Applications of Square Roots of Diffeomorphisms

Abstract

In this paper, we prove that on any contact manifold $(M,\xi )$ 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 paper is the first step to study the topology of $Con{t}_0(M,\xi )\setminus \mathrm{Aut}(M,\xi )$. As an application, we also prove a similar result for the diffeomorphism group $\mathrm{Diff}(M)$ for any smooth manifold M.

1. Introduction

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

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

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

$$exp:\mathcal{U}⟶\mathrm{Diff}(M)$$

is a diffeomorphism onto an open neighborhood of $\mathrm{Id}\in \mathrm{Diff}(M)$. However, this is far from true ([1], Warning 1.6). So one might expect that the set of “autonomous” diffeomorphisms

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

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

For a symplectic manifold $(M,\omega )$, the set of Hamiltonian diffeomorphisms ${\mathrm{Ham}}^c(M,\omega )$ contains “autonomous” subset $\mathrm{Aut}(M,\omega )$ which is defined by

$$\mathrm{Aut}(M,\omega )=\{exp(X)|\begin{array}{c}X\mathrm{is}\mathrm{a}\mathrm{time}-\mathrm{independent}\mathrm{Hamiltonian}\mathrm{vector}\mathrm{field}\\ \mathrm{whose}\mathrm{support}\mathrm{is}\mathrm{compact}\end{array}\}.$$

In [2], 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 ${\mathrm{Ham}}^c(M,\omega )\setminus \mathrm{Aut}(M,\omega )$.

Polterovich and Shelukhin used spectral spread of Floer homology and Conley conjecture to prove that ${\mathrm{Ham}}^c(M,\omega )\setminus \mathrm{Aut}(M,\omega )\subset {\mathrm{Ham}}^c(M,\omega )$ 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 ${\mathrm{Cont}}_0^c(M,\xi )\setminus \mathrm{Aut}(M,\xi )$. So, this paper is a contact manifold version of [2]. As an application, we prove that there exists an arbitrary $C^{\infty }$-small diffeomorphism in ${\mathrm{Diff}}_0^c(M)$ not admitting a square root. This also implies that there exists an arbitrary $C^{\infty }$-small diffeomorphism in ${\mathrm{Diff}}_0^c(M)\setminus \mathrm{Aut}(M)$.

2. Main Result

Let M be a smooth $(2n+1)$-dimensional manifold without boundary. A 1-form $\alpha$ on M is called contact if $(\alpha \wedge {(d\alpha )}^n)(p)\ne 0$ 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 $\varphi \in \mathrm{Diff}(M)$ is called contactomorphism if ${\varphi }_{*}\xi =\xi$ holds (i.e., $\varphi$ preserves the contact structure $\xi$). Let ${\mathrm{Cont}}_0^c(M,\xi )$ be the set of compactly supported contactomorphisms which are isotopic to $\mathrm{Id}$ through compactly supported contactomorphisms. In other words, ${\mathrm{Cont}}_0^c(M,\xi )$ is a connected component of compactly supported contactomorphisms (${\mathrm{Cont}}^c(M,\xi )$) which contains $\mathrm{Id}$.

$$Con{t}_0^c(M,\xi )=\left\{{\varphi }_1\right|\begin{array}{c}{\varphi }_t(t\in [0,1])\mathrm{is}\mathrm{an}\mathrm{isotopy}\mathrm{of}\mathrm{contactomorphisms}\\ {\varphi }_0=\mathrm{Id},{\cup }_{t\in [0,1]}\mathrm{supp}({\varphi }_t)\mathrm{is}\mathrm{compact}\end{array}\}$$

Let $X\in {\mathsf{\Gamma }}^c(TM)$ be a compactly 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 ${\mathsf{\Gamma }}_{\xi }^c(TM)$ be the set of compactly supported contact vector fields on M and let $\mathrm{Aut}(M,\xi )$ be their images

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

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 $\mathrm{Id}\in {\mathit{Cont}}_0^c(M,\xi )$. Then, there exists $\varphi \in \mathcal{W}$ such that

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

holds for any $\psi \in {\mathit{Cont}}_0^c(M,\xi )$. In particular, $\mathcal{W}\setminus \mathit{Aut}(M,\xi )$ is not empty.

Remark 1.

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

Corollary 1.

The exponential map $exp:{\mathsf{\Gamma }}_{\xi }^c(TM)\to {\mathit{Cont}}_0^c(M,\xi )$ is not surjective.

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

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

Let ${\mathrm{Diff}}_0^c(M)$ be the connected component of ${\mathrm{Diff}}^c(M)$ (i.e., any element of ${\mathrm{Diff}}_0^c(M)$ is isotopic to $\mathrm{Id}$). We define the set of autonomous diffeomorphisms by

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

By combining the arguments in this paper and in [2], 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 $\mathrm{Id}\in {\mathit{Diff}}_0^c(M)$. Then, there exists $\varphi \in \mathcal{W}$ such that

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

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

Corollary 2.

The exponential map $exp:{\mathsf{\Gamma }}^c(TM)\to {\mathit{Diff}}_0^c(M)$ is not surjective.

3. Milnor’s Criterion

In [1], Milnor gave a criterion for the existence of a square root of a diffeomorphism. We use this criterion later. We fix $l\in {\mathbb{N}}_{\ge 2}$ and a diffeomorphism $\varphi \in \mathrm{Diff}(M)$. Let $P^l(\varphi )$ be the set of “l-periodic orbits” which is defined by

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

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

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

Proposition 1

(Milnor [1], Albers-Frauenfelder [2]). Assume that $\varphi \in \mathit{Diff}(M)$ has a square root (i.e., there exists $\psi \in \mathit{Diff}(M)$ such that $\varphi ={\psi }^2$ holds). Then, there exists a free $\mathbb{Z}/2\mathbb{Z}$-action on $P^{2k}(\varphi )$ ($k\in \mathbb{N}$). In particular, $♯P^{2k}(\varphi )$ is even if $♯P^{2k}(\varphi )$ is finite.

4. Proof of Theorem 1

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

$$\begin{array}{c}\alpha (R_{\alpha })=1\\ d\alpha (R_{\alpha },·)=0.\end{array}$$

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

$$\begin{array}{c}{X}_h=h·R_{\alpha }+Z\mathrm{where}Z\in \xi .\end{array}$$

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,

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

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

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

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

$$\begin{array}{c}F:V⟶U\end{array}$$

(1)

which satisfies

$${\xi |}_U=ker({(F^{-1})}^{*}{\alpha }_0).$$

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

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

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

$$\begin{array}{c}{X}_f(x_1,\cdots ,z)=\sum _{1\le i\le n}(-\frac{\partial f}{\partial y_i}+\frac{x_i}{2}\frac{\partial f}{\partial z})\frac{\partial }{\partial x_i}\\ +\sum _{1\le i\le n}(\frac{\partial f}{\partial x_i}+\frac{y_i}{2}\frac{\partial f}{\partial z})\frac{\partial }{\partial y_i}\\ +(f-\sum _{1\le i\le n}\frac{x_i}{2}\frac{\partial f}{\partial x_i}-\sum _{1\le i\le n}\frac{y_i}{2}\frac{\partial f}{\partial y_i})\frac{\partial }{\partial z}.\end{array}$$

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

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

We define a contact Hamiltonian function h on V by

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

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.

• $\mathrm{supp}(\rho )\subset [0,\frac{R^2}{2}]$
• $\rho (r)\ge {\rho }^{\prime }(r)·r$, $-\frac{\pi }{2k}<{\rho }^{\prime }(r)\le \frac{\pi }{2k}$
• There exists an unique $a\in [0,\frac{R^2}{2}]$ which satisfies the following conditions

  $$\left\{\begin{array}{c}{\rho }^{\prime }(r)=\frac{\pi }{2k}⟺r=a\hfill \\ \rho (a)=\frac{\pi }{2k}·a\hfill \end{array}\right..$$
• $\mathrm{supp}(\beta )\subset [-\frac{R}{2},\frac{R}{2}]$
• $\beta (0)=1$, ${\beta }^{-1}(1)=0$

Then, we can prove the following lemma.

Lemma. 1.

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

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

holds if and only if

$$q\in \{(x_1,y_1,0,\cdots ,0)\in V|x_1^2+y_1^2=a\}\stackrel{\mathrm{def}.}{=}{S}_a$$

holds.

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

$$\begin{array}{c}\frac{d}{dt}(z({\varphi }_h^t(q)))=h-\sum _{1\le i\le n}\frac{x_i}{2}\frac{\partial h}{\partial x_i}-\sum _{1\le i\le n}\frac{y_i}{2}\frac{\partial h}{\partial y_i}\\ =\beta (z)\{\rho (e)-\sum _{1\le i\le n}\frac{x_i}{2}\frac{\partial }{\partial x_i}(\rho (e))-\sum _{1\le i\le n}\frac{y_i}{2}\frac{\partial }{\partial y_i}(\rho (e))\}\\ =\beta (z)\{\rho (e)-{\rho }^{\prime }(e)·e\}\ge 0\end{array}$$

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

$${\varphi }_h^{2k}(q)=q⟹\frac{d}{dt}(z({\varphi }_h^t(q)))=0$$

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

$$\begin{array}{c}{\pi }_i:{\mathbb{R}}^{2n+1}⟶{\mathbb{R}}^2.\\ (x_1,y_1,\cdots ,x_n,y_n,z)\mapsto (x_i,y_i)\end{array}$$

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

$$\begin{array}{c}{Y}_h^{i,\theta }(x_1,y_1,\cdots ,z)=-\frac{\partial h}{\partial y_i}\frac{\partial }{\partial x_i}+\frac{\partial h}{\partial x_i}\frac{\partial }{\partial y_i}\\ Y_h^{i,r}(x_1,y_1,\cdots ,z)=(\frac{1}{2}\frac{\partial h}{\partial z})(x_i\frac{\partial }{\partial x_i}+y_i\frac{\partial }{\partial y_i}).\end{array}$$

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,

$$arg(w_i)⟶arg(w_i)+2{\rho }^{\prime }(e(x_1,\cdots ,y_n))\beta (z)C_{i}t$$

$$C_i=\left\{\begin{array}{cc}1\hfill & i=1\hfill \\ \frac{1}{2}\hfill & 2\le i\le n\hfill \end{array}\right..$$

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

$$x_1=rcos\theta ,y_1=rsin\theta .$$

We fix ${ϵ}_k>0$. Then ${ϵ}_k(1-cos(k\theta ))$ 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

$$\begin{array}{c}{X}_{{ϵ}_k(1-cos(k\theta ))}=-\frac{{ϵ}_{k}k}{r}sin(k\theta )\frac{\partial }{\partial r}+{ϵ}_k(1-cos(k\theta ))\frac{\partial }{\partial z}.\end{array}$$

So ${\varphi }_{{ϵ}_k(1-cos(k\theta ))}$ only changes the r of $(x_1,y_1)$-coordinate and z-coordinate as follows

$$(r,\theta ,x_2,y_2,\cdots ,x_n,y_n,z)\mapsto (\sqrt{r^2-2{ϵ}_{k}ksin(k\theta )},\theta ,x_2,\cdots ,y_n,z+{ϵ}_k(1-cos(k\theta ))).$$

We fix two small open neighborhoods of the circle $S_a$ as follows

$$\begin{array}{c}{S}_a\subset W_1\subset W_2\subset {\mathbb{R}}^2\setminus (0,0)\times {\mathbb{R}}^{2n-1}\\ X_h(p)\ne 0\mathrm{on}p\in W_2.\end{array}$$

We also fix a cut-off function $\eta :{\mathbb{R}}^{2n+1}\to [0,1]$ which satisfies the following conditions

$$\begin{array}{c}\eta ((x_1,\cdots ,z))=1((x_1,\cdots ,z)\in W_1)\\ \eta ((x_1,\cdots ,z))=0((x_1,\cdots ,z)\in {\mathbb{R}}^{2n+1}\setminus W_2)\\ {\varphi }_h^j({\mathbb{R}}^{2n+1}\setminus W_2)\cap \mathrm{supp}(\eta )=\varnothing (1\le j\le 2k).\end{array}$$

We will use the last condition in the proof of Lemma 2. Then, $\eta (x_1.\cdots ,z)·{ϵ}_k(1-cos(k\theta ))$ is defined on ${\mathbb{R}}^{2n+1}$. We denote this contact Hamiltonian function by $g_{{ϵ}_k}$. We define ${\varphi }_{{ϵ}_k}\in {\mathrm{Cont}}_0^c({\mathbb{R}}^{2n+1},ker({\alpha }_0))$ by the composition ${\varphi }_{g_{{ϵ}_k}}\circ {\varphi }_h$.

Lemma. 2.

We take ${ϵ}_k>0$ sufficiently small. We define $2k$ points ${\left\{a_i\right\}}_{1\le i\le 2k}$ by

$$a_i=(\sqrt{a}cos(\frac{i\pi }{k}),\sqrt{a}sin(\frac{i\pi }{k}),0,\cdots ,0))\in S_a.$$

Then $P^{2k}({\varphi }_{{ϵ}_k})$ has only one point $[a_1,a_2,\cdots ,a_{2k}]$.

Proof of Lemma 2.

The proof of this lemma is as follows. On $W_1$, ${\varphi }_{g_{{ϵ}_k}}$ only changes the r-coordinate of $(x_1,y_1)$ and z-coordinate. So, ${\varphi }_{{ϵ}_k}$ 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 ${\varphi }_{g_{{ϵ}_k}}$ are 2k points $\left\{a_i\right\}$. From the arguments in the proof of Lemma 1, this implies that

$$[a_1,a_2,\cdots ,a_{2k}]\in P^{2k}({\varphi }_{{ϵ}_k})$$

holds and this is the only element of $P^{2k}({\varphi }_{{ϵ}_k})$ on $W_1$. So, it suffices to prove that this is the only element in $P^{2k}({\varphi }_{{ϵ}_k})$ if ${ϵ}_k>0$ is sufficiently small. We prove this by contradiction. Let ${\{{ϵ}_k^{(j)}>0\}}_{j\in \mathbb{N}}$ be a sequence which satisfies ${ϵ}_k^{(j)}\to 0$. We assume that there exists a sequence

$$[b_1^{(j)},\cdots ,b_{2k}^{(j)}]\in P^{2k}({\varphi }_{{ϵ}_k^{(j)}})\setminus [a_1,a_2,\cdots ,a_{2k}].$$

We may assume without loss of generality that $b_1^{(j)}\notin W_1$ holds because

$$(b_1^{(j)},\cdots ,b_{2k}^{(j)})\notin W_1^{2k}$$

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 ${\varphi }_{{ϵ}_k^{(N)}}^j(b_1^{(N)})={\varphi }_h^j(b_1^{(N)})$ 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 ${ϵ}_k>0$ is sufficiently small so that the conclusion of Lemma 2 holds and we define ${\varphi }_k$ by ${\varphi }_k={\varphi }_{{ϵ}_k}$. 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 ${ϵ}_k\to 0$ 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 $(1)$.

$${\psi }_k(x)=\left\{\begin{array}{cc}F\circ {\varphi }_k\circ F^{-1}(x)\hfill & x\in U\hfill \\ x\hfill & x\in M\setminus U\hfill \end{array}\right.$$

Lemma 2 implies that

$$P^{2k}({\psi }_k)=\left\{[F(a_1),\cdots ,F(a_{2k})]\right\}$$

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. □

5. Proof of Theorem 2

Proof. 

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

$$F:V⟶U.$$

In order to prove Theorem 2, it suffices to prove that there exists a sequence ${\psi }_k$$(k\in \mathbb{N})$ so that

• ${\psi }_k$ does not admit a square root
• $\mathrm{supp}({\psi }_k)\subset U$
• ${\psi }_k⟶\mathrm{Id}$ as $k⟶+\infty$

hold.

First, assume that m is odd ($m=2n+1$). In this case, ${\alpha }_0$ is a contact form on V. Let ${\varphi }_k$ be a contactomorphism which we constructed in the proof of Theorem 1

• ${\varphi }_k\in {\mathrm{Cont}}_o^c(V,ker({\alpha }_0))$
• $♯P^{2k}({\varphi }_k)=1$.

We define ${\psi }_k\in {\mathrm{Diff}}_0^c(M)$ by

$${\psi }_k(x)=\left\{\begin{array}{cc}F\circ {\varphi }_k\circ F^{-1}(x)\hfill & x\in U\hfill \\ x\hfill & x\in M\setminus U\hfill \end{array}\right..$$

Then, $♯P^{2k}({\psi }_k)=1$ holds and this implies that ${\psi }_k$ does not admit a square root and satisfies the above conditions. So, we proved Theorem 2 if m is odd.

Next, assume that m is even ($m=2n$). Let ${\omega }_0$ be a standard symplectic form on $(x_1,y_1,\cdots ,x_n,y_n)\in {\mathbb{R}}^{2n}$ which is defined by

$${\omega }_0=\sum _{1\le i\le n}d{x}_i\wedge d{y}_i.$$

By using the arguments in [2], we can construct a sequence ${\varphi }_k\in {\mathrm{Ham}}^c(V,{\omega }_0)$ for $k\in \mathbb{N}$ which satisfies the following conditions

• $♯P^{2k}({\varphi }_k)=1$
• ${\varphi }_k⟶\mathrm{Id}$ as $k⟶+\infty$.

We define ${\psi }_k\in {\mathrm{Diff}}_0^c(M)$ by

$${\psi }_k=\left\{\begin{array}{cc}F\circ {\varphi }_k\circ F^{-1}\hfill & x\in U\hfill \\ x\hfill & x\in M\setminus U\hfill \end{array}\right..$$

Then $♯P^{2k}({\psi }_k)=1$ holds and this implies that ${\psi }_k$ does not admit a square root and satisfies the above conditions. Hence, we have proved Theorem 2. □