Buliga. .Sub Riemannian.Geometry.&.Lie.Groups part.1.(2001).[sharethefiles.com]

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�
t 0
dt
Then the flow t �! �t which satisfies (3.3.14) and �0 = idR2n is the Hamiltonian flow
generated by H.
Conversely, for any Hamiltonian flow t �! �t, generated by H, the vertical flow
t �! �v satisfies the equation (3.3.15).
t
�
� � �t
Proof. Write the lifts �t and �h, compute then the differential of the quantity �t -�h
t
and show that it equals the differential of H.
3 THE HEISENBERG GROUP 39
Flows of volume preserving diffeomorphisms We want to know if there is any
nontrivial smooth (according to Pansu differentiability) flow of volume preserving dif-
feomorphisms.
�
Proposition 3.6 Suppose that t �! �t " Diff2(H(n), vol) is a flow such that
- is C2 in the classical sense with respect to (x, t),
�
- is horizontal, that is t �! �t(x) is a horizontal curve for any x.
Then the flow is constant.
�
Proof. By direct computation, involving second order derivatives. Indeed, let �t(x, x) =
�
�
(�t(x), x + Ft(x)). From the condition �t " Diff2(H(n), vol) we obtain
�
"Ft 1 "�t 1
y = �(�t(x), (x)y) - �(x, y)
"x 2 "x 2
�
and from the hypothesis that t �! �t(x) is a horizontal curve for any x we get
dFt 1
�
(x) = �(�t(x), �t(x))
dt 2
Equal now the derivative of the RHS of the first relation with respect to t with the
derivative of the RHS of the second relation with respect to x. We get the equality, for
any y " R2n:
1 "�t �
0 = �( (x)y, �t(x))
2 "x
�
therefore �t(x) = 0.
One should expect such a result to be true, based on two remarks. The first,
general remark: take a flow of left translations in a Carnot group, that is a flow t �!
�t(x) = xtx. We can see directly that each �t is smooth, because the distribution is left
invariant. But the flow is not horizontal, because the distribution is not right invariant.
The second, particular remark: any flow which satisfies the hypothesis of proposition
3.6 corresponds to a Hamiltonian flow with null Hamiltonian function, hence the flow
is constant.
At a first glance it is disappointing to see that the group of volume preserving
diffeomorphisms contains no smooth paths according to the intrinsic calculus on Carnot
groups. But this makes the richness of such groups of homeomorphisms, as we shall
see.
3.4 Volume preserving bi-Lipschitz maps
We shall work with the following groups of homeomorphisms.
Definition 3.7 The group Hom(H(n), vol, Lip) is formed by all locally bi-Lipschitz,
volume preserving homeomorphisms of H(n), which have the form:
�
�(x, x) = (�(x), x + F (x))
� �
3 THE HEISENBERG GROUP 40
The group Sympl(R2n, Lip) of locally bi-Lipschitz symplectomorphisms of R2n, in
the sense that for a.e. x " R2n the derivative D�(x) (which exists by classical Rademacher
theorem) is symplectic.
�
Given A �" R2n, we denote by Hom(H(n), vol, Lip)(A) the group of maps � which
belong to Hom(H(n), vol, Lip) such that � has compact support in A (i.e. it differs
from identity on a compact set relative to A).
The group Sympl(R2n, Lip)(A) is defined in an analogous way.
�
Remark that any element � " Hom(H(n), vol, Lip) preserves the vertical left
invariant distribution
(xx) �! (x, x)Z(H(n))
� �
for a.e. (x, x) " H(n).
�
We shall prove that any locally bi-Lipschitz volume preserving homeomorphism of
H(n) belongs to Hom(H(n), vol, Lip).
�
Theorem 3.8 Take any � locally bi-Lipschitz volume preserving homeomorphism of
�
H(n). Then � has the form:
�
�(x, x) = (�(x), x + F (x))
� �
Moreover
� " Sympl(R2n, Lip)
F : R2n �! R is Lipschitz and for almost any point (x, x) " H(n) we have:
�
1 1
DF (x)y = �(�(x), D�(x)y) - �(x, y)
2 2
� �
Proof. Set �(x) = (�(x), �(x)). Also denote by
� � �
| (y, y) |2 = | y |2 + | y |
� �
�
By theorem 2.21 � is almost everywhere derivable and the derivative can be written
in the particular form:
A 0
0 1
with A = A(x, x) " Sp(n, R).
�
�
For a.e. x " R2n the function � is derivable. The derivative has the form:
�
D�(x, x)(y, y) = (Ay, y)
� � �
where by definition
-1
� �
D�(x, x)(y, y) = lim �� (x, x)-1 �((x, x)��(y, y))
� � � � �
��!0
Let us write down what Pansu derivability means.
3 THE HEISENBERG GROUP 41
For the R2n component we have: the limit
1
lim | [�((x, x)��(y, y)) - �(x, x)] - A | = 0 (3.4.16)
� � �
��!0
�
is uniform with respect to � " B(0, 1). We shall rescale all for the ball B(0, R) " H(n).
Relation (3.4.16) means: for any � > 0 there exists �0(�) > 0 such that for any R > 0 [ Pobierz całość w formacie PDF ]

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