A dynamical system related to GIT
Nolan R. Wallach
June 4,2015
N. Wallach ()
A dynamical system
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A gradient system
Let φ 2 R[x1 , ..., xn ] be a polynomial that is homogeneous of degree
m such that φ(x ) 0 for all x 2 Rn . We consider the gradient
system
dx
= r φ (x )
dt
N. Wallach ()
A dynamical system
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A gradient system
Let φ 2 R[x1 , ..., xn ] be a polynomial that is homogeneous of degree
m such that φ(x ) 0 for all x 2 Rn . We consider the gradient
system
dx
= r φ (x )
dt
Note that
hrφ(x ), x i = mφ(x )
Denoting by F (t, x ) the solution to the system near t = 0 with
F (0, x ) = x. Then
d
hF (t, x ), F (t, x )i =
dt
=
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2 hrφ(F (t, x )), F (t, x )i
2mφ(F (t, x ))
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0.
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This implies kF (t, x )k kx k where de…ned for t
F (t, x ) is de…ned for all t 0.
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A dynamical system
0 and hence
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This implies kF (t, x )k kx k where de…ned for t
F (t, x ) is de…ned for all t 0.
0 and hence
The formula
hrφ(x ), x i = mφ(x )
combined with the Schwarz inequality implies that
krφ(x )k kx k
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A dynamical system
mφ(x ).
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This implies kF (t, x )k kx k where de…ned for t
F (t, x ) is de…ned for all t 0.
0 and hence
The formula
hrφ(x ), x i = mφ(x )
combined with the Schwarz inequality implies that
krφ(x )k kx k
mφ(x ).
The Lojasiewicz gradient inequality implies the following
1
improvement. There exists 0 < ε
m 1 and C > 0 both depending
only on φ such that
krφ(x )k1 +ε kx k1
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(m 1 ) ε
A dynamical system
C φ (x ).
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We take ε and C as above (but allow ε = 0 which is easy). If we
write F for F (t, X ) and H (t ) = φ(F (t, x )) then we have
H 0 (t ) =
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d φ(F )rφ(F ) =
A dynamical system
krφ(F )k2 .
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We take ε and C as above (but allow ε = 0 which is easy). If we
write F for F (t, X ) and H (t ) = φ(F (t, x )) then we have
H 0 (t ) =
If t
0 and kx k
krφ(F )k1 +ε r 1
N. Wallach ()
d φ(F )rφ(F ) =
krφ(F )k2 .
r
(m 1 ) ε
krφ(F )k1 +ε kF k1
A dynamical system
(m 1 ) ε
C φ (x ).
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We take ε and C as above (but allow ε = 0 which is easy). If we
write F for F (t, X ) and H (t ) = φ(F (t, x )) then we have
H 0 (t ) =
If t
0 and kx k
krφ(F )k1 +ε r 1
d φ(F )rφ(F ) =
krφ(F )k2 .
r
(m 1 ) ε
krφ(F )k1 +ε kF k1
(m 1 ) ε
C φ (x ).
We will now run through what has come to be called “the Lojasiewicz
argument” which I learned from a beautiful exposition of Neeman’s
theorem by Gerry Schwarz.
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krφ(F )k1 +ε
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C
r 1 3ε
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φ (F ).
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C
krφ(F )k1 +ε
krφ(F )k
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2
r 1 3ε
C
r 1 3ε
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φ (F ).
2
1 +ε
2
φ (F ) 1 +ε .
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C
krφ(F )k1 +ε
krφ(F )k
0
jH (t )j
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1
2
C
2
C
r 1 3ε
r 1 3ε
2
1 +ε
2
φ (F ) 1 +ε .
r 1 3ε
2
1 +ε
φ (F ).
2
2
φ ( F ) 1 + ε = C1 ( r ) H ( t ) 1 + ε .
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C
krφ(F )k1 +ε
krφ(F )k
0
jH (t )j
1
2
r 1 3ε
C
r 1 3ε
2
1 +ε
2
φ (F ) 1 +ε .
r 1 3ε
2
2
φ ( F ) 1 + ε = C1 ( r ) H ( t ) 1 + ε .
Since H 0 (t ) 0 for t 0 we have
Assuming H (t ) > 0 we have
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2
1 +ε
C
2
φ (F ).
H 0 (t )
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2
C1 ( r ) H ( t ) 1 + ε .
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C
krφ(F )k1 +ε
krφ(F )k
0
jH (t )j
1
2
r 1 3ε
2
1 +ε
C
r 1 3ε
N. Wallach ()
1 ε
1 +ε
=
2
φ (F ) 1 +ε .
r 1 3ε
2
2
φ ( F ) 1 + ε = C1 ( r ) H ( t ) 1 + ε .
Since H 0 (t ) 0 for t 0 we have
Assuming H (t ) > 0 we have
d
H (t )
dt
2
1 +ε
C
2
φ (F ).
H 0 (t )
1 ε H 0 (t )
1 + ε H (t ) 1+2 ε
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2
C1 ( r ) H ( t ) 1 + ε .
C1 ( r )
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H (t )
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1 ε
1 +ε
C1 (r )t.
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H (t )
H (t )
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C2 ( r ) t
1 ε
1 +ε
C1 (r )t.
(1 + ε )
1 ε
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C2 ( r ) t
(1 + ε )
,
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H (t )
H (t )
C2 ( r ) t
1 ε
1 +ε
C1 (r )t.
(1 + ε )
1 ε
C2 ( r ) t
(1 + ε )
,
This is true if H (t ) = 0 so the formula is valid for all t > 0.
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H (t )
H (t )
C2 ( r ) t
1 ε
1 +ε
C1 (r )t.
(1 + ε )
1 ε
C2 ( r ) t
(1 + ε )
,
This is true if H (t ) = 0 so the formula is valid for all t > 0.
This is the …rst half of the calculus part of the Lojasiewicz argument.
The …rst implication needs only the easy case ε = 0. If kx k r then
φ(F (t, x ))
C (r )
t
so limt !+∞ φ(F (t, x )) = 0 uniformly for x in compacta. We now do
the rest of the Lojasiewicz argument which uses the existence of
ε > 0.
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Let f (t ) = t 1 +δ with 0 < δ < ε then for t > 0
0 < H (t )f 0 (t )
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C2 (r )(1 + δ)t
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1 (ε δ)
.
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Let f (t ) = t 1 +δ with 0 < δ < ε then for t > 0
0 < H (t )f 0 (t )
H (s )f (s )
Z s
t
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C2 (r )(1 + δ)t
H (t )f (t ) =
Z s
d
t
0
H (u )f (u )du +
Z s
t
A dynamical system
du
1 (ε δ)
.
(H (u )f (u ))du =
H 0 (u )f (u )du.
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Let f (t ) = t 1 +δ with 0 < δ < ε then for t > 0
0 < H (t )f 0 (t )
H (s )f (s )
Z s
H (t )f (t ) =
H (u )f (u )du =
0
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0
H (u )f (u )du +
0
t
Z s
d
t
t
Z s
C2 (r )(1 + δ)t
Z s
t
H (s )f (s )
Z s
t
du
1 (ε δ)
(H (u )f (u ))du =
H 0 (u )f (u )du.
H (u )f 0 (u )du + H (t )f (t )
C2 ( r ) s
(1 + ε ) 1 + δ
A dynamical system
.
s
= C2 ( r ) s
H (s )f (s ).
(ε δ)
.
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Let f (t ) = t 1 +δ with 0 < δ < ε then for t > 0
0 < H (t )f 0 (t )
H (s )f (s )
Z s
H (t )f (t ) =
H (u )f (u )du =
H (s )f (s )
Z s
s !+∞ t
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Z s
t
0
lim
0
H (u )f (u )du +
0
t
Z s
d
t
t
Z s
C2 (r )(1 + δ)t
Z s
t
du
1 (ε δ)
(H (u )f (u ))du =
H 0 (u )f (u )du.
H (u )f 0 (u )du + H (t )f (t )
C2 ( r ) s
H 0 (u ) f (u )du =
(1 + ε ) 1 + δ
s
Z ∞
t
.
= C2 ( r ) s
H (s )f (s ).
(ε δ)
.
H (u )f 0 (u )du + H (t )f (t ).
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Thus
p
jH 0 (u )j f (u ) is in L2 ([t, +∞)) for all t > 0 and so
q
q
(1 + δ )
jH 0 (u )j = jH 0 (u )j f (u )u 2 2 L1 ([t, +∞)).
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Thus
p
jH 0 (u )j f (u ) is in L2 ([t, +∞)) for all t > 0 and so
q
q
(1 + δ )
jH 0 (u )j = jH 0 (u )j f (u )u 2 2 L1 ([t, +∞)).
Theorem. If t > 0 then
Z +∞
t
converges uniformly for kx k
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d
F (u, x ) du
du
r.
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Thus
p
jH 0 (u )j f (u ) is in L2 ([t, +∞)) for all t > 0 and so
q
q
(1 + δ )
jH 0 (u )j = jH 0 (u )j f (u )u 2 2 L1 ([t, +∞)).
Theorem. If t > 0 then
Z +∞
t
converges uniformly for kx k
d
F (u, x ) du
du
r.
Z ∞
d
F (u, x )du
du
converges absolutely and uniformly for kx k
t
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A dynamical system
r.
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Thus
p
jH 0 (u )j f (u ) is in L2 ([t, +∞)) for all t > 0 and so
q
q
(1 + δ )
jH 0 (u )j = jH 0 (u )j f (u )u 2 2 L1 ([t, +∞)).
Theorem. If t > 0 then
Z +∞
t
converges uniformly for kx k
d
F (u, x ) du
du
r.
Z ∞
d
F (u, x )du
du
converges absolutely and uniformly for kx k r .
Noting that if s > t then
Z s
d
F (u, x )du = F (s, x ) F (t, x )
t du
we have for t > 0
Z ∞
d
lim F (s, x ) =
F (u, x )du + F (t, x ).
s !∞
du
t
t
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Finally, set L(t, x ) = F ( 1 t t , x ) and de…ne L(1, x ) by the limit above
then L : [0, 1] Rn ! Rn is continuous and since
rφ(x ) = 0 () φ(x ) = 0
we have
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Finally, set L(t, x ) = F ( 1 t t , x ) and de…ne L(1, x ) by the limit above
then L : [0, 1] Rn ! Rn is continuous and since
rφ(x ) = 0 () φ(x ) = 0
we have
Theorem. L : [0, 1] Rn ! Rn de…nes a strong deformation
retraction of Rn onto Y = fx 2 Rn jφ(x ) = 0g.
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Finally, set L(t, x ) = F ( 1 t t , x ) and de…ne L(1, x ) by the limit above
then L : [0, 1] Rn ! Rn is continuous and since
rφ(x ) = 0 () φ(x ) = 0
we have
Theorem. L : [0, 1] Rn ! Rn de…nes a strong deformation
retraction of Rn onto Y = fx 2 Rn jφ(x ) = 0g.
Corollary. If Z
Rn is closed and such that F (t, z ) 2 Z for t 0
and z 2 Z then H : [0, 1] Z ! Z de…nes a strong deformation
retraction of Z onto Z \ Y .
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Kempf-Ness over the reals
Let G be an open subgroup of a Zariski closed subgroup of GL(n, R)
that is closed under real adjoint relative to the standard inner
product,h..., ...i, g ! g . Let K = G \ O (n ). Then K is a maximal
compact subgroup of G . On g = Lie (G ) we put the inner product
hX , Y i = tr (XY ), Set p =Lie (K )? relative to this inner product.
N. Wallach ()
A dynamical system
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Kempf-Ness over the reals
Let G be an open subgroup of a Zariski closed subgroup of GL(n, R)
that is closed under real adjoint relative to the standard inner
product,h..., ...i, g ! g . Let K = G \ O (n ). Then K is a maximal
compact subgroup of G . On g = Lie (G ) we put the inner product
hX , Y i = tr (XY ), Set p =Lie (K )? relative to this inner product.
We say that an element v 2 Rn is G -critical if for any X 2 Lie (G ),
hXv , v i = 0. The following is an extension of the Kempf-Ness
Theorem …rst observed by Richardson and Slodoway.
N. Wallach ()
A dynamical system
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Kempf-Ness over the reals
Let G be an open subgroup of a Zariski closed subgroup of GL(n, R)
that is closed under real adjoint relative to the standard inner
product,h..., ...i, g ! g . Let K = G \ O (n ). Then K is a maximal
compact subgroup of G . On g = Lie (G ) we put the inner product
hX , Y i = tr (XY ), Set p =Lie (K )? relative to this inner product.
We say that an element v 2 Rn is G -critical if for any X 2 Lie (G ),
hXv , v i = 0. The following is an extension of the Kempf-Ness
Theorem …rst observed by Richardson and Slodoway.
Theorem. Let G , K be as above. Let v 2 Rn .
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A dynamical system
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Kempf-Ness over the reals
Let G be an open subgroup of a Zariski closed subgroup of GL(n, R)
that is closed under real adjoint relative to the standard inner
product,h..., ...i, g ! g . Let K = G \ O (n ). Then K is a maximal
compact subgroup of G . On g = Lie (G ) we put the inner product
hX , Y i = tr (XY ), Set p =Lie (K )? relative to this inner product.
We say that an element v 2 Rn is G -critical if for any X 2 Lie (G ),
hXv , v i = 0. The following is an extension of the Kempf-Ness
Theorem …rst observed by Richardson and Slodoway.
Theorem. Let G , K be as above. Let v 2 Rn .
1
If v is critical if and only if kgv k
N. Wallach ()
kv kfor all g 2 G .
A dynamical system
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Kempf-Ness over the reals
Let G be an open subgroup of a Zariski closed subgroup of GL(n, R)
that is closed under real adjoint relative to the standard inner
product,h..., ...i, g ! g . Let K = G \ O (n ). Then K is a maximal
compact subgroup of G . On g = Lie (G ) we put the inner product
hX , Y i = tr (XY ), Set p =Lie (K )? relative to this inner product.
We say that an element v 2 Rn is G -critical if for any X 2 Lie (G ),
hXv , v i = 0. The following is an extension of the Kempf-Ness
Theorem …rst observed by Richardson and Slodoway.
Theorem. Let G , K be as above. Let v 2 Rn .
1
2
kv kfor all g 2 G .
If v is critical and if w 2 Gv is such that kv k = kw kthen w 2 Kv .
If v is critical if and only if kgv k
N. Wallach ()
A dynamical system
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Kempf-Ness over the reals
Let G be an open subgroup of a Zariski closed subgroup of GL(n, R)
that is closed under real adjoint relative to the standard inner
product,h..., ...i, g ! g . Let K = G \ O (n ). Then K is a maximal
compact subgroup of G . On g = Lie (G ) we put the inner product
hX , Y i = tr (XY ), Set p =Lie (K )? relative to this inner product.
We say that an element v 2 Rn is G -critical if for any X 2 Lie (G ),
hXv , v i = 0. The following is an extension of the Kempf-Ness
Theorem …rst observed by Richardson and Slodoway.
Theorem. Let G , K be as above. Let v 2 Rn .
1
2
3
kv kfor all g 2 G .
If v is critical and if w 2 Gv is such that kv k = kw kthen w 2 Kv .
If Gv is closed then there exists a critical element in Gv .
If v is critical if and only if kgv k
N. Wallach ()
A dynamical system
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Kempf-Ness over the reals
Let G be an open subgroup of a Zariski closed subgroup of GL(n, R)
that is closed under real adjoint relative to the standard inner
product,h..., ...i, g ! g . Let K = G \ O (n ). Then K is a maximal
compact subgroup of G . On g = Lie (G ) we put the inner product
hX , Y i = tr (XY ), Set p =Lie (K )? relative to this inner product.
We say that an element v 2 Rn is G -critical if for any X 2 Lie (G ),
hXv , v i = 0. The following is an extension of the Kempf-Ness
Theorem …rst observed by Richardson and Slodoway.
Theorem. Let G , K be as above. Let v 2 Rn .
1
2
3
4
kv kfor all g 2 G .
If v is critical and if w 2 Gv is such that kv k = kw kthen w 2 Kv .
If Gv is closed then there exists a critical element in Gv .
If v is critical then Gv is closed.
If v is critical if and only if kgv k
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We set V = Rn as a G –module and CritG (V ) equal to the set of all
critical vectors. If X1 , ..., Xr is an orthonormal basis of p then
φ (v ) =
∑ h Xj v , v i 2
is non-negative homogeneous polynomial of degree 4 de…ning
CritG (V ).
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We set V = Rn as a G –module and CritG (V ) equal to the set of all
critical vectors. If X1 , ..., Xr is an orthonormal basis of p then
φ (v ) =
∑ h Xj v , v i 2
is non-negative homogeneous polynomial of degree 4 de…ning
CritG (V ).
We consider Rn as n 1 columns and thus if v 2 V then v is v as a
row vector. So for v , w 2 V , vw is an n n matrix and
hXv , w i = trXvw .
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We set V = Rn as a G –module and CritG (V ) equal to the set of all
critical vectors. If X1 , ..., Xr is an orthonormal basis of p then
φ (v ) =
∑ h Xj v , v i 2
is non-negative homogeneous polynomial of degree 4 de…ning
CritG (V ).
We consider Rn as n 1 columns and thus if v 2 V then v is v as a
row vector. So for v , w 2 V , vw is an n n matrix and
hXv , w i = trXvw .
Let Pg be the orthogonal projection of Mn (R) onto g then
rφ(v ) = 4Pg (vv )v 2 Tv (Gv ).
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We set V = Rn as a G –module and CritG (V ) equal to the set of all
critical vectors. If X1 , ..., Xr is an orthonormal basis of p then
φ (v ) =
∑ h Xj v , v i 2
is non-negative homogeneous polynomial of degree 4 de…ning
CritG (V ).
We consider Rn as n 1 columns and thus if v 2 V then v is v as a
row vector. So for v , w 2 V , vw is an n n matrix and
hXv , w i = trXvw .
Let Pg be the orthogonal projection of Mn (R) onto g then
rφ(v ) = 4Pg (vv )v 2 Tv (Gv ).
Also note that rφ(kv ) = k rφ(v ) for k 2 K .
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Let F (t, x ) be the gradient ‡ow corresponding to φ. Then we have
shown using freshman calculus that for t > 0 and kx k r
φ(F (t, x ))
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A dynamical system
C (r )
.
t
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Let F (t, x ) be the gradient ‡ow corresponding to φ. Then we have
shown using freshman calculus that for t > 0 and kx k r
C (r )
.
t
In addition if Z
V is closed and G –invariant then F (t, Z )
2 in the real Kempf-Ness theorem implies:
φ(F (t, x ))
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A dynamical system
Z and
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Let F (t, x ) be the gradient ‡ow corresponding to φ. Then we have
shown using freshman calculus that for t > 0 and kx k r
C (r )
.
t
In addition if Z
V is closed and G –invariant then F (t, Z ) Z and
2 in the real Kempf-Ness theorem implies:
Theorem. Setting L(t, Kv ) = KF ( 1 t t , v ) 0 t < 1 then
limt !1 L(t, Kv ) converges uniformly on compacta and this yields a
strict deformation retraction of Z /K to (CritG (V ) \ Z ) /K for any
G –invariant closed subset of V .
φ(F (t, x ))
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A dynamical system
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Let F (t, x ) be the gradient ‡ow corresponding to φ. Then we have
shown using freshman calculus that for t > 0 and kx k r
C (r )
.
t
In addition if Z
V is closed and G –invariant then F (t, Z ) Z and
2 in the real Kempf-Ness theorem implies:
Theorem. Setting L(t, Kv ) = KF ( 1 t t , v ) 0 t < 1 then
limt !1 L(t, Kv ) converges uniformly on compacta and this yields a
strict deformation retraction of Z /K to (CritG (V ) \ Z ) /K for any
G –invariant closed subset of V .
The statement of the next result is simpli…ed.
φ(F (t, x ))
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A dynamical system
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Let F (t, x ) be the gradient ‡ow corresponding to φ. Then we have
shown using freshman calculus that for t > 0 and kx k r
C (r )
.
t
In addition if Z
V is closed and G –invariant then F (t, Z ) Z and
2 in the real Kempf-Ness theorem implies:
Theorem. Setting L(t, Kv ) = KF ( 1 t t , v ) 0 t < 1 then
limt !1 L(t, Kv ) converges uniformly on compacta and this yields a
strict deformation retraction of Z /K to (CritG (V ) \ Z ) /K for any
G –invariant closed subset of V .
The statement of the next result is simpli…ed.
Corollary. If Z is Zariski closed in V then the GIT quotient, Z //G ,
of Z is a strict deformation retract of Z /K .
φ(F (t, x ))
N. Wallach ()
A dynamical system
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Let F (t, x ) be the gradient ‡ow corresponding to φ. Then we have
shown using freshman calculus that for t > 0 and kx k r
C (r )
.
t
In addition if Z
V is closed and G –invariant then F (t, Z ) Z and
2 in the real Kempf-Ness theorem implies:
Theorem. Setting L(t, Kv ) = KF ( 1 t t , v ) 0 t < 1 then
limt !1 L(t, Kv ) converges uniformly on compacta and this yields a
strict deformation retraction of Z /K to (CritG (V ) \ Z ) /K for any
G –invariant closed subset of V .
The statement of the next result is simpli…ed.
Corollary. If Z is Zariski closed in V then the GIT quotient, Z //G ,
of Z is a strict deformation retract of Z /K .
This is a very useful result since if G is connected K is connected and
this implies that Z //G has path lifting. In the complex case this is
an important result of Kraft, Petrie and Randall.
φ(F (t, x ))
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A dynamical system
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Let F (t, x ) be the gradient ‡ow corresponding to φ. Then we have
shown using freshman calculus that for t > 0 and kx k r
C (r )
.
t
In addition if Z
V is closed and G –invariant then F (t, Z ) Z and
2 in the real Kempf-Ness theorem implies:
Theorem. Setting L(t, Kv ) = KF ( 1 t t , v ) 0 t < 1 then
limt !1 L(t, Kv ) converges uniformly on compacta and this yields a
strict deformation retraction of Z /K to (CritG (V ) \ Z ) /K for any
G –invariant closed subset of V .
The statement of the next result is simpli…ed.
Corollary. If Z is Zariski closed in V then the GIT quotient, Z //G ,
of Z is a strict deformation retract of Z /K .
This is a very useful result since if G is connected K is connected and
this implies that Z //G has path lifting. In the complex case this is
an important result of Kraft, Petrie and Randall.
We now consider the result implied by using the deep results of
Lojasiewicz.
φ(F (t, x ))
N. Wallach ()
A dynamical system
6/4
12 / 18
The Lojasiewicz argument implies that if we set L(t, v ) = F ( 1 t t , v )
then limt !1 H (t, v ) converges uniformly on compacta.
N. Wallach ()
A dynamical system
6/4
13 / 18
The Lojasiewicz argument implies that if we set L(t, v ) = F ( 1 t t , v )
then limt !1 H (t, v ) converges uniformly on compacta.
Theorem. Let Z
V be closed and G invariant then
L : [0, 1] Z ! Z de…nes a strong, K –equivariant deformation
retraction of Z onto Z \ CritG (V ).
N. Wallach ()
A dynamical system
6/4
13 / 18
The Lojasiewicz argument implies that if we set L(t, v ) = F ( 1 t t , v )
then limt !1 H (t, v ) converges uniformly on compacta.
Theorem. Let Z
V be closed and G invariant then
L : [0, 1] Z ! Z de…nes a strong, K –equivariant deformation
retraction of Z onto Z \ CritG (V ).
Over C this result is due to Neeman.
N. Wallach ()
A dynamical system
6/4
13 / 18
Cn = V iV so as a real vector space we write it as V V = R2n .
The real part of the standard Hermitian inner product on Cn becomes
the standard inner product on R2n . Mn (C) becomes the algebra of
2 2 block n n matrices
X
Y
Y
X
.
Adjoint in Mn (C) becomes transpose in M2n (R).
N. Wallach ()
A dynamical system
6/4
14 / 18
Cn = V iV so as a real vector space we write it as V V = R2n .
The real part of the standard Hermitian inner product on Cn becomes
the standard inner product on R2n . Mn (C) becomes the algebra of
2 2 block n n matrices
X
Y
Y
X
.
Adjoint in Mn (C) becomes transpose in M2n (R).
If X
Cn is Zariski closed and de…ned by f1 , ..., fk in C[x1 , ..., xn ]
then it is de…ned by φ(x, y ) = ∑ jfj (x + iy )j2 as a real variety.
N. Wallach ()
A dynamical system
6/4
14 / 18
Cn = V iV so as a real vector space we write it as V V = R2n .
The real part of the standard Hermitian inner product on Cn becomes
the standard inner product on R2n . Mn (C) becomes the algebra of
2 2 block n n matrices
X
Y
Y
X
.
Adjoint in Mn (C) becomes transpose in M2n (R).
If X
Cn is Zariski closed and de…ned by f1 , ..., fk in C[x1 , ..., xn ]
then it is de…ned by φ(x, y ) = ∑ jfj (x + iy )j2 as a real variety.
If G
GL(n, C) is a Zariski closed subgroup invariant under adjoint
then G as a subgroup of GL(2n, R) is invariant under transpose.
Furthermore, if we de…ne the critical set for the action of G on Cn to
be
fv 2 Cn j hXv , v i = 0, X 2 Lie (G )g
then this set is exactly CritG (R2n ).
N. Wallach ()
A dynamical system
6/4
14 / 18
Cn = V iV so as a real vector space we write it as V V = R2n .
The real part of the standard Hermitian inner product on Cn becomes
the standard inner product on R2n . Mn (C) becomes the algebra of
2 2 block n n matrices
X
Y
Y
X
.
Adjoint in Mn (C) becomes transpose in M2n (R).
If X
Cn is Zariski closed and de…ned by f1 , ..., fk in C[x1 , ..., xn ]
then it is de…ned by φ(x, y ) = ∑ jfj (x + iy )j2 as a real variety.
If G
GL(n, C) is a Zariski closed subgroup invariant under adjoint
then G as a subgroup of GL(2n, R) is invariant under transpose.
Furthermore, if we de…ne the critical set for the action of G on Cn to
be
fv 2 Cn j hXv , v i = 0, X 2 Lie (G )g
then this set is exactly CritG (R2n ).
The original Kempf-Ness theorem is now a special case of the real
Kempf-Ness theorem since Zariski closure of complex orbits is the
same as the closure in the metric topology of R2n .
N. Wallach ()
A dynamical system
6/4
14 / 18
The system in the abstract for my talk is just the case of GL(n, C)
acting on Mn (C) by conjugation. Yielding the gradient system
Ẋ =
N. Wallach ()
4[[X , X ], X ].
A dynamical system
6/4
15 / 18
The system in the abstract for my talk is just the case of GL(n, C)
acting on Mn (C) by conjugation. Yielding the gradient system
Ẋ =
4[[X , X ], X ].
Writing F∞ (X ) = limt !+∞ F (t, X ) then F∞ (X ) is a normal operator
with the same eigenvalues as X .
N. Wallach ()
A dynamical system
6/4
15 / 18
N. Wallach ()
A dynamical system
6/4
16 / 18
N. Wallach ()
A dynamical system
6/4
17 / 18
N. Wallach ()
A dynamical system
6/4
18 / 18