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We define a Hofer-type norm for the Hamiltonian map on regular Poisson
manifold and prove that it is nondegenerate. We show that the

This paper is devoted to establishing an invariant norm for Hamiltonian maps on the Poisson manifold. When

A smooth diffeomorphism

Given

For

Now one can define the energy of

Let

Here a Poisson manifold is called regular if the rank of the Poisson manifold is constant for all point. If one replaces the

For

Let

For a

For a

Moreover, if the path is length-minimizing, one has the following corollary.

If the

In this section, we recall the construction of Hofer-norm on

For

We just need to show the triangle inequality holds. Let

From the definition we can see that

The new pseudonorm

First, note that

So

We get that

Now consider a Hamiltonian function

This length is well defined; that is, it is independent the choice of the Hamiltonian functions. This is because of our pseudonorm vanishing on

Now one can define the energy of

The energy function

where

To prove this, we need to investigate the Hamiltonian functions of the Hamiltonian flows. Similar to the symplectic case, for the symplectic case, see page 144 of [

If

If

To prove this, we need the following fact.

If

For any function

The third formula is just the transition law of Hamiltonian vector. We know that if

We need to show that

By the property of the Poisson diffeomorphism, we get that the second term is

We are now ready to prove Proposition

From Proposition

To prove the last one, we note that

Now we can define

The proof of this theorem is a consequence of Proposition

We consider the trivial Poisson manifold

Theorem

Next we can get an estimate for the commutators in

From Proposition

If

Define

For a subset

Following [

A map

For any

The proof is a consequence of the definitions of

Let

Assume that we can extend the Hofer metric to the groups of Poisson maps; we still denote the metric by

Now we consider the geodesic under the above norm in

Let

First, by the definition we have

So

For the time-dependent case, we have a similar result. First we recall the definition of quasiautonomous function.

A function

Let

We first adopt the transformation in [

By Theorem

If the Poisson manifold is symplectic, then Theorems

Let

If all

We employ the method of Oh in [

Assume that

then

We assume that

If

then

If one replaces the

For a Hamiltonian function

The energy of

Recall that in the symplectic case, Polterovich proved that the

We first show the following results which will be useful in the proof.

We just show that this is true on each symplectic leaf, but in the symplectic case, the Casimir functions are constants; this finishes the proof.

Let

First, by Proposition

Let

By the above proposition, we now compute

Now we define

Here every

Fixed a regular point

the vectors

Now choose

Take the corresponding

But these vectors generates the whole

For

For

Let

In this section, we briefly introduce the Poisson reduction. Let

The Poisson structure induced by the bracket

So we have a well-defined homomorphism

More details can be found in [

Now we can give the proof of Theorem

From the above discussion, we know that For any

Let

By the definition of the norm, we have

Moreover, if the path is length-minimizing, that is,

If the Poisson manifold is symplectic, then the pseudonorm is the Hofer norm. Give a Hamiltonian

The authors declare that they have no conflicts of interest regarding this work.

The authors would like to express their deep gratitude to Professor Yiming Long for many valuable discussions. The research was supported by TianYuan Program of National Natural Science Foundation of China (11226158), Natural Science Foundation of Henan (2011B110011), and Doctor Fund of Henan University of Technology.

^{∞}-geometry: energy and stability of Hamiltonian flows, part I

^{∞}-geometry: energy and stability of Hamiltonian flows, part II