axioms

Article Applications of Square Roots of

Yoshihiro Sugimoto Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan; [email protected]

 Received: 09 March 2019; Accepted: 09 April 2019; Published: 11 April 2019 

Abstract: In this paper, we prove that on any contact manifold (M, ξ) there exists an arbitrary C∞-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C∞-small contactomorphism which is not “autonomous”. This paper is the first step to study the topology of Cont0(M, ξ)\Aut(M, ξ). As an application, we also prove a similar result for the group Diff(M) for any smooth manifold M.

Keywords: diffeomorphism; contactomorphism;

1. Introduction For any closed manifold M, the set of diffeomorphisms Diff(M) forms a group and any one-parameter subgroup f : R → Diff(M) can be written in the following form

f (t) = exp(tX).

Here, X ∈ Γ(TM) is a vector field and exp : Γ(TM) → 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 U ⊂ Γ(TM) such that exp : U −→ Diff(M) is a diffeomorphism onto an open neighborhood of Id ∈ Diff(M). However, this is far from true ([1], Warning 1.6). So one might expect that the set of “autonomous” diffeomorphisms

Aut(M) = exp(Γ(TM)) is a small subset of Diff(M). For a (M, ω), the set of Hamiltonian diffeomorphisms Hamc(M, ω) contains “autonomous” subset Aut(M, ω) which is defined by

  X is a time-independent Hamiltonian vector field Aut(M, ω) = exp(X) . whose support is compact

In [2], Albers and Frauenfelder proved that on any symplectic manifold there exists an arbitrary C∞-small Hamiltonian diffeomorphism not admitting a square root. In particular, there exists an arbitrary C∞-small Hamiltonian diffeomorphism in Hamc(M, ω)\Aut(M, ω). Polterovich and Shelukhin used spectral spread of Floer homology and Conley conjecture to prove that Hamc(M, ω)\Aut(M, ω) ⊂ Hamc(M, ω) is C∞-dense and dense in the topology induced from Hofer’s metric if (M, ω) 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∞-small contactomorphism not admitting a square root. In particular,

Axioms 2019, 8, 43; doi:10.3390/axioms8020043 www.mdpi.com/journal/axioms Axioms 2019, 8, 43 2 of 8

∞ c there exists an arbitrary C -small contactomorphism in Cont0(M, ξ)\Aut(M, ξ). So, this paper is a contact manifold version of [2]. As an application, we prove that there exists an arbitrary C∞-small c diffeomorphism in Diff0(M) not admitting a square root. This also implies that there exists an arbitrary ∞ c C -small diffeomorphism in Diff0(M)\Aut(M).

2. Main Result Let M be a smooth (2n + 1)-dimensional manifold without boundary. A 1-form α on M is called contact if (α ∧ (dα)n)(p) 6= 0 holds on any p ∈ M. A codimension 1 tangent distribution ξ on M is called contact structure if it is locally defined by ker(α) for some (locally defined) contact form α. A diffeomorphism φ ∈ Diff(M) is called contactomorphism if φ∗ξ = ξ holds (i.e., φ preserves the c contact structure ξ). Let Cont0(M, ξ) be the set of compactly supported contactomorphisms which c are isotopic to Id through compactly supported contactomorphisms. In other words, Cont0(M, ξ) is a connected component of compactly supported contactomorphisms (Contc(M, ξ)) which contains Id.   c φt (t ∈ [0, 1]) is an isotopy of contactomorphisms Cont0(M, ξ) = φ1 φ0 = Id, ∪t∈[0,1]supp(φt) is compact Let X ∈ Γc(TM) be a compactly supported vector field on M. X is called contact vector field if c the flow of X preserves the contact structure ξ (i.e., exp(X)∗ξ = ξ holds). Let Γξ (TM) be the set of compactly supported contact vector fields on M and let Aut(M, ξ) be their images

c Aut(M, ξ) = {exp(X) | X ∈ Γξ (TM)}.

We prove the following theorem.

Theorem 1. Let (M, ξ) be a contact manifold without boundary. Let W be any C∞-open neighborhood of c Id ∈ Cont0(M, ξ). Then, there exists φ ∈ W such that

φ 6= ψ2

c holds for any ψ ∈ Cont0(M, ξ). In particular, W\Aut(M, ξ) is not empty.

1 Remark 1. If φ is autonomous (φ = exp(X)), φ has a square root ψ = exp( 2 X).

c c Corollary 1. The exponential map exp : Γξ (TM) → Cont0(M, ξ) is not surjective.

We also consider the diffeomorphism version of this theorem and corollary. Let M be a smooth manifold without boundary and let Diffc(M) be the set of compactly supported diffeomorhisms

Diffc(M) = {φ ∈ Diff(M) | supp(φ) is compact}.

c c c Let Diff0(M) be the connected component of Diff (M) (i.e., any element of Diff0(M) is isotopic to Id). We define the set of autonomous diffeomorphisms by

Aut(M) = {exp(X) | X ∈ Γc(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 W be any C∞-open neighborhood of c Id ∈ Diff0(M). Then, there exists φ ∈ W such that

φ 6= ψ2 Axioms 2019, 8, 43 3 of 8 holds for any ψ ∈ Diffc(M). In particular, W\Aut(M) is not empty.

c c Corollary 2. The exponential map exp : Γ (TM) → Diff0(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 l criterion later. We fix l ∈ N≥2 and a diffeomorphism φ ∈ Diff(M). Let P (φ) be the set of “l-periodic orbits” which is defined by

l j−1 P (φ) = {(x1, ··· , xl) | xi 6= xj(i 6= j), xj = φ (x1), x1 = φ(xl)}/ ∼ .

This equivalence relation ∼ is given by the natural Z/lZ-action

(x1, ··· , xl) → (xl, x1, ··· , xl−1).

Proposition 1 (Milnor [1], Albers-Frauenfelder [2]). Assume that φ ∈ Diff(M) has a square root (i.e., there exists ψ ∈ Diff(M) such that φ = ψ2 holds). Then, there exists a free Z/2Z-action on P2k(φ) (k ∈ N). In particular, ]P2k(φ) is even if ]P2k(φ) is finite.

4. Proof of Theorem1 Proof. Before stating the proof of Theorem1, we introduce the notion of a contact Hamiltonian function. Let M be a smooth manifold without boundary and let α ∈ Ω1(M) be a contact form on M (ξ = ker(α)). A Reeb vector field Rα ∈ Γ(TM) is the unique vector field which satisfies

α(Rα) = 1

dα(Rα, ·) = 0.

∞ c For any smooth function h ∈ Cc (M), there exists only one contact vector field Xh ∈ Γξ (TM) which satisfies

Xh = h · Rα + Z where Z ∈ ξ.

L ( )| = L In fact, Xh is a contact vector field if and only if Xh α ξ 0 holds ( is the ). So,

L ( )( ) = ( ) + ( ) = ( ) + ( ) = Xh α Y dh Y dα Xh, Y dh Y dα Z, Y 0 holds for any Y ∈ ξ. Because dα is non-degenerate on ξ, above equation determines Z ∈ ξ uniquely. Xh is the contact vector field associated to the contact Hamiltonian function h. We denote the time t t flow of Xh by φh and time 1 flow of Xh by φh. Let (M, ξ) be a contact manifold without boundary. We fix a point p ∈ (M, ξ) and a 2n+1 sufficiently small open neighborhood U ⊂ M of p. Let (x1, y1, ··· , xn, yn, z) be a coordinate of R . 1 2n+1 Let α0 ∈ Ω (R ) be a contact form

1 α0 = ∑ (xidyi − yidxi) + dz 2 1≤i≤n

+ on R2n 1. By using the famous Moser’s arguments, we can assume that there exists an open + neighborhood of the origin V ⊂ R2n 1 and a diffeomorphism

F : V −→ U (1) Axioms 2019, 8, 43 4 of 8 which satisfies −1 ∗ ξ|U = ker((F ) α0).

So, we first prove the theorem for (V, ker(α0)) and apply this to (M, ξ). We fix k ∈ N≥1 and R > 0 so that

2n+1 {(x1, y1, ··· , z) ∈ R | |(x1, ··· , yn)| < R, |z| < R} ⊂ V

∞ holds. Let f ∈ Cc (V) be a contact Hamiltonian function. Then its contact Hamiltonian vector field X f can be written in the following form

∂ f xi ∂ f ∂ X f (x1, ··· , z) = ∑ (− + ) 1≤i≤n ∂yi 2 ∂z ∂xi ∂ f y ∂ f ∂ + ∑ ( + i ) 1≤i≤n ∂xi 2 ∂z ∂yi x ∂ f y ∂ f ∂ +( f − ∑ i − ∑ i ) . 1≤i≤n 2 ∂xi 1≤i≤n 2 ∂yi ∂z

Let e : R2n −→ R be a quadric function

2 2 2 2 xi + yi e(x1, y1, ··· , xn, yn) = x1 + y1 + ∑ . 2≤i≤n 2

We define a contact Hamiltonian function h on V by

h(x1, y1, ··· , xn, yn, z) = β(z)ρ(e(x1, y1, ··· , xn, yn)).

Here, β : R → [0, 1] and ρ : R≥0 → R≥0 are smooth functions which satisfy the following five conditions.

R2 1. supp(ρ) ⊂ [0, 2 ] 0 π 0 π 2. ρ(r) ≥ ρ (r) · r, − 2k < ρ (r) ≤ 2k R2 3. There exists an unique a ∈ [0, 2 ] which satisfies the following conditions ( ρ0(r) = π ⇐⇒ r = a 2k . π ρ(a) = 2k · a

R R 4. supp(β) ⊂ [− 2 , 2 ] 5. β(0) = 1, β−1(1) = 0

Then, we can prove the following lemma.

∞ Lemma 1. Let h ∈ Cc (V) be a contact Hamiltonian function as above. Then,

2k−1 2k [q, φh(q), ··· , φh (q)] ∈ P (φh) holds if and only if 2 2 def. q ∈ {(x1, y1, 0, ··· , 0) ∈ V | x1 + y1 = a} = Sa holds. Axioms 2019, 8, 43 5 of 8

Proof of Lemma1. In order to prove this lemma, we first calculate the behavior of the function t 2n+1 z(φh(q)) for a fixed q ∈ V (Here, z is the (2n + 1)-th coordinate of R ).

d t xi ∂h yi ∂h (z(φh(q))) = h − ∑ − ∑ dt 1≤i≤n 2 ∂xi 1≤i≤n 2 ∂yi x ∂ y ∂ = β(z){ρ(e) − ∑ i (ρ(e)) − ∑ i (ρ(e))} 1≤i≤n 2 ∂xi 1≤i≤n 2 ∂yi = β(z){ρ(e) − ρ0(e) · e} ≥ 0

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

d φ2k(q) = q =⇒ (z(φt (q))) = 0 h dt h holds. t t Next, we study the behavior of xi(φh(q)) and yi(φh(q)). Let πi be the projection

2n+1 2 πi : R −→ R .

(x1, y1, ··· , xn, yn, z) 7→ (xi, yi)

i i,θ Then, Yh = πi(Xh) can be decomposed into the angular component Yh and the radius component i,r Yh as follows

i,θ ∂h ∂ ∂h ∂ Yh (x1, y1, ··· , z) = − + ∂yi ∂xi ∂xi ∂yi i,r 1 ∂h ∂ ∂ Yh (x1, y1, ··· , z) = ( )(xi + yi ). 2 ∂z ∂xi ∂yi √ Let wi be the complex coordinate of (xi, yi) (wi = xi + −1yi). Then, the angular component causes the following rotation on wi, if we ignore the z-coordinate,

0 arg(wi) −→ arg(wi) + 2ρ (e(x1, ··· , yn))β(z)Cit ( 1 i = 1 C = . i 1 2 2 ≤ i ≤ n 0 2π By conditions 2, 3, and 5 in the definition of β and ρ, |2ρ (e(x1, ··· , yn))β(z)Ci| is at most 2k and 2π the equality holds if and only if (x1, y1, ··· , xn, yn, z) ∈ Sa holds. On the circle Sa, φh is the 2k -rotation of the circle Sa. This implies that Lemma1 holds.

2 Next, we perturb the contactomorphism φh. Let (r, θ) be a coordinate of (x1, y1) ∈ R \(0, 0) as follows x1 = r cos θ, y1 = r sin θ. 2 2n−1 We fix ek > 0. Then ek(1 − cos(kθ)) is a contact Hamiltonian function on R \(0, 0) × R and its contact Hamiltonian vector field can be written in the following form

e k ∂ ∂ X = − k sin(kθ) + e (1 − cos(kθ)) . ek(1−cos(kθ)) r ∂r k ∂z

So φ only changes the r of (x , y )-coordinate and z-coordinate as follows ek(1−cos(kθ)) 1 1 q 2 (r, θ, x2, y2, ··· , xn, yn, z) 7→ ( r − 2ekk sin(kθ), θ, x2, ··· , yn, z + ek(1 − cos(kθ))). Axioms 2019, 8, 43 6 of 8

We fix two small open neighborhoods of the circle Sa as follows

2 2n−1 Sa ⊂ W1 ⊂ W2 ⊂ R \(0, 0) × R

Xh(p) 6= 0 on p ∈ W2.

+ We also fix a cut-off function η : R2n 1 → [0, 1] which satisfies the following conditions

η((x1, ··· , z)) = 1 ((x1, ··· , z) ∈ W1) 2n+1 η((x1, ··· , z)) = 0 ((x1, ··· , z) ∈ R \W2) j 2n+1 φh(R \W2) ∩ supp(η) = ∅ (1 ≤ j ≤ 2k).

We will use the last condition in the proof of Lemma2. Then, η(x1. ··· , z) · ek(1 − cos(kθ)) 2n+1 is defined on R . We denote this contact Hamiltonian function by gek . We define c 2n+1 φ ∈ Cont ( , ker(α )) by the composition φg ◦ φ . ek 0 R 0 ek h

Lemma 2. We take ek > 0 sufficiently small. We define 2k points {ai}1≤i≤2k by

√ iπ √ iπ a = ( a cos( ), a sin( ), 0, ··· , 0)) ∈ S . i k k a

2k Then P (φek ) has only one point [a1, a2, ··· , a2k].

Proof of Lemma2. The proof of this lemma is as follows. On W , φg only changes the r-coordinate 1 ek ( ) ( ) 2π of x1, y1 and z-coordinate. So, φek increases the angle of each xi, yi coordinate at most 2k and the equality holds on only Sa. On the circle Sa, the fixed points of φg are 2k points {a }. From the ek i arguments in the proof of Lemma1, this implies that

2k [a1, a2, ··· , a2k] ∈ P (φek ) 2k holds and this is the only element of P (φek ) on W1. So, it suffices to prove that this is the only 2k( ) > { (j) > } element in P φek if ek 0 is sufficiently small. We prove this by contradiction. Let ek 0 j∈N be (j) a sequence which satisfies ek → 0. We assume that there exists a sequence

[ (j) ··· (j)] ∈ 2k( )\[ ··· ] b1 , , b2k P φ (j) a1, a2, , a2k . ek

(j) We may assume without loss of generality that b1 ∈/ W1 holds because

( (j) ··· (j)) ∈ 2k b1 , , b2k / W1

(j) 2k holds. We may assume that b1 converges to a point b ∈/ W1. Then, φh (b) = b holds. If Xh(b) 6= 0, 2π 2k φh increases the angle of every (xi, yi) coordinate less than 2k and this contradicts φh (b) = b. Thus Xh(b) = 0 holds. Because we assumed Xh(p) 6= 0 on p ∈ W2, Xh(b) = 0 implies that b ∈/ W2 (N) j 2n+1 holds. Let N ∈ N be a large integer so that b1 ∈/ W2 holds. Then, φh(R \W2) ∩ supp(η) = ∅ ( ≤ ≤ ) j ( (N)) = j ( (N)) ≤ ≤ [ (N) ··· (N)] ∈ 2k( ) 1 j 2k implies that φ (N) b1 φh b1 holds for 1 j 2k and b1 , , b2k P φh ek (N) holds. This contradicts Lemma1 because b1 ∈/ Sa. So, we proved Lemma2.

We assume that ek > 0 is sufficiently small so that the conclusion of Lemma2 holds and we define c φk by φk = φek . Thus, we have constructed φk ∈ Cont0(V, Ker(α0)) which does not admit a square root for each k ∈ N. Without loss of generality, we may assume that ek → 0 holds. Then φk converges to Id. Axioms 2019, 8, 43 7 of 8

c Finally, we prove Theorem1. We define ψk ∈ Cont0(M, ξ) for k ∈ N as follows. Recall that F is a diffeomorphism which was defined in Equation (1).

( −1 F ◦ φk ◦ F (x) x ∈ U ψk(x) = x x ∈ M\U

Lemma2 implies that 2k P (ψk) = {[F(a1), ··· , F(a2k)]} holds. Proposition1 implies that ψk does not admit a square root. Because p ∈ M is any point and U is any small open neighborhood of p, we proved Theorem1.

5. Proof of Theorem2 Proof. Let M be a m-dimensional smooth manifold without boundary. We fix a point p ∈ M. Let U be an open neighborhood of p and let V ⊂ Rm be an open neighborhood of the origin such that there is a diffeomorphism F : V −→ U.

In order to prove Theorem2, it suffices to prove that there exists a sequence ψk (k ∈ N) so that

• ψk does not admit a square root • supp(ψk) ⊂ U • ψk −→Id as k −→ +∞ hold. First, assume that m is odd (m = 2n + 1). In this case, α0 is a contact form on V. Let φk be a contactomorphism which we constructed in the proof of Theorem1

c • φk ∈ Conto(V, ker(α0)) 2k • ]P (φk) = 1 . c We define ψk ∈ Diff0(M) by

( −1 F ◦ φk ◦ F (x) x ∈ U ψk(x) = . x x ∈ M\U

2k Then, ]P (ψk) = 1 holds and this implies that ψk does not admit a square root and satisfies the above conditions. So, we proved Theorem2 if m is odd. Next, assume that m is even (m = 2n). Let ω0 be a standard symplectic form on 2n (x1, y1, ··· , xn, yn) ∈ R which is defined by

ω0 = ∑ dxi ∧ dyi. 1≤i≤n

c By using the arguments in [2], we can construct a sequence φk ∈ Ham (V, ω0) for k ∈ N which satisfies the following conditions

2k • ]P (φk) = 1 • φk −→ Id as k −→ +∞ . c We define ψk ∈ Diff0(M) by

( −1 F ◦ φk ◦ F x ∈ U ψk = . x x ∈ M\U Axioms 2019, 8, 43 8 of 8

2k Then ]P (ψk) = 1 holds and this implies that ψk does not admit a square root and satisfies the above conditions. Hence, we have proved Theorem2.

Funding: This research received no external funding. Acknowledgments: The author thanks Kaoru Ono and Urs Frauenfelder for many useful comments, discussion and encouragement. Conflicts of Interest: The author declares no conflict of interest.

References

1. Milnor, J. Remarks on infinite-dimensional Lie groups. In Relativity, Groups and Topology II; Elsevier Science Ltd.: Amsterdam, The Netherlands, 1984. 2. Albers, P.; Frauenfelder, U. Square roots of Hamiltonian diffeomorphisms. J. Symplectic Geom. 2014, 12, 427–434. [CrossRef] 3. Polterovich, L.; Schelukhin, E. Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules. Sel. Math. 2016, 22, 227–296. [CrossRef] 4. Sugimoto, Y. Spectral spread and non-autonomous Hamiltonian diffeomorphisms. Manuscr. Math. 2018. [CrossRef]

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