Remark that the group

Homv (H(n), vol, Lip)(A) ≡ {id} — Homv (H(n), vol, Lip)(A)

is normal in HAM (H(n))(A). We denote by

HAM (H(n))(A)/Homv (H(n), vol, Lip)(A)

the factor group. This is isomorphic with the group Sympl(R2n , Lip)(A).

Consider now the natural action of Homv (H(n), vol, Lip)(A) on AZ. The space of

orbits AZ/Homv (H(n), vol, Lip)(A) is nothing but A, with the Euclidean metric.

46

3 THE HEISENBERG GROUP

Factorize now the action of HAM (H(n))(A) on AZ, by the group

Homv (H(n), vol, Lip)(A)

We obtain the action of HAM (H(n))(A)/Homv (H(n), vol, Lip)(A) on

AZ/Homv (H(n), vol, Lip)(A)

Elementary examination shows this is the action of the symplectomorphisms group on

R2n .

All in all we have the following (for the de¬nition of the Hofer distance see next

section):

Theorem 3.14 The action of HAM (H(n))(A) on AZ descends after reduction with

the group Homv (H(n), vol, Lip) to the action of symplectomorphisms group with com-

pact support in A.

The distance dist on HAM (H(n)) descends to the Hofer distance on the connected

component of identity of the symplectomorphisms group.

3.5 Symplectomorphisms, capacities and Hofer distance

Symplectic capacities are invariants under the action of the symplectomorphisms group.

Hofer geometry is the geometry of the group of Hamiltonian di¬eomorphisms, with

respect to the Hofer distance. For an introduction into the subject see Hofer, Zehnder

[16] chapters 2,3 and 5, and Polterovich [28], chapters 1,2.

A symplectic capacity is a map which associates to any symplectic manifold (M, ω)

a number c(M, ω) ∈ [0, +∞]. Symplectic capacities are special cases of conformal

symplectic invariants, described by:

A1. Monotonicity: c(M, ω) ¤ c(N, „ ) if there is a symplectic embedding from M to

N,

A2. Conformality: c(M, µω) =| µ | c(M, ω) for any ± ∈ R, ± = 0.

We can see a conformal symplectic invariant from another point of view. Take a

symplectic manifold (M, ω) and consider the invariant de¬ned over the class of Borel

sets B(M ), (seen as embedded submanifolds). In the particular case of R2n with the

standard symplectic form, an invariant is a function c : B(R2n ) ’ [0, +∞] such that:

B1. Monotonicity: c(M ) ¤ c(N ) if there is a symplectomorphism φ such that φ(M ) ‚

N,

B2. Conformality: c(µM ) = µ2 c(M ) for any µ ∈ R.

An invariant is nontrivial if it takes ¬nite values on sets with in¬nite volume, like

cylinders:

Z(R) = x ∈ R2n : x2 + x2 < R

1 2

There exist highly nontrivial invariants, as the following theorem shows:

47

3 THE HEISENBERG GROUP

Theorem 3.15 (Gromov™s squeezing theorem) The ball B(r) can be symplectically em-

bedded in the cylinder Z(R) if and only if r ¤ R.

This theorem permits to de¬ne the invariant:

c(A) = sup R2 : ∃φ(B(R)) ‚ A

called Gromov™s capacity.

Another important invariant is Hofer-Zehnder capacity. In order to introduce this

we need the notion of a Hamiltonian ¬‚ow.

A ¬‚ow of symplectomorphisms t ’ φt is Hamiltonian if there is a function H :

M — R ’ R such that for any time t and place x we have

™

ω(φt (x), v) = dH(φt (x), t)v

for any v ∈ Tφt (x) M .

Let H(R2n ) be the set of compactly supported Hamiltonians. Given a set A ‚ R2n ,

the class of admissible Hamiltonians is H(A), made by all compactly supported maps

in A such that the generated Hamiltonian ¬‚ow does not have closed orbits of periods

smaller than 1. Then the Hofer-Zehnder capacity is de¬ned by:

hz(A) = sup { H : H ∈ H(A)}

∞

Let us denote by Ham(A) the class of Hamiltonian di¬eomorphisms compactly

supported in A. A Hamiltonian di¬eomorphism is the time one value of a Hamiltonian

¬‚ow. In the case which interest us, that is R2n , Ham(A) is the connected component

of the identity in the group of compactly supported symplectomorphisms.

A curve of Hamiltonian di¬eomorphisms (with compact support) is a Hamiltonian

¬‚ow. For any such curve t ’ c(t) we shall denote by t ’ Hc (t, ·) the associated

Hamiltonian function (with compact support).

On the group of Hamiltonian di¬eomorphisms there is a bi-invariant distance intro-

duced by Hofer. This is given by the expression:

1

dt : c : [0, 1] ’ Ham(R2n )

dH (φ, ψ) = inf Hc (t) (3.5.20)

∞,R2n

0

It is easy to check that d is indeed bi-invariant and it satis¬es the triangle property.

It is a deep result that d is non-degenerate, that is d(id, φ) = 0 implies φ = id.

With the help of the Hofer distance one can de¬ne another symplectic invariant,

called displacement energy. For a set A ‚ R2n the displacement energy is:

de(A) = inf dH (id, φ) : φ ∈ Ham(R2n ) , φ(A) © A = …

A displacement energy can be associated to any group of transformations endowed

with a bi-invariant distance (see Eliashberg & Polterovich [7], section 1.3). The fact that

the displacement energy is a nontrivial invariant is equivalent with the non-degeneracy

of the Hofer distance. Section 2. Hofer & Zehnder [16] is dedicated to the implications

of the existence of a non-trivial capacity. All in all the non-degeneracy of the Hofer

distance (proved for example in the Section 5. Hofer & Zehnder [16]) is the cornerstone

of symplectic rigidity theory.

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3 THE HEISENBERG GROUP

3.6 Hausdor¬ dimension and Hofer distance

We shall give in this section a metric proof of the non-degeneracy of the Hofer distance.

A ¬‚ow of symplectomorphisms t ’ φt with compact support in A is Hamiltonian

if it is the projection onto R2n of a ¬‚ow in Hom(H(n), vol, Lip)(A).

Consider such a ¬‚ow which joins identity id with φ. Take the lift of the ¬‚ow

˜

t ∈ [0, 1] ’ φt ∈ Hom(H(n), vol, Lip)(A).

˜

Proposition 3.16 The curve t ’ φt (x, 0) has Hausdor¬ dimension 2 and measure

1

˜

2

H t ’ φt (x, x) | Ht (φt (x)) | dt

¯ = (3.6.21)

0

Proof. The curve is not horizontal. The tangent has a vertical part (equal to the

Hamiltonian). Use the de¬nition of the (spherical) Hausdor¬ measure H2 and the

Ball-Box theorem 2.9 to obtain the formula (3.6.21).

We shall prove now that the Hofer distance (3.5.20) is non-degenerate. For given

φ with compact support in A and generating function F , look at the one parameter

family:

˜

φa (x, x) = (φ(x), x + F (x) + a)

¯ ¯

The volume of the cylinder

{(x, z) : x ∈ A, z between 0 and F (x) + a}

attains the minimum V (φ, A) for an a0 ∈ R.

˜ ˜

Given an arbitrary ¬‚ow t ∈ [0, 1] ’ φt ∈ Hom(H(n), vol, Lip)(A) such that φ0 =

˜ ˜

id and φ1 = φa , we have the following inequality:

˜

H2 ( t ’ φt (x, 0)

V (φ, A) ¤ dx (3.6.22)

A

˜

Indeed, the family of curves t ’ φt (x, 0) provides a foliation of the set

˜

φt (x, 0) : x ∈ A

The volume of this set is lesser than the RHS of inequality (3.6.22):

˜ ˜

H2 t ’ φt (x, 0)

φt (x, 0) : x ∈ A ¤

C V ol dx (3.6.23)

A

where C > 0 is an universal positive constant (that is it depends only on the algebraic

structure of H(n)). This is a statement which is local in nature; it is simply saying that

the area of the parallelogram de¬ned by the vectors a, b is smaller than the product

of the norms a b . We give further a proof of this inequality.

˜

Let A ‚ H(n) be a set with ¬nite Hausdor¬ measure Q = 2n + 2 which contains a

˜

set B = B — {0} such that there is a ¬‚ow

t ∈ [0, 1] ’ φt : H(n) ’ H(n)

with the properties:

49

3 THE HEISENBERG GROUP

(a) φ0 (x) = x for any x ∈ B,

(b) for almost any t φt preserves the Hausdor¬ measure Q,

(c) for any t ∈ [0, 1] φt is bi-Lipschitz and the Lipschitz constants of φt , φ’1 are

t

upper bounded by a positive constant Const,

˜

(d) A = {φt (x) : t ∈ [0, 1] , x ∈ B},

(e) for H2n almost any x ∈ B the ¬ber fx = {φt (x) : t ∈ [0, 1]} has ¬nite Hausdor¬

dimension P = 2.

Then

˜

C HQ (A) ¤ H2 (fx ) d(x)

B

Indeed, we start from the isodiametric inequality in the Heisenberg group:

Q

˜ ˜

≥ c HQ (D)

diam D

We can rescale the Hausdor¬ measure Q in order to have c = 1. Consider also the

projection π : H(n) ’ R2n , π(x, x) = x. We can rescale the Lebesgue measure on

¯

2n such that the volume of the projection of the ball B CC (˜, 1) equals 1. Here we

R x

CC (˜, R) is the ball in the Carnot-Carath´odory distance in

have used the notation: B x e

H(n), and B E (x, R) is the Euclidean ball of radius R in R2n .

Fix µ 0 small and cover B with Euclidean balls of radius µ B E (xj , µ), j ∈ Jµ , in an

e¬cient way, i.e. such that Jµ µ2n tends to vol(B) when µ ’ 0. We want now to cover

˜

A with sets of the form

˜

φtk (B CC ((xj , 0), µ))

where k depends on xj . How many sets do we need?

˜

From the isodiametric inequality and the fact that φt is volume preserving, we see

that

˜

diam φt (B CC ((xj , 0), µ)) ≥ µ

Put this together with the fact that the ¬ber fxj has Hausdor¬ dimension P = 2

and get that we need about HP (fxj )/µP such sets to cover the ¬ber fxj . The sum

of volumes of the cover is greater then the volume of A. Tend µ to 0 and obtain the

claimed inequality (3.6.23). The constant C comes from the rescalings of the measures.

Without rescalling we have:

vol(B CC (0, 1))

C=

vol(B E (0, 1)

We go back now to the main stream of the proof and remark that by continuity of

the ¬‚ow the volume of the set

˜

φt (x, 0) : x ∈ A

is greater than V (φ, A).

50

3 THE HEISENBERG GROUP

For any curve of the ¬‚ow we have the uniform obvious estimate:

1

˜

2

H ( t ’ φt (x, 0) ¤ Ht (φt (x)) dt (3.6.24)

A,∞

0

Put together inequalities (3.6.22), (3.6.24) and get

1

C V (φ, A) ¤ Ht (φt (x)) dt (3.6.25)

A,∞

0

Use the de¬nition of the Hofer distance (3.5.20) to obtain the inequality

C V (φ, A) ¤ vol(A) dH (id, φ) (3.6.26)

This proves the non-degeneracy of the Hofer distance, because if the RHS of (3.6.26)

equals 0 then V (φ, A) is 0, which means that the generating function of φ is almost

everywhere constant, therefore φ is the identity everywhere in R2n .

We close with the translation of the inequality (3.6.25) in symplectic terms.