µ

with arbitrary | y |< 1. We get: for any » > 0 there is µ(») = µ0 (»)/2 > 0 such that

for any µ < µ(») and any | y |< 1 we have

1

| (φ(x + µy, x) ’ φ(x, x)) ’ Ay | < 2»

¯ ¯

µ

Therefore x ’ φ(x, x) is derivable a.e. with (locally) bounded derivative. It is therefore

¯

(locally) Lipschitz.

Let us now return to (3.4.16) and choose y = 0. We get: the map

¯

1

y ’ φ(x + y, x + ω(x, y))

¯

2

is derivable in y = 0 and the derivative equals A. But we have just proven that the

derivative of φ(·, x) in x equals A. We get: φ is constant with respect to x.

¯ ¯

We have proven until now that

˜¯ ¯¯

φ(x, x) = (φ(x), φ(x, x))

with φ ∈ Sympl(R2n , Lip).

Let us use this information in the de¬nition of the center component of the Pansu

derivative. We have:

1 1

¯x ˜ ¯x

| φ(˜δµ y ) ’ φ(˜) ’ ω(φ(x), φ(x + µy)) ’ y ’

¯ (3.4.18)

µ2 2

42

3 THE HEISENBERG GROUP

11

’ ω( (φ(x + µy) ’ φ(x)), Ay) | ’ 0

2µ

as µ ’ 0, uniformly with respect to | y |< 1. From previously proved facts we obtain

˜

that (3.4.18) is equivalent to

1 1

¯x ˜ ¯x

| φ(˜δµ y ) ’ φ(˜) ’ ω(φ(x), φ(x + µy) ’ φ(x)) ’ y | ’ 0

¯

µ2 2

¯¯

as µ ’ 0, uniformly with respect to y . We quickly get that φ(x, x) = x + F (x).

˜ ¯

All in all the Pansu derivability (center component) reads

1 1 1 1

lim | [F (x + µy) ’ F (x)] + ω(x, y) ’ ω(φ(x), 2 [φ(x + µy) ’ φ(x)]) | = 0 (3.4.19)

µ2 2µ 2 µ

µ’0

This means that for almost any x ∈ R2n

1 the function φ is derivable and the derivative is equal to A = A(x),

2 the function F is derivable and is connected to φ by the relation from the con-

clusion of the proposition.

˜

We have seen that φ is locally Lipschitz implies that x ∈ R2n ’ A(x) is locally

bounded, therefore φ is locally Lipschitz. In the same way we obtain that F is locally

Lipschitz.

The ¬‚ows in the group Hom(H(n), vol, Lip) are de¬ned further.

De¬nition 3.9 A ¬‚ow in the group Hom(H(n), vol, Lip) is a curve

˜

t ’ φt ∈ Hom(H(n), vol, Lip)

such that for a.e. x ∈ R2n the curve t ’ φt (x) ∈ R2n is (locally) Lipschitz.

For any ¬‚ow we can de¬ne the horizontal lift of this ¬‚ow like this: a.e. curve

t ’ φt (x) lifts to a horizontal curve t ’ φh (x, 0). De¬ne then

t

φh (x, x) = φh (x, 0)(0, x)

¯ ¯

t t

˜

If the ¬‚ow is smooth (in the classical sense) and φt ∈ Dif f 2 (H(n), vol) then this

lift is the same as the one described in de¬nition 3.4. We can de¬ne now the vertical

¬‚ow by the formula (3.3.14), that is

˜

φv = φ’1 —¦ φh

There is an analog of proposition 3.6. In the proof we shall need lemma 3.11, which

comes after.

˜

Proposition 3.10 Let t ’ φt ∈ Hom(H(n), vol, Lip) be a curve such that the function

˜x ˜

t ’ ¦(˜, t) = (φt (˜), t) is locally Lipschitz from H(n) — R to itself. Then t ’ φt is a

x

constant curve.

43

3 THE HEISENBERG GROUP

Proof. By Rademacher theorem 2.15 for the group H(n) — R we obtain that ¦ is

almost everywhere derivable. Use now lemma 3.11 to deduce the claim.

A short preparation is needed in order to state the lemma 3.11. Let N be a non-

commutative Carnot group. We shall look at the group N —R with the group operation

de¬ned component wise. This is also a Carnot group. Indeed, consider the family of

dilatations

δµ (x, t) = (δµ (x), µt)

which gives to N — R the structure of a CC group. The left invariant distribution on

the group which generates the distance is (the left translation of) W1 = V1 — R.

Lemma 3.11 Let N be a noncommutative Carnot group which admits the orthogonal

decomposition

N = V1 + [N, N ]

and satis¬es the condition

V1 © Z(N ) = 0

The group of linear transformations of N — R is then

A0

HL(N — R) = : A ∈ HL(N ) , c ∈ V1 , d ∈ R

cd

Proof. We shall proceed as in the proof of proposition 3.1. We are looking ¬rst at

the Lie algebra isomorphisms of N — R, with general form

Ab

cd

We obtain the conditions:

(i) c orthogonal on [N, N ],

(ii) b commutes with the image of A: [b,Ay] = 0, for any y ∈ N ,

(iii) A is an algebra isomorphism of N .

From (ii), (iii) we deduce that b is in the center of N and from (i) we see that c ∈ V1 .

We want now the isomorphism to commute with dilatations. This condition gives:

(iv) b ∈ V1 ,

(v) A commutes with the dilatations of N .

(iii) and (v) imply that A ∈ HL(N ) and (iv) that b = 0.

44

3 THE HEISENBERG GROUP

Hamiltonian di¬eomorphisms: more structure In this section we look closer

to the structure of the group of volume preserving homeomorphisms of the Heisenberg

group.

Let Homh (H(n), vol, Lip)(A) be the group of time one homeomorphisms t ’ φh ,

for all curves t ’ φt ∈ Sympl(R2n , Lip)(A) such that (x, t) ’ (φt (x), t) is locally

Lipschitz.

The elements of this group are also volume preserving, but they are not smooth

with respect to the Pansu derivative.

˜

De¬nition 3.12 The group Hom(H(n), vol)(A) contains all maps φ which have a.e.

the form:

˜¯

φ(x, x) = (φ(x), x + F (x))

¯

where φ ∈ Sympl(R2n , Lip)(A) and F : R2n ’ R is locally Lipschitz and constant

outside a compact set included in the closure of A (for short: with compact support in

A).

This group contains three privileged subgroups:

- Hom(H(n), vol, Lip)(A),

- Homh (H(n), vol, Lip)(A)

- and Homv (H(n), vol, Lip)(A).

The last is the group of vertical homeomorphisms, any of which has the form:

φv (x, x) = (x, x + F (x))

¯ ¯

with F locally Lipschitz, with compact support in A.

˜

Take a one parameter subgroup t ’ φt ∈ Hom(H(n), vol, Lip)(A), in the sense

of the de¬nition 3.9. We know that it cannot be smooth as a curve in in the group

Hom(H(n), vol, Lip)(A), but we also know that there are vertical and horizontal ¬‚ows

˜

t ’ φv , φh such that we have the decomposition φt —¦ φv = φh . Unfortunately none

t t t t

v , φh are one parameter groups.

of the ¬‚ows t ’ φt t

There is more structure here that it seems. Consider the class

HAM (H(n))(A) = Hom(H(n), vol, Lip)(A) — Homv (H(n), vol, Lip)(A)

˜

For any pair in this class we shall use the notation (φ, φv ) This class forms a group

with the (semidirect product) operation:

˜ ˜ ˜˜ ˜ ˜

(φ, φv )(ψ, ψ v ) = (φ —¦ ψ, φv —¦ φ —¦ ψ v —¦ φ’1 )

˜

Proposition 3.13 If t ’ φt ∈ Hom(H(n), vol, Lip)(A) is an one parameter group

˜

then t ’ (φt , φv ) ∈ HAM (H(n)) is an one parameter group.

t

45

3 THE HEISENBERG GROUP

Proof. The check is left to the reader. Use de¬nition and proposition 1, chapter 5,

Hofer & Zehnder [16], page 144.

We can (indirectly) put a distribution on the group HAM (H(n))(A) by specifying

˜

the class of horizontal curves. Such a curve t ’ (φt , φv ) in the group HAM (H(n))(A)

t

˜t to a ¬‚ow in Hom(H(n), vol, Lip)(A). De¬ne for

projects in the ¬rst component t ’ φ

˜

any (φ, φv ) ∈ HAM (H(n))(A) the function

˜

φh = φ —¦ φv

and say that the curve

˜

t ’ (φt , φv ) ∈ HAM (H(n))(A)

t

is horizontal if

t ’ φh (x, x)

¯

t

is horizontal for any (x, x) ∈ H(n).

¯

We introduce the following length function for horizontal curves:

1

˜ ™

L t ’ (φt , φv ) φv

= dt

L∞ (A)

t t

0

With the help of the length we are able to endow the group HAM (H(n))(A) as a path

metric space. The distance is de¬ned by:

˜ ˜ ˜

dist (φI , φv ), (φII , φv ) = inf L t ’ (φt , φv )

I II t

˜

over all horizontal curves t ’ (φt , φv ) such that

t

˜ ˜

(φ0 , φv ) = (φI , φv )

0 I

˜ ˜

(φ1 , φv ) = (φII , φv )

1 II

The distance is not de¬ned for any two points in HAM (H(n))(A). In principle the

distance can be degenerated.

The group HAM (H(n))(A) acts on AZ (where Z is the center of H(n)) by:

˜ ˜¯

(φ, φv )(x, x) = φv —¦ φ(x, x)

¯