**Proof.** Before stating the proof of Theorem 1, we introduce the notion of a 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

${R}_{\alpha}\in \mathsf{\Gamma}(TM)$ is the unique vector field which satisfies

For any smooth function

$h\in {C}_{c}^{\infty}(M)$, there exists only one contact vector field

${X}_{h}\in {\mathsf{\Gamma}}_{\xi}^{c}(TM)$ which satisfies

In fact,

${X}_{h}$ is a contact vector field if and only if

${\mathcal{L}}_{{X}_{h}}{(\alpha )|}_{\xi}=0$ holds (

$\mathcal{L}$ is the Lie derivative). So,

holds for any

$Y\in \xi $. Because

$d\alpha $ is non-degenerate on

$\xi $, above equation determines

$Z\in \xi $ uniquely.

${X}_{h}$ is the contact vector field associated to the contact Hamiltonian function

h. We denote the time

t flow of

${X}_{h}$ by

${\varphi}_{h}^{t}$ and time 1 flow of

${X}_{h}$ by

${\varphi}_{h}$.

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 a contact form

on

${\mathbb{R}}^{2n+1}$. By using the 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

which satisfies

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

We fix

$k\in {\mathbb{N}}_{\ge 1}$ and

$R>0$ so that

holds. Let

$f\in {C}_{c}^{\infty}(V)$ be a contact Hamiltonian function. Then its contact Hamiltonian vector field

${X}_{f}$ can be written in the following form

Let

$e:{\mathbb{R}}^{2n}\u27f6\mathbb{R}$ be a quadric function

We define a contact Hamiltonian function

h on

V by

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

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

$\rho (r)\ge {\rho}^{\prime}(r)\xb7r$, $-\frac{\pi}{2k}<{\rho}^{\prime}(r)\le \frac{\pi}{2k}$

There exists an unique

$a\in [0,\frac{{R}^{2}}{2}]$ which satisfies the following conditions

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

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

Then, we can prove the following lemma.

**Lemma.** **1.** Let $h\in {C}_{c}^{\infty}(V)$ be a contact Hamiltonian function as above. Then,holds if and only ifholds. **Proof of** **Lemma 1.** In order to prove this lemma, we first calculate the behavior of the function

$z({\varphi}_{h}^{t}(q))$ for a fixed

$q\in V$ (Here,

z is the

$(2n+1)$-th coordinate of

${\mathbb{R}}^{2n+1}$).

In the last inequality, we used the condition 2. So, this inequality implies that

holds.

Next, we study the behavior of

${x}_{i}({\varphi}_{h}^{t}(q))$ and

${y}_{i}({\varphi}_{h}^{t}(q))$. Let

${\pi}_{i}$ be the projection

Then,

${Y}_{h}^{i}={\pi}_{i}({X}_{h})$ can be decomposed into the angular component

${Y}_{h}^{i,\theta}$ and the radius component

${Y}_{h}^{i,r}$ as follows

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,

By conditions 2, 3, and 5 in the definition of $\beta $ and $\rho $, $|2{\rho}^{\prime}(e({x}_{1},\cdots ,{y}_{n}))\beta (z){C}_{i}|$ is at most $\frac{2\pi}{2k}$ and the equality holds if and only if $({x}_{1},{y}_{1},\cdots ,{x}_{n},{y}_{n},z)\in {S}_{a}$ holds. On the circle ${S}_{a}$, ${\varphi}_{h}$ is the $\frac{2\pi}{2k}$-rotation of the circle ${S}_{a}$. This implies that Lemma 1 holds. □

Next, we perturb the contactomorphism

${\varphi}_{h}$. Let

$(r,\theta )$ be a coordinate of

$({x}_{1},{y}_{1})\in {\mathbb{R}}^{2}\setminus (0,0)$ as follows

We fix

${\u03f5}_{k}>0$. Then

${\u03f5}_{k}(1-cos(k\theta ))$ is a contact Hamiltonian function on

${\mathbb{R}}^{2}\setminus (0,0)\times {\mathbb{R}}^{2n-1}$ and its contact Hamiltonian vector field can be written in the following form

So

${\varphi}_{{\u03f5}_{k}(1-cos(k\theta ))}$ only changes the

r of

$({x}_{1},{y}_{1})$-coordinate and

z-coordinate as follows

We fix two small open neighborhoods of the circle

${S}_{a}$ as follows

We also fix a cut-off function

$\eta :{\mathbb{R}}^{2n+1}\to [0,1]$ which satisfies the following conditions

We will use the last condition in the proof of Lemma 2. Then, $\eta ({x}_{1}.\cdots ,z)\xb7{\u03f5}_{k}(1-cos(k\theta ))$ is defined on ${\mathbb{R}}^{2n+1}$. We denote this contact Hamiltonian function by ${g}_{{\u03f5}_{k}}$. We define ${\varphi}_{{\u03f5}_{k}}\in {\mathrm{Cont}}_{0}^{c}({\mathbb{R}}^{2n+1},ker({\alpha}_{0}))$ by the composition ${\varphi}_{{g}_{{\u03f5}_{k}}}\circ {\varphi}_{h}$.

**Lemma.** **2.** We take ${\u03f5}_{k}>0$ sufficiently small. We define $2k$ points ${\left\{{a}_{i}\right\}}_{1\le i\le 2k}$ by Then ${P}^{2k}({\varphi}_{{\u03f5}_{k}})$ has only one point $[{a}_{1},{a}_{2},\cdots ,{a}_{2k}]$.

**Proof of** **Lemma 2.** The proof of this lemma is as follows. On

${W}_{1}$,

${\varphi}_{{g}_{{\u03f5}_{k}}}$ only changes the

r-coordinate of

$({x}_{1},{y}_{1})$ and

z-coordinate. So,

${\varphi}_{{\u03f5}_{k}}$ increases the angle of each

$({x}_{i},{y}_{i})$ coordinate at most

$\frac{2\pi}{2k}$ and the equality holds on only

${S}_{a}$. On the circle

${S}_{a}$, the fixed points of

${\varphi}_{{g}_{{\u03f5}_{k}}}$ are 2

k points

$\left\{{a}_{i}\right\}$. From the arguments in the proof of Lemma 1, this implies that

holds and this is the only element of

${P}^{2k}({\varphi}_{{\u03f5}_{k}})$ on

${W}_{1}$. So, it suffices to prove that this is the only element in

${P}^{2k}({\varphi}_{{\u03f5}_{k}})$ if

${\u03f5}_{k}>0$ is sufficiently small. We prove this by contradiction. Let

${\{{\u03f5}_{k}^{(j)}>0\}}_{j\in \mathbb{N}}$ be a sequence which satisfies

${\u03f5}_{k}^{(j)}\to 0$. We assume that there exists a sequence

We may assume without loss of generality that

${b}_{1}^{(j)}\notin {W}_{1}$ holds because

holds. We may assume that

${b}_{1}^{(j)}$ converges to a point

$b\notin {W}_{1}$. Then,

${\varphi}_{h}^{2k}(b)=b$ holds. If

${X}_{h}(b)\ne 0$,

${\varphi}_{h}$ increases the angle of every

$({x}_{i},{y}_{i})$ coordinate less than

$\frac{2\pi}{2k}$ and this contradicts

${\varphi}_{h}^{2k}(b)=b$. Thus

${X}_{h}(b)=0$ holds. Because we assumed

${X}_{h}(p)\ne 0$ on

$p\in {W}_{2}$,

${X}_{h}(b)=0$ implies that

$b\notin {W}_{2}$ holds. Let

$N\in \mathbb{N}$ be a large integer so that

${b}_{1}^{(N)}\notin {W}_{2}$ holds. Then,

${\varphi}_{h}^{j}({\mathbb{R}}^{2n+1}\setminus {W}_{2})\cap \mathrm{supp}(\eta )=\varnothing $$(1\le j\le 2k)$ implies that

${\varphi}_{{\u03f5}_{k}^{(N)}}^{j}({b}_{1}^{(N)})={\varphi}_{h}^{j}({b}_{1}^{(N)})$ holds for

$1\le j\le 2k$ and

$[{b}_{1}^{(N)},\cdots ,{b}_{2k}^{(N)}]\in {P}^{2k}({\varphi}_{h})$ holds. This contradicts Lemma 1 because

${b}_{1}^{(N)}\notin {S}_{a}$. So, we proved Lemma 2. □

We assume that ${\u03f5}_{k}>0$ is sufficiently small so that the conclusion of Lemma 2 holds and we define ${\varphi}_{k}$ by ${\varphi}_{k}={\varphi}_{{\u03f5}_{k}}$. Thus, we have constructed ${\varphi}_{k}\in {\mathrm{Cont}}_{0}^{c}(V,\mathrm{Ker}({\alpha}_{0}))$ which does not admit a square root for each $k\in \mathbb{N}$. Without loss of generality, we may assume that ${\u03f5}_{k}\to 0$ holds. Then ${\varphi}_{k}$ converges to $\mathrm{Id}$.

Finally, we prove Theorem 1. We define

${\psi}_{k}\in {\mathrm{Cont}}_{0}^{c}(M,\xi )$ for

$k\in \mathbb{N}$ as follows. Recall that

F is a diffeomorphism which was defined in Equation

$(1)$.

Lemma 2 implies that

holds. Proposition 1 implies that

${\psi}_{k}$ does not admit a square root. Because

$p\in M$ is any point and

U is any small open neighborhood of

p, we proved Theorem 1. □