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Open AccessArticle

Applications of Square Roots of Diffeomorphisms

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Axioms 2019, 8(2), 43; https://doi.org/10.3390/axioms8020043
Received: 9 March 2019 / Revised: 8 April 2019 / Accepted: 9 April 2019 / Published: 11 April 2019
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

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 C o n t 0 ( M , ξ ) Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M.
Keywords: diffeomorphism; contactomorphism; symplectomorphism diffeomorphism; contactomorphism; symplectomorphism

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 ( t X ) .
Here, X Γ ( T M ) is a vector field and exp : Γ ( T M ) 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 ( T M ) 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 ( T M ) )
is a small subset of Diff ( M ) .
For a symplectic manifold ( M , ω ) , the set of Hamiltonian diffeomorphisms Ham c ( M , ω ) contains “autonomous” subset Aut ( M , ω ) which is defined by
Aut ( M , ω ) = { exp ( X ) | X is a time independent Hamiltonian vector field 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 Ham c ( M , ω ) Aut ( M , ω ) .
Polterovich and Shelukhin used spectral spread of Floer homology and Conley conjecture to prove that Ham c ( M , ω ) Aut ( M , ω ) Ham c ( 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, there exists an arbitrary C -small contactomorphism in Cont 0 c ( 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 diffeomorphism in Diff 0 c ( M ) not admitting a square root. This also implies that there exists an arbitrary C -small diffeomorphism in Diff 0 c ( M ) Aut ( M ) .

2. Main Result

Let M be a smooth ( 2 n + 1 ) -dimensional manifold without boundary. A 1-form α on M is called contact if ( α ( d α ) n ) ( p ) 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 contact structure ξ ). Let Cont 0 c ( M , ξ ) be the set of compactly supported contactomorphisms which are isotopic to Id through compactly supported contactomorphisms. In other words, Cont 0 c ( M , ξ ) is a connected component of compactly supported contactomorphisms ( Cont c ( M , ξ ) ) which contains Id .
C o n t 0 c ( M , ξ ) = { ϕ 1 | ϕ t ( t [ 0 , 1 ] ) is an isotopy of contactomorphisms ϕ 0 = Id , t [ 0 , 1 ] supp ( ϕ t ) is compact }
Let X Γ c ( T M ) be a compactly supported vector field on M. X is called contact vector field if the flow of X preserves the contact structure ξ (i.e., exp ( X ) * ξ = ξ holds). Let Γ ξ c ( T M ) be the set of compactly supported contact vector fields on M and let Aut ( M , ξ ) be their images
Aut ( M , ξ ) = { exp ( X ) | X Γ ξ c ( T M ) } .
We prove the following theorem.
Theorem 1.
Let ( M , ξ ) be a contact manifold without boundary. Let W be any C -open neighborhood of Id Cont 0 c ( M , ξ ) . Then, there exists ϕ W such that
ϕ ψ 2
holds for any ψ Cont 0 c ( M , ξ ) . In particular, W Aut ( M , ξ ) is not empty.
Remark 1.
If ϕ is autonomous ( ϕ = exp ( X ) ), ϕ has a square root ψ = exp ( 1 2 X ) .
Corollary 1.
The exponential map exp : Γ ξ c ( T M ) Cont 0 c ( 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 Diff c ( M ) be the set of compactly supported diffeomorhisms
Diff c ( M ) = { ϕ Diff ( M ) | supp ( ϕ ) is compact } .
Let Diff 0 c ( M ) be the connected component of Diff c ( M ) (i.e., any element of Diff 0 c ( M ) is isotopic to Id ). We define the set of autonomous diffeomorphisms by
Aut ( M ) = { exp ( X ) | X Γ c ( T M ) } .
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 Id Diff 0 c ( M ) . Then, there exists ϕ W such that
ϕ ψ 2
holds for any ψ Diff c ( M ) . In particular, W Aut ( M ) is not empty.
Corollary 2.
The exponential map exp : Γ c ( T M ) 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 N 2 and a diffeomorphism ϕ Diff ( M ) . Let P l ( ϕ ) be the set of “l-periodic orbits” which is defined by
P l ( ϕ ) = { ( x 1 , , x l ) | x i x j ( i j ) , x j = ϕ j 1 ( x 1 ) , x 1 = ϕ ( x l ) } / .
This equivalence relation ∼ is given by the natural Z / l Z -action
( x 1 , , x l ) ( x l , x 1 , , x l 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 = ψ 2 holds). Then, there exists a free Z / 2 Z -action on P 2 k ( ϕ ) ( k N ). In particular, P 2 k ( ϕ ) is even if P 2 k ( ϕ ) 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 α Ω 1 ( M ) be a contact form on M ( ξ = ker ( α ) ). A Reeb vector field R α Γ ( T M ) is the unique vector field which satisfies
α ( R α ) = 1 d α ( R α , · ) = 0 .
For any smooth function h C c ( M ) , there exists only one contact vector field X h Γ ξ c ( T M ) which satisfies
X h = h · R α + Z where Z ξ .
In fact, X h is a contact vector field if and only if L X h ( α ) | ξ = 0 holds ( L is the Lie derivative). So,
L X h ( α ) ( Y ) = d h ( Y ) + d α ( X h , Y ) = d h ( Y ) + d α ( Z , Y ) = 0
holds for any Y ξ . Because d α is non-degenerate on ξ , above equation determines Z ξ 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 ϕ h t and time 1 flow of X h by ϕ h .
Let ( M , ξ ) be a contact manifold without boundary. We fix a point p ( M , ξ ) and a sufficiently small open neighborhood U M of p. Let ( x 1 , y 1 , , x n , y n , z ) be a coordinate of R 2 n + 1 . Let α 0 Ω 1 ( R 2 n + 1 ) be a contact form
α 0 = 1 2 1 i n ( x i d y i y i d x i ) + d z
on R 2 n + 1 . By using the famous Moser’s arguments, we can assume that there exists an open neighborhood of the origin V R 2 n + 1 and a diffeomorphism
F : V U
which satisfies
ξ | U = ker ( ( F 1 ) * α 0 ) .
So, we first prove the theorem for ( V , ker ( α 0 ) ) and apply this to ( M , ξ ) .
We fix k N 1 and R > 0 so that
{ ( x 1 , y 1 , , z ) R 2 n + 1 | | ( x 1 , , y n ) | < R , | z | < R } V
holds. Let f C c ( V ) be a contact Hamiltonian function. Then its contact Hamiltonian vector field X f can be written in the following form
X f ( x 1 , , z ) = 1 i n ( f y i + x i 2 f z ) x i + 1 i n ( f x i + y i 2 f z ) y i + ( f 1 i n x i 2 f x i 1 i n y i 2 f y i ) z .
Let e : R 2 n R be a quadric function
e ( x 1 , y 1 , , x n , y n ) = x 1 2 + y 1 2 + 2 i n x i 2 + y i 2 2 .
We define a contact Hamiltonian function h on V by
h ( x 1 , y 1 , , x n , y n , z ) = β ( z ) ρ ( e ( x 1 , y 1 , , x n , y n ) ) .
Here, β : R [ 0 , 1 ] and ρ : R 0 R 0 are smooth functions which satisfy the following five conditions.
  • supp ( ρ ) [ 0 , R 2 2 ]
  • ρ ( r ) ρ ( r ) · r , π 2 k < ρ ( r ) π 2 k
  • There exists an unique a [ 0 , R 2 2 ] which satisfies the following conditions
    ρ ( r ) = π 2 k r = a ρ ( a ) = π 2 k · a .
  • supp ( β ) [ R 2 , R 2 ]
  • β ( 0 ) = 1 , β 1 ( 1 ) = 0
Then, we can prove the following lemma.
Lemma. 1.
Let h C c ( V ) be a contact Hamiltonian function as above. Then,
[ q , ϕ h ( q ) , , ϕ h 2 k 1 ( q ) ] P 2 k ( ϕ h )
holds if and only if
q { ( x 1 , y 1 , 0 , , 0 ) V | x 1 2 + y 1 2 = a } = def . S a
holds.
Proof of Lemma 1.
In order to prove this lemma, we first calculate the behavior of the function z ( ϕ h t ( q ) ) for a fixed q V (Here, z is the ( 2 n + 1 ) -th coordinate of R 2 n + 1 ).
d d t ( z ( ϕ h t ( q ) ) ) = h 1 i n x i 2 h x i 1 i n y i 2 h y i = β ( z ) { ρ ( e ) 1 i n x i 2 x i ( ρ ( e ) ) 1 i n y i 2 y i ( ρ ( e ) ) } = β ( z ) { ρ ( e ) ρ ( e ) · e } 0
In the last inequality, we used the condition 2. So, this inequality implies that
ϕ h 2 k ( q ) = q d d t ( z ( ϕ h t ( q ) ) ) = 0
holds.
Next, we study the behavior of x i ( ϕ h t ( q ) ) and y i ( ϕ h t ( q ) ) . Let π i be the projection
π i : R 2 n + 1 R 2 . ( x 1 , y 1 , , x n , y n , z ) ( x i , y i )
Then, Y h i = π i ( X h ) can be decomposed into the angular component Y h i , θ and the radius component Y h i , r as follows
Y h i , θ ( x 1 , y 1 , , z ) = h y i x i + h x i y i Y h i , r ( x 1 , y 1 , , z ) = ( 1 2 h z ) ( x i x i + y i y i ) .
Let w i be the complex coordinate of ( x i , y i ) ( w i = x i + 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 ρ ( e ( x 1 , , y n ) ) β ( z ) C i t
C i = 1 i = 1 1 2 2 i n .
By conditions 2, 3, and 5 in the definition of β and ρ , | 2 ρ ( e ( x 1 , , y n ) ) β ( z ) C i | is at most 2 π 2 k and the equality holds if and only if ( x 1 , y 1 , , x n , y n , z ) S a holds. On the circle S a , ϕ h is the 2 π 2 k -rotation of the circle S a . This implies that Lemma 1 holds. □
Next, we perturb the contactomorphism ϕ h . Let ( r , θ ) be a coordinate of ( x 1 , y 1 ) R 2 ( 0 , 0 ) as follows
x 1 = r cos θ , y 1 = r sin θ .
We fix ϵ k > 0 . Then ϵ k ( 1 cos ( k θ ) ) is a contact Hamiltonian function on R 2 ( 0 , 0 ) × R 2 n 1 and its contact Hamiltonian vector field can be written in the following form
X ϵ k ( 1 cos ( k θ ) ) = ϵ k k r sin ( k θ ) r + ϵ k ( 1 cos ( k θ ) ) z .
So ϕ ϵ k ( 1 cos ( k θ ) ) only changes the r of ( x 1 , y 1 ) -coordinate and z-coordinate as follows
( r , θ , x 2 , y 2 , , x n , y n , z ) ( r 2 2 ϵ k k sin ( k θ ) , θ , x 2 , , y n , z + ϵ k ( 1 cos ( k θ ) ) ) .
We fix two small open neighborhoods of the circle S a as follows
S a W 1 W 2 R 2 ( 0 , 0 ) × R 2 n 1 X h ( p ) 0 on p W 2 .
We also fix a cut-off function η : R 2 n + 1 [ 0 , 1 ] which satisfies the following conditions
η ( ( x 1 , , z ) ) = 1 ( ( x 1 , , z ) W 1 ) η ( ( x 1 , , z ) ) = 0 ( ( x 1 , , z ) R 2 n + 1 W 2 ) ϕ h j ( R 2 n + 1 W 2 ) supp ( η ) = ( 1 j 2 k ) .
We will use the last condition in the proof of Lemma 2. Then, η ( x 1 . , z ) · ϵ k ( 1 cos ( k θ ) ) is defined on R 2 n + 1 . We denote this contact Hamiltonian function by g ϵ k . We define ϕ ϵ k Cont 0 c ( R 2 n + 1 , ker ( α 0 ) ) by the composition ϕ g ϵ k ϕ h .
Lemma. 2.
We take ϵ k > 0 sufficiently small. We define 2 k points { a i } 1 i 2 k by
a i = ( a cos ( i π k ) , a sin ( i π k ) , 0 , , 0 ) ) S a .
Then P 2 k ( ϕ ϵ k ) has only one point [ a 1 , a 2 , , a 2 k ] .
Proof of Lemma 2.
The proof of this lemma is as follows. On W 1 , ϕ g ϵ k only changes the r-coordinate of ( x 1 , y 1 ) and z-coordinate. So, ϕ ϵ k increases the angle of each ( x i , y i ) coordinate at most 2 π 2 k and the equality holds on only S a . On the circle S a , the fixed points of ϕ g ϵ k are 2k points { a i } . From the arguments in the proof of Lemma 1, this implies that
[ a 1 , a 2 , , a 2 k ] P 2 k ( ϕ ϵ k )
holds and this is the only element of P 2 k ( ϕ ϵ k ) on W 1 . So, it suffices to prove that this is the only element in P 2 k ( ϕ ϵ k ) if ϵ k > 0 is sufficiently small. We prove this by contradiction. Let { ϵ k ( j ) > 0 } j N be a sequence which satisfies ϵ k ( j ) 0 . We assume that there exists a sequence
[ b 1 ( j ) , , b 2 k ( j ) ] P 2 k ( ϕ ϵ k ( j ) ) [ a 1 , a 2 , , a 2 k ] .
We may assume without loss of generality that b 1 ( j ) W 1 holds because
( b 1 ( j ) , , b 2 k ( j ) ) W 1 2 k
holds. We may assume that b 1 ( j ) converges to a point b W 1 . Then, ϕ h 2 k ( b ) = b holds. If X h ( b ) 0 , ϕ h increases the angle of every ( x i , y i ) coordinate less than 2 π 2 k and this contradicts ϕ h 2 k ( b ) = b . Thus X h ( b ) = 0 holds. Because we assumed X h ( p ) 0 on p W 2 , X h ( b ) = 0 implies that b W 2 holds. Let N N be a large integer so that b 1 ( N ) W 2 holds. Then, ϕ h j ( R 2 n + 1 W 2 ) supp ( η ) = ( 1 j 2 k ) implies that ϕ ϵ k ( N ) j ( b 1 ( N ) ) = ϕ h j ( b 1 ( N ) ) holds for 1 j 2 k and [ b 1 ( N ) , , b 2 k ( N ) ] P 2 k ( ϕ h ) holds. This contradicts Lemma 1 because b 1 ( N ) 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 ϕ k by ϕ k = ϕ ϵ k . Thus, we have constructed ϕ k Cont 0 c ( V , Ker ( α 0 ) ) which does not admit a square root for each k N . Without loss of generality, we may assume that ϵ k 0 holds. Then ϕ k converges to Id .
Finally, we prove Theorem 1. We define ψ k Cont 0 c ( M , ξ ) for k N as follows. Recall that F is a diffeomorphism which was defined in Equation ( 1 ) .
ψ k ( x ) = F ϕ k F 1 ( x ) x U x x M U
Lemma 2 implies that
P 2 k ( ψ k ) = { [ F ( a 1 ) , , F ( a 2 k ) ] }
holds. Proposition 1 implies that ψ k does not admit a square root. Because p 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 M . Let U be an open neighborhood of p and let V 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 ψ k ( k N ) so that
  • ψ k does not admit a square root
  • supp ( ψ k ) U
  • ψ k Id as k +
hold.
First, assume that m is odd ( m = 2 n + 1 ). In this case, α 0 is a contact form on V. Let ϕ k be a contactomorphism which we constructed in the proof of Theorem 1
  • ϕ k Cont o c ( V , ker ( α 0 ) )
  • P 2 k ( ϕ k ) = 1 .
We define ψ k Diff 0 c ( M ) by
ψ k ( x ) = F ϕ k F 1 ( x ) x U x x M U .
Then, P 2 k ( ψ k ) = 1 holds and this implies that ψ 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 = 2 n ). Let ω 0 be a standard symplectic form on ( x 1 , y 1 , , x n , y n ) R 2 n which is defined by
ω 0 = 1 i n d x i d y i .
By using the arguments in [2], we can construct a sequence ϕ k Ham c ( V , ω 0 ) for k N which satisfies the following conditions
  • P 2 k ( ϕ k ) = 1
  • ϕ k Id as k + .
We define ψ k Diff 0 c ( M ) by
ψ k = F ϕ k F 1 x U x x M U .
Then P 2 k ( ψ k ) = 1 holds and this implies that ψ k does not admit a square root and satisfies the above conditions. Hence, we have proved Theorem 2. □

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

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