Applications of square roots of diffeomorphisms

TL;DR

This paper demonstrates that on contact manifolds, there are arbitrarily small contactomorphisms without square roots, highlighting the complexity of non-autonomous contactomorphisms and extending the result to diffeomorphism groups.

Contribution

It proves the existence of small contactomorphisms without square roots on contact manifolds and extends this to diffeomorphism groups, advancing understanding of non-autonomous contact dynamics.

Findings

Existence of small contactomorphisms without square roots

Extension of results to diffeomorphism groups

Implications for the topology of non-autonomous contactomorphisms

Abstract

In this paper, we prove that on any contact manifold, there exists an arbitrary C^{\infty}-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C^{\infty}-small contactomorphism which is not "autonomous". This result is the first step to study the topology of non-autonomous contactomorphisms. As an application, we also prove a similar result for the diffeomorphism group for any smooth manifold.

Equations

$$f(t)=\exp (tX)$$

$$\exp :\mathcal{U}⟶\text{Diff}(M)$$

$$\text{Aut}(M)=\exp (\Gamma (TM))$$

$$\textrm{Aut}(M,\omega)=\biggl{\{}\exp(X)\ \biggl{|}\ \begin{matrix}X\ \textrm{is \ a \ time-independent \ Hamiltonian \ vector \ field}\\ \textrm{whose \ support\ is \ compact}\end{matrix}\biggl{\}}$$

$$Cont^{c}_{0}(M,\xi)=\biggl{\{}\phi_{1}\ \biggl{|}\ \begin{matrix}\phi_{t}\ (t\in[0,1])\ \textrm{is \ an \ isotopy \ of \ contactomorphisms}\\ \phi_{0}=\textrm{Id},\ \cup_{t\in[0,1]}\textrm{supp}(\phi_{t})\ \textrm{is \ compact}\end{matrix}\biggl{\}}$$

$$\text{Aut}(M,\xi )=\{\exp (X)|X\in {\Gamma }_{\xi }^c(TM)\}$$

$$\varphi \ne {\psi }^2$$

$${\text{Diff}}^c(M)=\{\varphi \in \text{Diff}(M)|\text{supp}(\varphi )\text{is compact}\}$$

$$\text{Aut}(M)=\{\exp (X)|X\in {\Gamma }^c(TM)\}$$

$$\varphi \ne {\psi }^2$$

$$P^l(\varphi )=\{(x_1,\cdots ,x_l)|x_i\ne x_j(i\ne j),x_j={\varphi }^{j-1}(x_1),x_1=\varphi (x_l)\}/\sim$$

$$(x_1,\cdots ,x_l)\to (x_l,x_1,\cdots ,x_{l-1}).$$

$$[x_1,\cdots ,x_{2k}]⟶[\psi (x_1),\cdots ,\psi (x_{2k})]$$

$$\psi (x_1)=x_r={\varphi }^{(r-1)}(x_1)={\psi }^{2(r-1)}(x_1)$$

$$x_1={\psi }^{2r-3}(x_1)={\psi }^{2r-3}({\psi }^{2r-3}(x_1))={\varphi }^{2r-3}(x_1)$$

$$x_1={\varphi }^l(x_1)=x_{l+1}$$

$$\alpha (R)=1$$

$$d\alpha (R,\cdot )=0$$

$$X_h=h\cdot R+Z(Z\in \xi )$$

$${\mathcal{L}}_{X_h}(\alpha )(Y)=dh(Y)+d\alpha (X_h,Y)=dh(Y)+d\alpha (Z,Y)=0$$

$${\alpha }_0=\frac{1}{2}\sum _{1\le i\le n}(x_{i}d{y}_i-y_{i}d{x}_i)+dz$$

$$F:V⟶U$$

$$\xi {|}_U=\text{Ker}(F_{\ast }{\alpha }_0)$$

$$\{(x_1,y_1,\cdots ,z)\in {\mathbb{R}}^{2n+1}||(x_1,\cdots ,y_n)|<R,|z|<R\}\subset V$$

$$X_f(x_1,\cdots ,z)=\sum _{1\le i\le n}(-\frac{\partial f}{\partial y_i}+\frac{x_i}{2}\frac{\partial f}{\partial z})\frac{\partial }{\partial x_i}$$

$$+\sum _{1\le i\le n}(\frac{\partial f}{\partial x_i}+\frac{y_i}{2}\frac{\partial f}{\partial z})\frac{\partial }{\partial y_i}$$

$$+(f-\sum _{1\le i\le n}\frac{x_i}{2}\frac{\partial f}{\partial x_i}-\sum _{1\le i\le n}\frac{y_i}{2}\frac{\partial f}{\partial y_i})\frac{\partial }{\partial z}$$

$$e(x_1,y_1,\cdots ,x_n,y_n)=x_1^2+y_1^2+\sum _{2\le i\le n}\frac{x_i^2+y_i^2}{2}$$

$$h(x_1,y_1,\cdots ,x_n,y_n,z)=\beta (z)\rho (e(x_1,y_1,\cdots ,x_n,y_n))$$

$$\{\begin{array}{l}{\rho }^{\prime }(r)=\frac{\pi }{2k}⟺r=a\\ \rho (a)=\frac{\pi }{2k}\cdot a\end{array}$$

$$(q,{\varphi }_h(q),\cdots ,{\varphi }_h^{2k-1}(q))\in P^{2k}({\varphi }_h)$$

$$q\in \{(x_1,y_1,0,\cdots ,0)\in V|x_1^2+y_1^2=a\}=S_a$$

Full text

1 Introduction

For any closed manifold $M$, the set of diffeomorphisms ${\textrm{Diff}(M)}$ forms a group and any one-parameter subgroup ${f:\mathbb{R}\rightarrow\textrm{Diff}(M)}$ can be written in the following form.

[TABLE]

Here, ${X\in\Gamma(TM)}$ is a vector field and ${\exp:\Gamma(TM)\rightarrow\textrm{Diff}(M)}$ is a time $1$ flow of a vector field. From the inverse function theorem, one might expect that there exists an open neighborhood of the zero section ${\mathcal{U}\subset\Gamma(TM)}$ such that

[TABLE]

is a diffeomorphism onto an open neighborhood of ${\textrm{Id}\in\textrm{Diff}(M)}$. However, this is far from true ([2], Warning 1.6). So one might expect that the set of ”autonomous” diffeomorphisms

[TABLE]

is a small subset of ${\textrm{Diff}(M)}$.

For a symplectic manifold ${(M,\omega)}$, the set of Hamiltonian diffeomorphisms ${\textrm{Ham}^{c}(M,\omega)}$ contains ”autonomous” subset ${\textrm{Aut}(M,\omega)}$ as follows.

[TABLE]

In [1], Albers and Frauenfelder proved that on any symplectic manifold there exists an arbitrary ${C^{\infty}}$-small Hamiltonian diffeomorphism not admitting a square root. In particular, there exists an arbitrary ${C^{\infty}}$-small Hamiltonian diffeomorphism in ${\textrm{Ham}^{c}(M,\omega)\backslash\textrm{Aut}(M,\omega)}$.

Polterovich and Shelukhin used spectral spread of Floer homology and Conley conjecture to prove that ${\textrm{Ham}^{c}(M,\omega)\backslash\textrm{Aut}(M,\omega)\subset\textrm{Ham}^{c}(M,\omega)}$ is ${C^{\infty}}$-dense and dense in the topology induced from Hofer’s metric if ${(M,\omega)}$ is closed symplectically aspherical manifold ([3]). The author generalized this theorem to arbitrary closed symplectic manifolds and convex symplectic manifolds ([4]).

One might expect that ”contact manifold” version of these theorems hold. In this paper, we prove that there exists an arbitrary ${C^{\infty}}$-small contactomorphism not admitting a square root. In particular, there exists an arbitrary ${C^{\infty}}$-small contactomorphism in ${\textrm{Cont}^{c}_{0}(M,\xi)\backslash\textrm{Aut}(M,\xi)}$. So, this paper is a contact manifold version of [1]. As an application, we prove that there exists an arbitrary ${C^{\infty}}$-small diffeomorphism in ${\textrm{Diff}^{c}_{0}(M)}$ not admitting a square root. This also implies that there exists an arbitrary ${C^{\infty}}$-small diffeomorphism in ${\textrm{Diff}^{c}_{0}(M)\backslash\textrm{Aut}(M)}$.

Acknowledgement

The author thanks Kaoru Ono and Urs Frauenfelder for many useful comments, discussions and encouragement.

2 Main result

Let $M$ be a smooth ${(2n+1)}$-dimensional manifold without boundary. A $1$-form $\alpha$ on $M$ is called contact if ${(\alpha\wedge(d\alpha)^{n})(p)\neq 0}$ holds on any ${p\in M}$. A codimension $1$ tangent distribution $\xi$ on $M$ is called contact structure if it is locally defined by ${\ker(\alpha)}$ for some (locally defined) contact form $\alpha$. A diffeomorphism ${\phi\in\textrm{Diff}(M)}$ is called contactomorphism if ${\phi_{*}\xi=\xi}$ holds (i.e. ${\phi}$ preserves the contact structure $\xi$). Let ${\textrm{Cont}^{c}_{0}(M,\xi)}$ be the set of compact supported contactomorphisms which are isotopic to Id. In other words, ${\textrm{Cont}^{c}_{0}(M,\xi)}$ is a connected component of compact supported contactomorphisms(${\textrm{Cont}^{c}(M,\xi)}$) which contains Id.

[TABLE]

Let ${X\in\Gamma^{c}(TM)}$ be a compact supported vector field on $M$. $X$ is called contact vector field if the flow of $X$ preserves the contact structure $\xi$ (i.e. ${\exp(X)_{*}\xi=\xi}$ holds). Let ${\Gamma^{c}_{\xi}(TM)}$ be the set of compact supported contact vector fields on $M$ and let ${\textrm{Aut}(M,\xi)}$ be their images.

[TABLE]

We prove the following theorem.

Theorem 1

Let ${(M,\xi)}$ be a contact manifold without boundary. Let $\mathcal{W}$ be any ${C^{\infty}}$-open neighborhood of $\textrm{Id}\in\textrm{Cont}^{c}_{0}(M,\xi)$. Then, there exists ${\phi\in\mathcal{W}}$ such that

[TABLE]

holds for any ${\psi\in\textrm{Cont}^{c}_{0}(M,\xi)}$. In particular, ${\mathcal{W}\backslash\textrm{Aut}(M,\xi)}$ is not empty.

Remark 1

If $\phi$ is autonomous ($\phi=\exp(X)$), $\phi$ has a square root ${\psi=\exp(\frac{1}{2}X)}$.

Corollary 1

The exponential map ${\exp:\Gamma^{c}_{\xi}(TM)\rightarrow\textrm{Cont}^{c}_{\xi}(M,\xi)}$ is not surjective.

We also consider diffeomorphism version of this theorem and corollary. Let $M$ be a smooth manifold without boundary and let ${\textrm{Diff}^{c}(M)}$ be the set of compact supported diffeomorhisms which contains Id.

[TABLE]

Let ${\textrm{Diff}^{c}_{0}(M)}$ be the connected component of ${\textrm{Diff}^{c}(M)}$(i.e. any element of ${\textrm{Diff}^{c}_{0}(M)}$ is isotopic to Id.). We define the set of autonomous diffeomorphisms as follows.

[TABLE]

By combining the arguments in this paper and in [1], we can prove the following theorem.

Theorem 2

Let $M$ be a smooth manifold without boundary. Let $\mathcal{W}$ be any ${C^{\infty}}$-open neighborhood of $\textrm{Id}\in\textrm{Diff}^{c}_{0}(M)$. Then, there exists ${\phi\in\mathcal{W}}$ such that

[TABLE]

holds for any ${\psi\in\textrm{Diff}^{c}(M)}$. In particular, ${\mathcal{W}\backslash\textrm{Aut}(M)}$ is not empty.

Corollary 2

The exponential map ${\exp:\Gamma^{c}(TM)\rightarrow\textrm{Diff}^{c}_{0}(M)}$ is not surjective.

3 Milnor’s criterion

In this section, we review Milnor’s arguments in [2]. In [2], Milnor gave a criterion for the existence of square root of a diffeomorphism. We use this criterion later. We fix ${l\in\mathbb{N}_{\geq 2}}$ and a diffeomorphism ${\phi\in\textrm{Diff}(M)}$. Let ${P^{l}(\phi)}$ be the set of ”$l$-periodic orbits” as follows.

[TABLE]

This equivalence relation $\sim$ is given by the natural ${\mathbb{Z}/l\mathbb{Z}}$-action

[TABLE]

Proposition 1** (Milnor[2], Albers-Frauenfelder[1])**

Assume that ${\phi\in\textrm{Diff}(M)}$ has a square root (i.e. there exists ${\psi\in\textrm{Diff}(M)}$ such that ${\phi=\psi^{2}}$ holds). Then, there exists a free ${\mathbb{Z}/2\mathbb{Z}}$-action on ${P^{2k}(\phi)}$ (${k\in\mathbb{N}}$). In particular, ${\sharp P^{2k}(\phi)}$ is even if ${\sharp P^{2k}(\phi)}$ is finite.

The proof of this proposition is very simple. We explain the proof for the sake of self-containedness. Let ${\psi}$ be a square root of ${\phi}$ (${\phi=\psi^{2}}$). It is sufficient to prove that the natural ${\mathbb{Z}/2\mathbb{Z}}$-action

[TABLE]

on ${P^{2k}(\phi)}$ is free. We prove this by contradiction. So, we assume that ${[x_{1},\cdots,x_{2k}]}$ is a fixed point of this ${\mathbb{Z}/2\mathbb{Z}}$-action. Then there exists ${1\leq r\leq 2k}$ so that ${\psi(x_{1})=x_{r}}$ holds. This implies that

[TABLE]

holds. Let $1\leq l<2k$ be the greatest common divisor of ${2k}$ and ${2r-3}$. Then,

[TABLE]

holds. This is a contradiction.

$\Box$

4 Proof of Theorem 1

Before stating the proof of Theorem 1, we introduce the notion of contact Hamiltonian function. Let ${M}$ be a smooth manifold without boundary and let ${\alpha\in\Omega^{1}(M)}$ be a contact form on $M$ (${\xi=\ker(\alpha)}$). A Reeb vector field ${R\in\Gamma(TM)}$ is defined as follows.

[TABLE]

For any smooth function ${h\in C_{c}^{\infty}(M)}$, there exists an unique contact vector field ${X_{h}\in\Gamma_{\xi}^{c}(TM)}$ which satisfies the following condition.

[TABLE]

In fact, $X_{h}$ is contact vector field if and only if ${\mathcal{L}_{X_{h}}(\alpha)|_{\xi}=0}$ holds ($\mathcal{L}$ is the Lie derivative.). So,

[TABLE]

holds for any ${Y\in\xi}$. Because ${d\alpha}$ is non-degenerate on ${\xi}$, above equation determines ${Z\in\xi}$ uniquely. ${X_{h}}$ is a contact vector field associated to a contact Hamiltonian function $h$. We denote the time $t$ flow of ${X_{h}}$ by ${\phi^{t}_{h}}$ and time $1$ flow of ${X_{h}}$ by ${\phi_{h}}$.

Let ${(M,\xi)}$ be a contact manifold without boundary. We fix a point ${p\in(M,\xi)}$ and a sufficiently small open neighborhood ${U\subset M}$ of $p$. Let ${(x_{1},y_{1},\cdots,x_{n},y_{n},z)}$ be a coordinate of ${\mathbb{R}^{2n+1}}$. Let ${\alpha_{0}\in\Omega^{1}(\mathbb{R}^{2n+1})}$ be the following contact form on ${\mathbb{R}^{2n+1}}$.

[TABLE]

By using famous Moser’s arguments, we can assume that there exists an open neighborhood of the origin ${V\subset\mathbb{R}^{2n+1}}$ and a diffeomorphism

[TABLE]

which satisfies the following condition.

[TABLE]

So, we first prove the theorem for ${(M,\xi)=(V,\ker(\alpha_{0}))}$ and apply this to ${(M,\xi)}$.

We fix ${k\in\mathbb{N}_{\geq 1}}$ and ${R>0}$ so that

[TABLE]

holds. Let ${f\in C^{\infty}_{c}(V)}$ be a contact Hamiltonian function. Then its contact Hamiltonian vector field ${X_{f}}$ can be written in the following form.

[TABLE]

Let ${e:\mathbb{R}^{2n}\longrightarrow\mathbb{R}}$ be a quadric function as follows.

[TABLE]

We consider the following contact Hamiltonian function on ${\mathbb{R}^{2n+1}}$.

[TABLE]

Here, ${\beta:\mathbb{R}\rightarrow[0,1]}$ and ${\rho:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}}$ are smooth functions which satisfy the following conditions.

• •

${\textrm{supp}(\rho)\subset[0,\frac{R^{2}}{2}]}$

• •

${\rho(r)\geq\rho^{\prime}(r)\cdot r}$, ${-\frac{\pi}{2k}<\rho^{\prime}(r)\leq\frac{\pi}{2k}}$

• •

There exists an unique ${a\in[0,\frac{R^{2}}{2}]}$ which satisfies the following conditions.

[TABLE]

• •

$\textrm{supp}(\beta)\subset[-\frac{R}{2},\frac{R}{2}]$

• •

$\beta(0)=1$, $\beta^{\prime}(0)=0$, ${\beta^{-1}(1)=0}$

Then, we can prove the following lemma.

Lemma 1

Let ${h\in C^{\infty}_{c}(V)}$ be a contact Hamiltonian function as above. Then,

[TABLE]

holds if and only if

[TABLE]

holds.

In order to prove this lemma, we first calculate the behavior of the function ${z(\phi^{t}_{h}(q))}$ for fixed ${q\in V}$ (Here, $z$ is the ${(2n+1)}$-th coordinate of ${\mathbb{R}^{2n+1}}$.).

[TABLE]

So, this inequality implies that

[TABLE]

holds.

Next, we study the behavior of ${x_{i}(\phi^{t}_{h}(q))}$ and ${y_{i}(\phi^{t}_{h}(q))}$. Let ${\phi_{i}}$ be the following projection.

[TABLE]

Then, ${Y_{h}^{i}=\pi_{i}(X_{h})}$ can be decomposed into angular component ${Y_{h}^{i,\theta}}$ and radius component ${Y_{h}^{i,r}}$ as follows.

[TABLE]

Let ${w_{i}}$ be the complex coordinate of ${(x_{i},y_{i})}$ (${w_{i}=x_{i}+\sqrt{-1}y_{i}}$). Then, the angular component causes the following rotation on ${w_{i}}$ (If we ignore the $z$-coordinate).

[TABLE]

[TABLE]

By the assumptions of ${\rho}$ and ${\beta}$, ${|2\rho^{\prime}(e(x_{1},\cdots,y_{n}))\beta(z)C_{i}|}$ is at most ${\frac{2\pi}{2k}}$ and equality holds if and only if ${(x_{1},y_{1},\cdots,x_{n},y_{n},z)\in S_{a}}$ holds. On the circle ${q\in S_{a}}$, ${\phi_{h}}$ is the ${\frac{2\pi}{2k}}$-rotation of the circle ${S_{a}}$. This implies that the lemma holds.

$\Box$

Next, we perturb the contactomorphism ${\phi_{h}}$. Let ${(r,\theta)}$ be a coordinate of ${(x_{1},y_{1})\in\mathbb{R}^{2}\backslash(0,0)}$ as follows.

[TABLE]

We fix ${\epsilon>0}$. Then ${\epsilon(1-\cos(k\theta))}$ is a contact Hamiltonian function on ${\mathbb{R}^{2}\backslash(0,0)\times\mathbb{R}^{2n-1}}$ and its contact Hamiltonian vector field can be written in the following form.

[TABLE]

So ${\phi_{\epsilon(1-\cos(k\theta))}}$ only changes the $r$ of ${(x_{1},y_{1})}$-coordinate and ${z}$-coordinate as follows.

[TABLE]

We fix two small open neighborhood of the circle ${S_{a}}$ as follows.

[TABLE]

We also fix a cut-off function ${\eta:\mathbb{R}^{2n+1}\rightarrow[0,1]}$ as follows.

[TABLE]

Then, ${\eta(x_{1}.\cdots,z)\cdot\epsilon(1-\cos(k\theta))}$ is defined on ${\mathbb{R}^{2n+1}}$. We denote this contact Hamiltonian function by ${g_{\epsilon}}$. We define ${\phi_{\epsilon}\in\textrm{Cont}^{c}_{0}(\mathbb{R}^{2n+1},\ker(\alpha_{0}))}$ by the composition ${\phi_{g_{\epsilon}}\circ\phi_{h}}$.

Lemma 2

We take ${\epsilon>0}$ sufficiently small. We define ${2k}$ points ${\{a_{i}\}_{1\leq i\leq 2k}}$ as follows.

[TABLE]

Then ${P^{2k}(\phi_{\epsilon})}$ has only one point ${[a_{1},a_{2},\cdots,a_{2k}]}$.

The proof of this lemma is as follows. On $W_{1}$, $\phi_{g_{\epsilon}}$ only changes the $r$-coordinate of ${(x_{1},y_{1})}$ and ${z}$-coordinate. On the circle ${S_{a}}$, the fixed points of ${\phi_{g_{\epsilon}}}$ are 2k points ${\{a_{i}\}}$. This implies that on $W_{1}$, ${\phi_{\epsilon}}$ increases the angle ${\theta}$ at most ${\frac{2\pi}{2k}}$ and the equality holds on only 2k points ${\{a_{i}\}}$. From the arguments in the proof of Lemma 1, this implies that

[TABLE]

holds if $\epsilon$ is sufficiently small.

$\Box$

Finally, we prove Theorem 1. We define ${\psi_{\epsilon}\in\textrm{Cont}^{c}_{0}(M,\xi)}$ as follows.

[TABLE]

Lemma 2 implies that

[TABLE]

holds. Proposition 1 implies that ${\psi_{\epsilon}\in\textrm{Cont}^{c}_{0}(M,\xi)\backslash\textrm{Aut}(M,\xi)}$ holds. Because ${p\in M}$ is any point and ${U}$ is any small open neighborhood of $p$, we proved Theorem 1.

$\Box$

5 Proof of Theorem 2

Let $M$ be a $m$-dimensional smooth manifold without boundary. We fix a point ${p\in M}$. Let ${U}$ be an open neighborhood of $p$ and let ${V\subset\mathbb{R}^{m}}$ be an open neighborhood of the origin such that there is a diffeomorphism

[TABLE]

In order to prove Theorem 2, it suffices to prove that there exists a sequence ${\psi_{\epsilon}}$ so that

• •

${\psi_{\epsilon}\in\textrm{Diff}^{c}_{0}(M)\backslash\textrm{Aut}(M)}$

• •

${\textrm{supp}(\Psi_{\epsilon})\subset U}$

• •

${\psi_{\epsilon}\longrightarrow}\textrm{Id}$ as ${\epsilon\rightarrow 0}$

hold.

First, assume that ${m}$ is odd (${m=2n+1}$). In this case, $\alpha_{0}$ is a contact form on $V$. Let ${\phi_{\epsilon}}$ be a contactomorphism which we constructed in the proof of Theorem 1.

• •

${\phi_{\epsilon}\in\textrm{Cont}^{c}_{o}(\mathbb{R}^{2n+1},\ker(\alpha_{0}))}$

• •

${\textrm{supp}(\phi_{\epsilon})\subset V}$

We define ${\psi_{\epsilon}\in\textrm{Diff}^{c}_{0}(M)}$ as follows.

[TABLE]

Then, ${\sharp P^{2k}(\psi_{\epsilon})=1}$ holds and this implies that ${\psi_{\epsilon}\in\textrm{Diff}^{c}_{0}(M)\backslash\textrm{Aut}(M)}$ holds.

Next, assume that $m$ is even (${m=2n}$). Let ${\omega_{0}}$ be a standard symplectic form on ${(x_{1},y_{1},\cdots,x_{n},y_{n})\in\mathbb{R}^{2n}}$ as follows.

[TABLE]

As in the contact case, Hamiltonian function ${h\in C_{c}^{\infty}(V)}$ defines Hamiltonian vector field ${X_{h}}$ as follows.

[TABLE]

We denote the time $1$ flow of $X_{h}$ by ${\phi_{h}}$.

Remark 2

We can use the arguments in [1]. However, our perturbation is slightly different from that of [1] (Our perturbation is in the relatively compact domain U.). So, we explain our arguments in detail for the sake of self-containedness.

We assume that ${B(R)\subset V}$ holds (${B(R)}$ is a radius $R$-ball in ${\mathbb{R}^{2n}}$.). Let ${e:\mathbb{R}^{2n}\rightarrow\mathbb{R}}$ be a following quadric function.

[TABLE]

Let ${\rho:[0,\infty)\rightarrow\mathbb{R}_{\geq 0}}$ be a non-negative function which satisfies the following conditions.

• •

${\textrm{supp}(\rho)=[0,\frac{R^{2}}{2}]}$

• •

${-\frac{k}{2\pi}<\rho^{\prime}\leq\frac{k}{2\pi}}$, $\rho^{\prime}(r)=\frac{k}{2\pi}\Longleftrightarrow r=a$

Let ${h\in C^{\infty}_{c}(V)}$ be a Hamiltonian function as follows.

[TABLE]

The following lemma is an analogy of Lemma 1.

Lemma 3

Let $h\in C^{\infty}_{c}(V)$ be a Hamiltonian function as above. Then,

[TABLE]

holds if and only if

[TABLE]

holds.

The proof of this lemma is similar to that of Lemma 1. Let ${w_{i}}$ be a complex coordinate of ${(x_{i},y_{i})}$ (${w_{i}=x_{i}+\sqrt{-1}y_{i}}$). Then, ${X_{h}}$ causes the following rotaion on ${w_{i}}$.

[TABLE]

[TABLE]

${|2\rho^{\prime}(e)C_{i}|}$ is at most ${\frac{2\pi}{2k}}$ and equality holds if and only if ${(x_{1},\cdots,y_{n})\in S_{a}}$ holds. This implies that the lemma holds.

${\Box}$

Next we perturb this ${\phi_{h}\in\textrm{Ham}^{c}(V)}$. Let ${(r,\theta)}$ be a coordinate of ${\mathbb{R}^{2}\backslash(0,0)}$ as follows.

[TABLE]

We fix ${\epsilon>0}$, then ${\epsilon\cos(k\theta)}$ is a Hamiltonian function on ${\mathbb{R}^{2}\backslash(0,0)\times\mathbb{R}^{2n-2}}$. The Hamiltonian vector field of this Hamiltonian function can be written in the following form.

[TABLE]

So, ${\phi_{\epsilon\cos(k\theta)}}$ only changes the $r$ of ${(x_{1},y_{1})}$-coordinate.

[TABLE]

We fix two small open neighborhoods of ${S_{a}}$ and cut-off function ${\eta:\mathbb{R}^{2n}\rightarrow\mathbb{R}}$ as follows.

[TABLE]

We denote the Hamiltonian function ${\eta\cdot\epsilon\cos(k\theta)\in C^{\infty}_{c}(\mathbb{R}^{2n})}$ by ${g_{\epsilon}}$ and let ${\phi_{\epsilon}\in\textrm{Ham}^{c}(\mathbb{R}^{2n})}$ be the composition ${\phi_{g_{\epsilon}}\circ\phi_{h}}$. The next lemma is an analogy of Lemma 2 and its proof is almost the same. So we omit to prove it.

Lemma 4

We take ${\epsilon>0}$ sufficiently small. We define ${2k}$ points ${\{a_{i}\}_{1\leq i\leq 2k}}$ as follows.

[TABLE]

Then, ${P^{2k}(\phi_{\epsilon})}$ has only one point ${[a_{1},a_{2},\cdots,a_{2k}]}$.

We define ${\psi_{\epsilon}\in\textrm{Diff}^{c}_{0}(M)}$ as follows.

[TABLE]

Lemma 4 implies that ${\sharp P^{2k}(\psi_{\epsilon})=1}$ holds and this implies that

[TABLE]

holds. So we proved Theorem 2.

${\Box}$