Applications of square roots of diffeomorphisms
Yoshihiro Sugimoto

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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Topological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology
Applications of square roots of diffeomorphisms
Yoshihiro Sugimoto
Abstract
In this paper, we prove that on any contact manifold , there exists an arbitrary -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary -small contactomorphism which is not ”autonomous”. This paper is the first step to study the topology of . As an application, we also prove a similar result for the diffeomorphism group for any smooth manifold .
1 Introduction
For any closed manifold , the set of diffeomorphisms forms a group and any one-parameter subgroup can be written in the following form.
[TABLE]
Here, is a vector field and is a time flow of a vector field. From the inverse function theorem, one might expect that there exists an open neighborhood of the zero section such that
[TABLE]
is a diffeomorphism onto an open neighborhood of . 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 .
For a symplectic manifold , the set of Hamiltonian diffeomorphisms contains ”autonomous” subset as follows.
[TABLE]
In [1], Albers and Frauenfelder proved that on any symplectic manifold there exists an arbitrary -small Hamiltonian diffeomorphism not admitting a square root. In particular, there exists an arbitrary -small Hamiltonian diffeomorphism in .
Polterovich and Shelukhin used spectral spread of Floer homology and Conley conjecture to prove that is -dense and dense in the topology induced from Hofer’s metric if 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 -small contactomorphism not admitting a square root. In particular, there exists an arbitrary -small contactomorphism in . So, this paper is a contact manifold version of [1]. As an application, we prove that there exists an arbitrary -small diffeomorphism in not admitting a square root. This also implies that there exists an arbitrary -small diffeomorphism in .
Acknowledgement
The author thanks Kaoru Ono and Urs Frauenfelder for many useful comments, discussions and encouragement.
2 Main result
Let be a smooth -dimensional manifold without boundary. A -form on is called contact if holds on any . A codimension tangent distribution on is called contact structure if it is locally defined by for some (locally defined) contact form . A diffeomorphism is called contactomorphism if holds (i.e. preserves the contact structure ). Let be the set of compact supported contactomorphisms which are isotopic to Id. In other words, is a connected component of compact supported contactomorphisms() which contains Id.
[TABLE]
Let be a compact supported vector field on . is called contact vector field if the flow of preserves the contact structure (i.e. holds). Let be the set of compact supported contact vector fields on and let be their images.
[TABLE]
We prove the following theorem.
Theorem 1
Let be a contact manifold without boundary. Let be any -open neighborhood of . Then, there exists such that
[TABLE]
holds for any . In particular, is not empty.
Remark 1
If is autonomous (), has a square root .
Corollary 1
The exponential map is not surjective.
We also consider diffeomorphism version of this theorem and corollary. Let be a smooth manifold without boundary and let be the set of compact supported diffeomorhisms which contains Id.
[TABLE]
Let be the connected component of (i.e. any element of 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 be a smooth manifold without boundary. Let be any -open neighborhood of . Then, there exists such that
[TABLE]
holds for any . In particular, is not empty.
Corollary 2
The exponential map 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 and a diffeomorphism . Let be the set of ”-periodic orbits” as follows.
[TABLE]
This equivalence relation is given by the natural -action
[TABLE]
Proposition 1** (Milnor[2], Albers-Frauenfelder[1])**
Assume that has a square root (i.e. there exists such that holds). Then, there exists a free -action on (). In particular, is even if is finite.
The proof of this proposition is very simple. We explain the proof for the sake of self-containedness. Let be a square root of (). It is sufficient to prove that the natural -action
[TABLE]
on is free. We prove this by contradiction. So, we assume that is a fixed point of this -action. Then there exists so that holds. This implies that
[TABLE]
holds. Let be the greatest common divisor of and . Then,
[TABLE]
holds. This is a contradiction.
4 Proof of Theorem 1
Before stating the proof of Theorem 1, we introduce the notion of contact Hamiltonian function. Let be a smooth manifold without boundary and let be a contact form on (). A Reeb vector field is defined as follows.
[TABLE]
For any smooth function , there exists an unique contact vector field which satisfies the following condition.
[TABLE]
In fact, is contact vector field if and only if holds ( is the Lie derivative.). So,
[TABLE]
holds for any . Because is non-degenerate on , above equation determines uniquely. is a contact vector field associated to a contact Hamiltonian function . We denote the time flow of by and time flow of by .
Let be a contact manifold without boundary. We fix a point and a sufficiently small open neighborhood of . Let be a coordinate of . Let be the following contact form on .
[TABLE]
By using famous Moser’s arguments, we can assume that there exists an open neighborhood of the origin and a diffeomorphism
[TABLE]
which satisfies the following condition.
[TABLE]
So, we first prove the theorem for and apply this to .
We fix and so that
[TABLE]
holds. Let be a contact Hamiltonian function. Then its contact Hamiltonian vector field can be written in the following form.
[TABLE]
Let be a quadric function as follows.
[TABLE]
We consider the following contact Hamiltonian function on .
[TABLE]
Here, and are smooth functions which satisfy the following conditions.
- •
- •
,
- •
There exists an unique which satisfies the following conditions.
[TABLE]
- •
- •
, ,
Then, we can prove the following lemma.
Lemma 1
Let 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 for fixed (Here, is the -th coordinate of .).
[TABLE]
So, this inequality implies that
[TABLE]
holds.
Next, we study the behavior of and . Let be the following projection.
[TABLE]
Then, can be decomposed into angular component and radius component as follows.
[TABLE]
Let be the complex coordinate of (). Then, the angular component causes the following rotation on (If we ignore the -coordinate).
[TABLE]
[TABLE]
By the assumptions of and , is at most and equality holds if and only if holds. On the circle , is the -rotation of the circle . This implies that the lemma holds.
Next, we perturb the contactomorphism . Let be a coordinate of as follows.
[TABLE]
We fix . Then is a contact Hamiltonian function on and its contact Hamiltonian vector field can be written in the following form.
[TABLE]
So only changes the of -coordinate and -coordinate as follows.
[TABLE]
We fix two small open neighborhood of the circle as follows.
[TABLE]
We also fix a cut-off function as follows.
[TABLE]
Then, is defined on . We denote this contact Hamiltonian function by . We define by the composition .
Lemma 2
We take sufficiently small. We define points as follows.
[TABLE]
Then has only one point .
The proof of this lemma is as follows. On , only changes the -coordinate of and -coordinate. On the circle , the fixed points of are 2k points . This implies that on , increases the angle at most and the equality holds on only 2k points . From the arguments in the proof of Lemma 1, this implies that
[TABLE]
holds if is sufficiently small.
Finally, we prove Theorem 1. We define as follows.
[TABLE]
Lemma 2 implies that
[TABLE]
holds. Proposition 1 implies that holds. Because is any point and is any small open neighborhood of , we proved Theorem 1.
5 Proof of Theorem 2
Let be a -dimensional smooth manifold without boundary. We fix a point . Let be an open neighborhood of and let 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 so that
- •
- •
- •
as
hold.
First, assume that is odd (). In this case, is a contact form on . Let be a contactomorphism which we constructed in the proof of Theorem 1.
- •
- •
We define as follows.
[TABLE]
Then, holds and this implies that holds.
Next, assume that is even (). Let be a standard symplectic form on as follows.
[TABLE]
As in the contact case, Hamiltonian function defines Hamiltonian vector field as follows.
[TABLE]
We denote the time flow of by .
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 holds ( is a radius -ball in .). Let be a following quadric function.
[TABLE]
Let be a non-negative function which satisfies the following conditions.
- •
- •
,
Let be a Hamiltonian function as follows.
[TABLE]
The following lemma is an analogy of Lemma 1.
Lemma 3
Let 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 be a complex coordinate of (). Then, causes the following rotaion on .
[TABLE]
[TABLE]
is at most and equality holds if and only if holds. This implies that the lemma holds.
Next we perturb this . Let be a coordinate of as follows.
[TABLE]
We fix , then is a Hamiltonian function on . The Hamiltonian vector field of this Hamiltonian function can be written in the following form.
[TABLE]
So, only changes the of -coordinate.
[TABLE]
We fix two small open neighborhoods of and cut-off function as follows.
[TABLE]
We denote the Hamiltonian function by and let be the composition . 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 sufficiently small. We define points as follows.
[TABLE]
Then, has only one point .
We define as follows.
[TABLE]
Lemma 4 implies that holds and this implies that
[TABLE]
holds. So we proved Theorem 2.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] P. Albers, U. Frauenfelder. Square roots of Hamiltonian diffeomorphisms. J. Symplectic. Geom. Volume 12, Number 3 (2014), 427-434
- 2[2] J. Milnor. Remarks on infinite-dimensional Lie groups . relativity, groups and topology Ⅱ, COURSE 10
- 3[3] L. Polterovich, E. Schelukhin. Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules. Selecta Mathematica January 2016, Volume 22, Issue 1, pp 227-296
- 4[4] Y. Sugimoto. Spectral spread and non-autonomous Hamiltonian diffeomorphisms. manuscripta math. (2018). https://doi.org/10.1007/s 00229-018-1078-0
