,.
.
.
.
x

),
(x"

,XVI

,
. .
.,x

),
.
.
.,(x

,x

,. .
.,x

)
whe e
each
componen
in
he
i h
g oup,
say,
ei he
is
ze o
o
has
a
sign
(-1)i+1
.
F~z he mo e,
he e
mus be
a
leas
one
nonze o
componen
in
ai11
aá12
.
. .
ailm
y
i
1
ai
2
1
ai22
...
ai2m
y
i2
ai_+1
1
aim+12
.
.
ai~,
+1
m
.

Zm+1
DIFFEOMORPHISMS
WITH
OSCILLATORY
JACOBIANS

259
each
g oup
.
Unless
n
>
p
+
1,
which
we
assume
o
be
he
case,
he e
is
no hing
o
p o e
.
We
se
o
=
0,
p+1
=
m, and
cons uc
he
ec o s
and
o em
.
k
=
j= k-i+l
V
(i,

i2

. .
.

ip+1
1

2

. .
.

p+
1
)
_ji
1
Ixi
laiij
Qi=1
22= 1+1
whe e
aj
is
he j h
column
ec o
o
A
.
Then
_",
~1
IxjIaln+lj

.
.
.

Ej=
m
~,+1
Ixj
I
a
in+ij
21
22
. . .
2p+1
V
(1
2
.
.
.
p+1)
k=1,2,
. .
.,p+1
p+1
(
-
1)k+1

Ixjlaj
=(-1)k+l k
j=1
k=1
j= k-1+1
k=
m
j=
p
+
1
I
x
J
Iaáij
l
x
e1
1

X121
. . .
ix
Q
+
1
JA
~
zli2
. . .
p+1

)~
L

Q1~2
. . .
2P+1= n+i

~p+1
whe e
V
is
he
ma ix
whose
k h
column
ec o
is
k
.
By
he
na u e
o
he
cons uc ion
o
he
blocks
weknow
ha
he e
is
a
leas
one
selec ion
(~1,
~a,
. . . ,
Qp+1)
o
which
Ixel
1
X121
. . .
IxeP
1
>
0
.
Thus
all
he mino s
a e
posi i e
.
Now
y
=
l
:
P
Ci(-1)k+l
k
and
since
V
sa is ies
he hy-
po hesis
o
P oposi ion
2
.7,
we
ob ain
NM(y)
<
p-1-
1
=
N,,
(x)
.
Lemma
2
.8
will
be
c ucial in
he
p oo
o
he
spec al
heo em
o
oscilla o y
ma ices
.
Be o e
we
go
o his
spec al
heo em,
le
us
p o e
ano he
change-sign
esul ,
Lemma
2
.9,
use ul
o
he
ans e sali y
he-
2
.9
.
Lemma
.
Le
A
be a
nonsingula
o ally
posi i e
n
x n
ma ix
.
Then
NM(Ax)
<
Nm(x)
o
any x E
Rn
.

260

W
.
M
.
OLIVA,
N
.
M
.
KUHL,
L
.
T
.
MAGALHAES
0
and
le
K
1
be
he
se
0
1
E
0
J(E)
=

E

1

E

.
E
(0
E1
By
con inui y
and
by
2
.6
he e
exis s
Eo
>
0
such
ha
J(E)
is
oscilla o y
o
any 0
<
E
<
Eo
.
Hence,
by
2
.4
B(E)
=
J(E)n
-1
is
a
s ic ly
o ally
posi i e
ma ix
o
any 0
<
E
<
Eo
and
using
2
.3
and
con inui y
we
conclude
ha
C(E)
=
B(E)A
is
s ic ly
o ally
posi i e,
0
<
E
<
Eo,
and
lim,-o
C(E)
=
A
.
Thus,
o
E
su icien ly
small,
N,,,(Ax)
<
N
.
.(C(E)x)
<
NM(C(E)x)
and by
Lemma
2
.8,
applied
o
C(E),
we
ob ain
o
x
SA
0
N
.(Ax)
<
,.
.x)
.
Now,
since
A
is
nonsingula ,
he e
exis s
a
neighbou hood
U
o
x
such
ha
N
,(x)
=

max
N(x')
and
NM(Ax)
=

max
N(y)
.
x~EuniV

yEAUnN
Thus
NM(Ax)
=
N(A
.T)
o
some
xE
U
and
by
(*)
N(A
. )
<
N,,~(x)
<
Nm(x)-
Since
x E
U,
Nm(x)
<
Nm(x)
and
he
lemma
is
p o ed
.
Following
[FO-1]
we
in oduce
now
a
amily
o
" ones"
in
Rn
which
plays
an
impo an
ole in
he
spec al
and
ans e sali y
heo ems
.
2
.10
.
De ini ion
.
Fo
any
gi en
in ege
1
<
i
<
n
le
K
i
be
he
se
Kz={XERn
:NM(x)<i}
K1
=
{0}
U
in
Ki
=
{0}
U
{x
E
Rn
:
Nm(x)
<
i}
.
Rema k
.
No e
ha
by
Lemma
2
.9, i
A
is
o ally
posi i e
n
x
n
ma ix,
he
se s
K
i
and
{0}
U
(Rn K¡)
a e
in a ian
unde
A
and
A-1,
espec-
i ely
.
Also,
K1 1{0}U(R
n
Ki))
_
{0}
and
C$[KiU{
0
}UR
n Ki)]
=
Rn
.
Aswe
will
see,
he e
in a ian
and
ans e sali y p ope ies
will
be
use ul
o
p o e
he
ans e sali y
heo em
.
We
s a e
now,
wi hou
p oo ,
he
gene aliza ion
o
Pe on's
Theo em,
wi h
he
e sion
due
o
C
.
Fusco
and
W
.
M
.
Oli a,
which,
oge he
wi h
Lemma
2
.9,
will
be
he
ools o
p o e
he
spec al
heo em
o
oscilla o y
ma ices
.
P oo
..
Le
J(E)
be
he
n
x
n
Jacobi
ma ix
DIFFEOMORPHISMS
WITH
OSCILLATORY
JACOBIANS

261
2
.11
.
Theo em
([FO-2])
.
Le
E
be
an
n-dimensional
eal
ec o s
space,
le
K
C
E
be
a
closed
se
wi h
non emp y
in e io
and
le
T
be
a
linea
ans o ma ion
o
E
in o
E
.
Assume
ha
hl)
xEK,aER=>axEK
;
h2)
max{dim
W
:
W
a
subspace,
W
CK}=
d,
1 <_
d
<
n
;
and
h3)
T(K {0})
C
in K
.
Then
he e
exis
(unique)
subspaces
Wl,
W
2
such
ha
1)
W
l
n
W
2
=
{0},
dim
W
l
=
d,
dim
W
2
=
n-
d
;
2)
TW
j
C
W
j
,
j
=
1,
2
;
and
3)
W
l
C
{0}
U
in
K,
W
2
n
K
=
{0}
.
Mo eo e ,
i
oI(T),
Q2(T)
a e he
spec a
o
T
es ic ed
o
Wl,
W2,
hen,
be ween
al
(T)
and
Q2
(T)
he e
is
a
gap
A
E
u1(T),
E
0
`2(T)
==> JAJ
>
Iwl
.
2
.12
.
Theo em
([G
K ])
.
Le
A
be
an n x
n
oscilla o y
ma ix
.
Then,
(a)
all
he
eigen alues
o
A
a e
eal,
simple
and
posi i e
;
(b)
I
A1
>
A2
>
.
.
.
>
A
n
>
0
deno e
he
eigen alues
o
A
and
uá
is
an
eigen ec o
co esponding
o
ai,
hen
ui
E
V
and
N(ui)
=
i ;
and
(c)
I
w
=
j
:k
j=h
Ciu7,
whe e
1
<
h
<
k
<
n,
hen
w
7É
0
implies
h
<
Nm(w)
<
N
M
(w)
<
k
.
P oo
.
I
su ices
o
p o e
he
heo em
o
a
s ic ly
o ally
posi i e
ma ix
since
by
2
.4
A''
-i
and
A'
a e
s ic ly
o ally
posi i e
ma ices
.
We
e e
he
eade
o
[FO-2,
Theo em
2]
o
he
p oo ,
including
he
ac s
ha
he
se s
Ki,
1
<
i
<
n,
sa is y
hl)
and
h2),
wi h
d
=
i
.
Lemma
2
.8
implies
ha
A(K
i
{0})
C
in
Ki,
1
<_
i
<
n,
since
i
x E
Ki {0}
hen
N,,,,(x)
<
i
and
hence
NM(Ax)
<
N,,(x)
<
i
( emem-
be
ha
in K
i
=
{x
E
Rn
:
NM(x)
<
i})
.
The e o e
Theo em
2
.11
implies
ha ,
o
each
1
<
i
<
n,
he e
is
an
A-in a ian
¡-dimensional
0
subspace
Wi
C
Ki
and
an
A-in a ian
(n
-
i)-dimensional
subspace
W2,
WZ
n
Ki
=
{0}
wi h
co esponding
spec al
gap
.

Clea ly (wi h
W1
=W2
=
Rn)
we
ha e
Wi
C
Wi
+l
,
WZ
C
W2-1
,
1
<_ i
<n
.
Fo
each
1
<
i
<_
n
le
Vi
=
Wi
n
W2
-1
.
V
is
an
A-in a ian
sub-
space
and dim
Vi
=
1
because
Wi
n
W2i
=
{0}
.
I
is
also
clea
ha
span{Vl,
. . . ,
V
n
}
=
R
n
,
he e o e
he
spec um
o
A
is
he
se
o
he
262

W
.
M
.
OLIVA,
N
.
M
.
KUHL,
L
.
T
.
MAGALHAES
( eal)
eigen alues
{A1, A2,
.
.
.
.
A2}
co esponding
o
V
l
,. .
.
.
V

, .
Since
A
is
s ic ly
o ally
posi i e,
o
each
1 <_
<_
n
he
h
ex e io
powe
o
A
is
a
ma ix
wi h
posi i e
coe icien s
and by
he
classical
Pe -
on
Theo em
(which
is
a
pa icula
case
o
Theo em
2
.11)
one
easily
concludes
ha
all
he
ai's
a e
posi i e
.
By
he
spec al
gap
we
lla e
Al
>
A2
>
. . .
>
An
>
0
.
Le
u i
E
V
i
be
a
nonze o
ec o ,
hen
ui
E
in
KZ
and
he e o e
NM(ui)
_<
i
.
Also,
ui
1
Ki_1
and
he e o e
N,
(u
¡
)
>d-
1
.
I
ollows
ha
ui
E
N
and
N(ui)
=
i
.
The
las
s a e-
men
is
p o ed
simila ly,
jus
emembe ing
ha
w
=
_
k
j=h
Cjuj
0
0
belongs
o
he
subspaces
Wi
n
WZ
-1
.
Rema k
.
Le
A
be
an
nx
m
ma ix
.
We
say
ha
A
is
s ic ly
sign
egula
(see
zl
. . .
i

Q
[Ka])
i
o
each
1
<
<
min(m,
n),
all
he
mino s
A
~
ki
. . .
k,
a e
nonze o
and
lla e
he
same
sign
.
Lemma
2
.8
is
ue
o
hese
ma ices
and
hence
we
lla e
he
same
spec al
heo em
o
s ic ly
sign egula
squa e
ma ices
i
we
dis ega d he
posi i eness
o
he
eigen alues
and
conside
hem
in
dec easing o de
o hei
moduli
.
We
now
p o e
he
ans e sali y
heo em
:
2
.13
.
Theo em
.
Le
:
Rn
--,
Rn
be
a
Ck
di eomo phism,
k
>_ 1,
such
ha
:
(H1)
Fo
all
x,
yE
Rn,
Nm(
(x)
-
(y))
<
NM(x
-
y)
;
and
(H2)
Fo
all
x
E
Rn,
,
(x)
is
an
oscilla o y
ma ix
.
Then,
i
e
-
and
e+
a e
wo
hype bolic
ixed
poin s
o ,
he
uns able
mani old
W''(e
-
)
and
he
s able
mani old
Ws(e+)
a e
ans e sal
.
P oo£
Le
x
E
W'u(e
-
)
n
Ws(e+)
.
Conside
he
sequence
By
(H
1),
NM(u
+
)
<
NM(u-)
.
k+l(x)__

k(x)
uk

11 k+i(x)
_
k(x)I1
Taking
a
subsequence
i
necessa y
we
lla e
:
lim
uk
=
U+E
T
e
+W
s
(e
+
)
k-
.+oo
li
m

uk
=
u-
E
T
e
-W
s
(e
-
)
.
k
-~
_oo
k
E Z
.
DIFFEOMORPHISMS
WITH
OSCILLATORY
JACOBIANS

263
Le
{u
}i=1,
.
.
.,n
be
a
basis
o med by
eigen ec o s
o
'(e
)
as
in
Theo em
2.12 and
le
m
:
~
be
he
dimension
o
W'(e
:
~)
.
Then,
u
-
á
u
i
c,uz
F om
Theo em
2
.12
NM(u+)
>_
m+
+
1
and
Nm(u
-
)
<_
m
-
;
since
NM(u+)<
Nm(u
-
),
we
ha e
m
+
<m
-
-l
.
Now
we
p oceed
as in
he
p oo
o
he
main
esul
o
[F0-11
.
In
ac ,
he
las
inequali y
implies ha
he
n-m
-
eigen ec o s
u
.-
+15
.
.
.
)un
o
'(e+)
belong
e
T
e
+W'(e+)
and by
Theo em
2.12
we
conclude
ha
E
=
span{u+m-+1,
.
.
.
,
uñ
} is
con ained
in
{0}
U
(Rn K
m
-
)
.
Since
R
n
K,
.-
is
an open
se
and
Ws(e+)
is
a
smoo h
mani old, he e
exis s
an
in ege
no
>
0
such ha
T
n
o
(,,)
W'
(e+)
con ains
a
(n
-
m
-
)-dimensional
ec o
subspace
E
con ained
in
{0}
U
(Rn K,
,-)
.
Le
E
o
be
he ec o
space
E
°
_
(
-no),
.o
(x))
E
.
Then,
dim
Eo
=
n-
m
-
and
by
(H2)
we
ha e
Eo
C
T,,
:W-1(e+)
n
({0}
U
(Rn K
m
-»
.
A
simila
a gumen
shows
ha
0

0
T
'
WU(e
-
)
C
K,
n-
.
Since
dim(T
S
Wu(e
-
))
=
m
-
and
since
K
m
-
n
({0}
U
(Rn K,

,-))
=
{0}
we
ha e
Eo
®T,,Wu(e+)
=
Rn
and
he e o e
he
heo em
is
p o ed
.
2
.14
.
Example
.
Le
A
be a
s ic ly
o ally
posi i e
ma ix
and
A
be
an
eigen alue
o
A
wi h
eigen ec o
u
.
De ine
g
:
Rn
__>
Rn
by
1
xi

xñ
g(x)
=
2A
~u2
+
x1,
.
. . ,
u2
+
xn
1

n
whe e x
=
(x1,
.
.
. ,
x
n
)
and u
=
(u1,
. . . ,
un)
.
g
is
a
C°°-di eomo phism wi h
de i a i e
3
CUI
z
+
1
J
o allxER'
.