On symplectic dynamics

Differential Geometry and its Applications 61 (2018) 170–196

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Differential Geometry and its Applications www.elsevier.com/locate/difgeo

On symplectic dynamics Stéphane Tchuiaga Department of Mathematics of the University of Buea, South West Region, Cameroon

a r t i c l e

i n f o

Article history: Received 22 October 2016 Received in revised form 14 December 2017 Available online xxxx Communicated by M. Gotay MSC: 53D35 53D05 57R52 53C21 Keywords: Differential forms Linear sections Isotopies Riemannian metric Symplectic diffeomorphisms Hofer-like geometries

a b s t r a c t This paper contributes to the foundation of various Hofer-like topologies of a closed symplectic manifold (M, ω). Here, considering an arbitrarily linear section of the natural projection of the space of closed 1-forms onto the first de Rham’s group, we study various Hofer-like metrics [2,7]. The outcome is that different splitting of the space of closed 1-forms lead to similar “symplectic analogues” of some results from Hamiltonian dynamics: Without appealing to the positivity result of any symplectic displacement energy, we point out an impact of the L∞ -Hoferlike lengths in the investigation of the symplectic nature of the uniform limits of sequences of symplectic diffeomorphisms isotopic to the identity map: This provides various symplectic analogues of a result that was proved by Hofer–Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on R2n ); and reformulated by Oh–Müller [13] for Hamiltonian diffeomorphisms in general. Moreover, we show that Polterovich’s regularization method for Hamiltonian paths extends over the whole group of symplectic paths (no matter the choice of the above section), and use this to prove the equality between the two versions of the corresponding Hofer-like metrics defined on the group of time-one maps of all symplectic isotopies: The symplectic analogues of the uniqueness result of Hofer’s geometry [14]. This includes some condition(s) that make the latter uniqueness result continues holds in the context of Bus–Leclercq [7], and a short proof of the non-degeneracy of various Hofer-like metrics. Finally, an alternate proof (without appealing to Floer’s theory) of a result from flux geometry found by McDuff–Salamon [12] is given, and various symplectic analogues of some approximation results found by Oh–Müller [13] are provided. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The Hofer geometry originated with the remarkable paper of Hofer that introduced the Hofer topologies on the group of Hamiltonian diffeomorphisms of a symplectic manifold (so-called Hofer metrics, [9]). In particular, Hofer–Zehnder [10] had elaborated almost all the basic formulas and perspectives for the subsequent development of Hamiltonian dynamics based on Hofer’s metrics. E-mail address: [email protected]. https://doi.org/10.1016/j.difgeo.2018.09.003 0926-2245/© 2018 Elsevier B.V. All rights reserved.

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Recently, Banyaga [2] and Bus–Leclercq [7] showed that on a closed symplectic manifold (M, ω), each Hofer’s metric generalizes over the whole group of time-one maps of all symplectic isotopies (so-called Hoferlike metrics, [2,7]). Here, generalization means: If the first de Rham cohomology group of M is trivial, then these Hofer-like metrics reduce to some metrics defined on the group of all Hamiltonian diffeomorphisms. Furthermore, if the above de Rham group is non-trivial, then it is known that the restriction of the L(1,∞) -version of Banyaga’s Hofer-like norm to the group of all Hamiltonian diffeomorphisms is equivalent to the usual Hofer norm (see [7] and [16]). However, we have some thorough discussions based on the usual Hofer’s metrics whose symplectic analogues are still unknown (see [10], [11], [13], and [14]). These facts seem to attest that to better understand the geometries behind the Hofer-like metrics (e.g. the study of geodesics in the group of all time-one maps of symplectic isotopies with respect to Hofer-like metrics), further investigations still need to be done. This is one of the main objectives of the present paper: Contribute to the foundation of various Hofer-like geometries. We organize this paper as follows. In Section 2, we study some relationships between symplectic isotopies and the splitting of the space of closed 1-forms Z 1 (M ) as a direct sum Z 1 (M ) = H1 (M, S) ⊕S B1 (M ),

(1.1)

with respect to an arbitrary linear section S of the natural projection π : Z 1 (M ) → H 1 (M, R), where H1 (M, S) is isomorphic to the first de Rham group H 1 (M, R), and B1 (M ) is contained in ker π: Subsection 2.4 deals with the S-decomposition of symplectic isotopies which generalizes the usual Hodge decomposition of symplectic isotopies; Subsection 2.5 illustrates some implications of Riemannian geometry in the study of various Hofer-like geometry, while in Subsection 2.10, we use the splitting in (1.1) to prove that Polterovich’s regularization method for Hamiltonian isotopies admits various symplectic analogues (no matter the choice of the linear section S): This general regularization process could play a paramount role in the study of geodesics with respect to various Hofer-like metrics. As a consequence of Proposition 2.4, we give an alternative proof (without appealing to Floer’s theory) of a result from flux geometry (Proposition 2.7) found by McDuff–Salamon [12]. Section 3 contains the main results of this paper: The first main result Theorem 3.3 shows in particular that for each section S, one can use the L∞ -Hofer-like lengths to investigate the symplectic nature of the C 0 -limit of sequences of symplectic diffeomorphisms isotopic to the identity map (without appealing to the positivity result of any symplectic displacement energy). Here, we exhibit the existence of a larger number of non-Hamiltonian symplectic isotopies whose Hofer-like lengths are symmetric (Example 3.1). The second main result Theorem 3.8 shows that the Hofer-like geometry resulting from each of the above splitting is independent to the choice of a corresponding version (L∞-version or L(1,∞) -version) of the Hofer-like metrics. This includes some condition(s) that make the latter uniqueness result holds in the context of Bus–Leclercq [7], and a short proof of the non-degeneracy of these Hofer-like metrics. The last Section deals with various symplectic analogues of some approximation lemmas found by Oh–Müller [13]. 2. Preliminaries Let M be a smooth closed manifold of dimension 2n. In brief, a 2-form ω on M is called a symplectic form if it is closed and non-degenerate. Then, a symplectic manifold is a manifold which can be equipped with a symplectic form. In particular, note that any symplectic manifold is oriented, and not all even dimensional manifolds can be equipped with a symplectic form. In the rest of this paper, we shall always assume that M is a closed manifold that admits a symplectic form ω, and we shall fix a Riemannian metric g on M (any differentiable manifold M can be equipped

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with a Riemannian metric). Furthermore, we shall write d to denote the distance induced on M by the Riemannian metric g. The metric topology induced by d on M coincides with the underlying topological structure of M . 2.1. Splittings of closed 1-forms Let H 1 (M, R) denote the first de Rham cohomology group (with real coefficients) of M , and let Z 1 (M ) denote the space of all closed 1-forms on M . Consider S : H 1 (M, R) → Z 1 (M ),

(2.1)

to be a fixed linear section of the natural projection π : Z 1 (M ) → H 1 (M, R).

(2.2)

α = S(π(α)) + (α − S(π(α))).

(2.3)

Each α ∈ Z 1 (M ) splits as:

In the above splitting, we shall call the 1-form (α − S(π(α))) the exact part of α, and throughout all the paper, for simplicity, when this will be necessary, the latter 1-form will be denoted dfα,S to mean that it is the differential of a certain smooth function that depends on α and S; while we shall call the 1-form S(π(α)), the S-form of α. Let H1 (M, S) denote the space of all S-forms, and B1 (M ) denote the set (Z 1 (M )\H1 (M, S)) ∪ {0}. We have that Z 1 (M ) = H1 (M, S) ⊕S B1 (M ).

(2.4)

If one chooses the above section such that each S-form is a harmonic 1-form with respect to some Riemannian metric g on M , then the corresponding splitting is called the Hodge decomposition, and H1 (M, S) is nothing than the space of harmonic 1-forms. 2.1.1. Norms on H1 (M, S), B1 (M ) and Z 1 (M ) Consider ν B to be any norm on B1 (M ). From now on, we equip H1 (M, S) with the L2 -norm .L2 for 1-forms induced by the Riemannian metric g. Note that on a closed manifold, the L2-norms induced by different Riemannian metrics are equivalent. Hence, for each non-negative λ, we define a semi-norm NS,λ on Z 1 (M ) as follows: NS,λ (α) = ν B ((α − S(π(α)))) + λS(π(α))L2 ,

(2.5)

for all α ∈ Z 1 (M ). Remark 2.1. For technical purpose, we shall sometimes choose the norm ν B to be equivalent to the following oscillation norm: If α ∈ Z 1 (M ) such that dfα,S := α − S(π(α)), then osc(f˜α,S ) := max f˜α,S (x) − max f˜α,S (x), x∈M

x∈M

(2.6)

where f˜α,S is the function fα,S normalized, i.e.,  f˜α,S = fα,S −

f ω M α,S ωn M

n

.

(2.7)

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We denote by PH1 (M, S) the space of all smooth mappings H : [0, 1] → H1 (M, S), and by PB1 (M ), we denote the space of all smooth mappings H : [0, 1] → B1 (M ). 2.1.2. Comparison of norms Consider α to be a differential 1-form on M : for each x ∈ M , we know that α induces a linear map αx : Tx M → R, whose norm is given by αx g = sup{|αx (X)| : X ∈ Tx M, Xg = 1},

(2.8)

where .g is the norm induced on each tangent space Tx M (at the point x) by the Riemannian metric g. Therefore, the uniform sup norm of α, say |.|0 is defined as |α|0 = sup αx g .

(2.9)

x∈M

Since both spaces H1 (M, S) and H 1 (M, R) are isomorphic, and H 1 (M, R) is a finite dimensional vector space whose dimension is the first Betti number b1 (M ), then H1 (M, S) is of finite dimension. Thus, there exists a positive constant K(g) which depends on the Riemannian metric g such that |α|0 ≤ K(g)αL2 ,

(2.10)

for all α ∈ H1 (M, S). 2.2. Symplectic vector fields The symplectic structure ω on M being non-degenerate, it induces an isomorphism between vector fields and 1-forms on M . This isomorphism is given by: to each vector field Y on M , one assigns the 1-form ι(Y )ω := ω(Y, .), where ι is the usual interior product. A vector field Y on M is symplectic if the 1-form ι(Y )ω is closed. A symplectic vector field Y is said to be a Hamiltonian vector field if the 1-form ι(Y )ω is exact. It follows from the definition of symplectic vector fields that if the first de Rham’s cohomology group of M is trivial, then any symplectic vector field on M is Hamiltonian. Note that the first de Rham’s group is a topological invariant, i.e., it does not depend on the differentiable structure on M and only depends on the underlying topological structure of M [17]. 2.3. Symplectic diffeomorphisms and symplectic isotopies A diffeomorphism φ : M → M , is called symplectic if it preserves the symplectic form ω, i.e. φ∗ (ω) = ω. We denote by Symp(M, ω), the group of all symplectic diffeomorphisms of (M, ω). An isotopy {φt } of a symplectic manifold (M, ω) is said to be symplectic if φt ∈ Symp(M, ω) for each t, or equivalently, the vector dφt ◦ φ−1 field φ˙ t := is symplectic for each t. In particular, a symplectic isotopy {ψt } is a Hamiltonian t dt dψt ◦ ψt−1 is Hamiltonian, i.e., there exists a smooth function isotopy if for each t, the vector field ψ˙ t := dt F : [0, 1] × M → R, called generating Hamiltonian such that ι(ψ˙ t )ω = dFt ,

(2.11)

for each t. Any Hamiltonian isotopy determines its generating Hamiltonian up to an additive constant. Throughout this paper we assume that every generating Hamiltonian F : [0, 1] × M → R is normalized, i.e.,  we require that M Ft ω n = 0, for every t.

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Let N ([0, 1] × M , R) denote the vector space of all smooth normalized Hamiltonians, let Iso(M, ω) denote the group of all symplectic isotopies of (M, ω), and let Symp0 (M, ω) denote the group of time-one maps of all symplectic isotopies. 2.4. S-decompositions of symplectic isotopies Given any symplectic isotopy Φ = {φt }, one derives from the S-decomposition that the closed 1-form ι(φ˙ t )ω decomposes in a unique way as: ι(φ˙ t )ω = S(π(ι(φ˙ t )ω)) + (ι(φ˙ t )ω − S(π(ι(φ˙ t )ω))).

(2.12)

Now, let {Xt } and {Yt } be two smooth families of symplectic vector fields defined respectively by hΦ t :=   ι(Xt )ω = S(π(ι(φ˙ t )ω)), and ι(Yt )ω = ρ∗t dUtΦ , for all t, where dUtΦ := ι(φ˙ t )ω − S(π(ι(φ˙ t )ω)) and {ρt } is the symplectic isotopy such that ∂ ρt (x) = Xt (ρt (x)), ∂t for all x ∈ M . We have the following fact. Proposition 2.2. For each fixed section S, any symplectic isotopy Φ = {φt } decomposes in a unique way as φt = ρt ◦S ψt ,

(2.13)

for each t, where {ρt } is called the S-isotopy, and {ψt } is a Hamiltonian isotopy. As in [5], denote by U the smooth family (UtΦ ), and by H the smooth family (hΦ t ). The Cartesian 1 1 1 product PB (M ) × PH (M, S) is isomorphic to N ([0, 1] × M , R) × PH (M, S) =: T(M, ω, S). We equip this Cartesian product with a group structure which makes the bijection AS : Iso(M, ω) → T(M, ω, S), Φ → (U, H)

(2.14)

a group isomorphism. Under this identification, any symplectic isotopy Φ is denoted by φ(U,H) to mean that AS maps Φ onto (U, H), and the pair (U, H) is called the “generator” of the symplectic isotopy Φ. For instance, a symplectic isotopy φ(0,H) , is an S-isotopy, and a symplectic isotopy φ(U,0) , is a Hamiltonian isotopy. 2.4.1. Group structure on T(M, ω, S) The product rule in T(M, ω, S) is given by, −1  (U, H) 1S (V, K) = (U + V ◦ φ−1 (U,H) + Δ(K, φ(U,H) ), H + K).

(2.15)

The inverse of (U, H), say (U, H) is given by  (U, H) = (−U ◦ φ(U,H) − Δ(H, φ(U,H) ), −H).  is the function Δt (α, Ψ) := In (2.15) and (2.16) the quantity Δ particular case where the S-forms are harmonic 1-forms). Here is a consequence of Proposition 2.2.

t 0

(2.16)

αt (ψ˙ s ) ◦ ψ s ds normalized (see [5], for the

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Corollary 2.3. Every S-generator (U, H) decomposes in a unique way as: (U, H) = (0, H) 1S (U ◦ φ(0,H) , 0).

(2.17)

Since the proof of the above corollary is easy, we leave it to readers. In fact, the S-decomposition generalizes the Hodge decomposition introduced in [2]. Hence, the S-decomposition can be used to generalize some (not to say almost all) results found in [2,5]. A part of this generalization is done in the present paper. 2.5. Scholium So far, we have introduced the function Δ(H, Φ) whenever H is a smooth family of closed 1-forms, and Φ = {φt } is an isotopy of a compact manifold M with φ0 = idM , but have not produced any of its algebraic, geometric or analytic properties, so one might be wondering whether they are rare or not. 2.5.1. Boundedness properties of Δ(H, Φ) Here, we use some well-known facts from Riemannian geometry to point out some boundedness properties of the Hofer norms of the functions Δ(H, Φ). Firstly, recall that for any smooth family of closed p-forms {Ωt }, and for any isotopy Φ = {φt } of a compact manifold, it is well-known that φ∗t (Ωt )

t − Ωt = d{

  φ∗s ι(φ˙ s )Ωt ds},

(2.18)

0

for each t, where d stands for the usual differential operator (see [2] for a quick proof). In particular, when {Ωt } is a smooth family of closed 1-forms, say H = (Ht ), it follows from (2.18) that φ∗t (Ht ) − Ht = d{Δt (H, Φ)},

(2.19)

for each t. Now fix a point x0 ∈ M , and for all x ∈ M pick a curve γx from x0 to x, then define a smooth function by setting  u ¯t (x) :=

(φ∗t (Ht ) − Ht ) ,

(2.20)

γx

for each t. The function u ¯ defined in (2.20) does not depend on the choice of the curve γx from x0 to x, and it follows from (2.19) that for each (t, x), we have u ¯t (x) = (Δt (H, Φ)) (x) − (Δt (H, Φ)) (x0 ),

(2.21)

i.e., both functions u ¯ and Δ(H, Φ) have the same Hofer norms. For all > 0, for each x ∈ M , choose the 1 curve γx such that its length l(γx ) := 0 γ˙ x (s)g ds satisfies l(γx ) ≤ (1 + )diam(M ),

(2.22)

where diam(M ) stands for the diameter of M with respect to the Riemannian metric g. For instance, for every t, consider y0 to be any point of M that realizes the supremum of the function x → |¯ ut (x)|, and derive from the triangle inequality that,

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 sup |¯ ut (x)| ≤ |

 Ht | + |

x

γy0

φ∗t (Ht )|

γy0

1 ≤ |Ht |0

1 γ˙ y0 (s)g ds + sup |Dφv (γy0 (s))||Ht |0

γ˙ y0 (s)g ds,

v,s

0

0

≤ diam(M )(1 + )(1 + sup |Dφv (γy0 (s))|)|Ht |0 , s,v

for every t, and where Dφv is the tangent map of φv . So, we obtain for all > 0, we have sup |¯ ut (x)| ≤ diam(M )(1 + )(1 + sup |Dφv (γy0 (s))|)|Ht |0 , x

(2.23)

s,v

for each t, and since we always have, osc(Δt (H, Φ)) = osc(¯ ut ) ≤ 2 sup |¯ ut (x)|,

(2.24)

x

for each t, then we deduce from (2.23) together with (2.24) that 1

1 osc(Δt (H, Φ))dt ≤ 2(1 + )diam(M )(1 + sup |Dφv (γy0 (s))|)

|Ht |0 dt,

s,v

0

(2.25)

0

max osc(Δt (H, Φ)) ≤ 2(1 + )diam(M )(1 + sup |Dφv (γy0 (s))|) max |Ht |0 .

t∈[0,1]

t∈[0,1]

s,v

2

(2.26)

2.5.2. Algebraic properties of Δ(H, Φ) For further survey of the class of functions Δ(H, Φ), let us consider the following Poincaré pairing: 

, P : H (M, R) × H 1

2n−1

(M, R) → R, ([α], [β]) →

α ∧ β,

(2.27)

M

where H ∗ (M, R) represents the ∗-th de Rham cohomology group with real coefficients, and [α] stands for the de Rham cohomology class of a closed differential form α. We have the following fact. Proposition 2.4. Let H = (Ht ) be a smooth family of closed 1-forms, and let Φ = {φt } be a symplectic isotopy. Then for each t ∈ [0, 1], we have  ¯ t ), [Ht ∧ ω (n−1) ]P , Δt (H, Φ)ω n = n F lux(Φ

(2.28)

M

¯ t is the isotopy s → φst , and F lux stands for the flux homomorphism. In particular, if Φ is a where Φ Hamiltonian isotopy then the right hand side in (2.28) vanishes. Proof. For each fixed t ∈ [0, 1], since the differential form Ht ∧ω n is of degree (2n +1) over a 2n-dimensional manifold, we derive that Ht ∧ ω n = 0. This implies that Ht (φ˙ s )ω n − nι(φ˙ s )ω ∧ Ht ∧ ω (n−1) = 0, for each s ∈ [0, t]. Composing the above equality in both sides by φ∗s yields,

(2.29)

S. Tchuiaga / Differential Geometry and its Applications 61 (2018) 170–196

    φ∗s Ht (φ˙ s ) ω n − nφ∗s ι(φ˙ s )ω ∧ φ∗s (Ht ) ∧ ω (n−1) = 0,

177

(2.30)

for each s ∈ [0, t]. Using (2.19), we derive that for each s ∈ [0, t], we have s φ∗s (Ht ) = Ht + df{φ , t },Ht

(2.31)

where s s f{φ t },Ht

Ht (φ˙ u ) ◦ φu du.

:=

(2.32)

0

Relations (2.30) and (2.31) immediately imply that,       s φ∗s Ht (φ˙ s ) ω n = nφ∗s ι(φ˙ s )ω ∧ Ht ∧ ω (n−1) + nφ∗s ι(φ˙ s )ω ∧ df{φ ∧ ω (n−1) , t },Ht

(2.33)

for each s ∈ [0, t]. That is, 

⎞ ⎞ ⎛ t ⎛  t      ⎝ φ∗s Ht (φ˙ s ) ds⎠ ω n = n ⎝ φ∗s ι(φ˙ s )ω ds⎠ ∧ Ht ∧ ω (n−1) 0

M

0

M

 +n

⎛ t     s ⎝ φ∗s ι(φ˙ s )ω ∧ df{φ



t },Ht

M

⎞ ds⎠ ∧ ω (n−1) .

(2.34)

0

  On the other hand, since the differential forms φ∗s ι(φ˙ s )ω , and ω (n−1) are closed, we compute  t

  s φ∗s ι(φ˙ s )ω ∧ df{φ ∧ ω (n−1) ds = t },Ht

M 0



t d[

M

  s f{φ φ∗ ι(φ˙ s )ω ∧ ω (n−1) ds], t },Ht s

(2.35)

0

and since the manifold M is without boundary, we derive from Stokes’ theorem that 

t d[

M

s f{φ φ∗ t },Ht s



 t

 ι(φ˙ s )ω ∧ ω (n−1) ds] =

0

  s f{φ φ∗ ι(φ˙ s )ω ∧ ω (n−1) ds = 0, t },Ht s

(2.36)

∂M 0

for each t. We combine (2.34), (2.35), and (2.36) to obtain  t

  φ∗s Ht (φ˙ s ) dsω n = n

M 0

 M

⎞ ⎛ t    ⎝ φ∗s ι(φ˙ s )ω ds⎠ ∧ Ht ∧ ω (n−1) .

(2.37)

0

Observe that for each t, the de Rham cohomology class of the 1-form ¯ t [1], i.e., (2.37) can be written as of the isotopy Φ

t 0

  φ∗s ι(φ˙ s )ω ds is exactly the flux

 ¯ t ), [Ht ∧ ω (n−1) ]P . Δt (H, Φ)ω n = n F lux(Φ

(2.38)

M

¯ t for each t, i.e., for each time t, the right hand In particular, if the isotopy Φ is Hamiltonian, then so is Φ side in (2.38) is zero. This completes the proof. 2

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Proposition 2.5. If Φ = {φt } and Ψ = {ψ t } are two isotopies, and H = (Ht ) is a smooth family of closed 1-forms, then we have Δt (H, Φ ◦ Ψ)(x) = Δt (H, Ψ)(x) + Δt (H, Φ) ◦ ψ t (x),

(2.39)

for each t and for all x ∈ M . Proof. Using (2.18) we derive that, dΔt (H, Φ ◦ Ψ) = (φt ◦ ψt )∗ Ht − Ht = ψt∗ (Ht + dΔt (H, Φ)) − Ht = ψt∗ (Ht ) + d(Δt (H, Φ) ◦ ψt ) − Ht = Ht + dΔt (H, Ψ) + d(Δt (H, Φ) ◦ ψt ) − Ht = dΔt (H, Ψ) + d(Δt (H, Φ) ◦ ψt ), for each t ∈ [0, 1]. Since M is connected, it follows from the above equalities that there exists a constant η such that: Δt (H, Φ ◦ Ψ)(x) = Δt (H, Ψ)(x) + Δt (H, Φ) ◦ ψt (x) + η,

(2.40)

for all x ∈ M . On the other hand, Proposition 2.4 implies that  Δt (H, Φ ◦ Ψ)ω n = n F lux(Φt ◦¯ Ψt ), [Ht ∧ ω (n−1) ]P M

= n F lux(Φ¯t ), [Ht ∧ ω (n−1) ]P + n F lux(Ψ¯t ), [Ht ∧ ω (n−1) ]P   n = Δt (H, Φ)ω + Δt (H, Ψ)ω n . M

That is,

 M

M

ηω n = 0, i.e., η = 0. This achieves the proof. 2

Proposition 2.4 and Proposition 2.5 generalize to volume-preserving isotopies [16]. Furthermore, it seems that on a product of two symplectic manifolds, combining Proposition 2.4 with the Künneth formula could yield a comparison between the flux group of the product manifold and the product of flux groups. 2.6. A geometric interpretation of the functions Δ(α, Φ) Given a smooth family of diffeomorphisms Φ = {φt }, by setting Xt (x) =

d φs (x), ds |s=t

(2.41)

for each t and for all x ∈ M , one defines a family of tangent vector fields (Xt ) on M : Xt (x) ∈ Tφt (x) M for each t. Now, set φ˙ t = Xt ◦ φ−1 t ,

(2.42)

for each t (see page 5 in [3] for more convenience). In particular, for each fixed (t, x) ∈ [0, 1] × M , if we consider the curve γx,t : s → φst (x), then

S. Tchuiaga / Differential Geometry and its Applications 61 (2018) 170–196

γ˙ x,t (s) =

d d φst (x) = t φu (x) = tXu (x)|u=st , ds du |u=st

179

(2.43)

for all s. Assume Φ is a symplectic isotopy, and α is a closed 1-form. We have 

1 α=

γx,t

1 αγx,t (s) (γ˙ x,t (s))ds =

0

φ∗st

  α(tφ˙ st ) (x)ds =

0

t

  φ∗u α(φ˙ u ) (x)du.

(2.44)

0

On the other hand, if {θt } is the symplectic flow generated by α, then we derive as in [12] that 

 α= γx,t

γx,t

⎞ ⎛ 1  1 1 ⎝ ι(θ˙u )ωdu⎠ = ωγx,t (s) (θ˙u (γx,t (s)), γ˙ x,t (s))duds, 0

0

(2.45)

0

i.e., 

 α= γx,t

(Θγx,t )∗ ω,

(2.46)

[0,1]×[0,1]

where Θγx,t : [0, 1] × [0, 1] → M, (u, s) → θu (φst (x)).

(2.47)

Geometrically, the equalities (2.44) and (2.46) tell us that for each t, the real number Δt (α, Φ)(x) can be interpreted as the signed symplectic area of the 2-chain {θu (φst (x))|0 ≤ u, s ≤ 1}. In particular, each zero of the function Δt (α, Φ) gives rise to a 2-chain whose symplectic area is trivial. The following consequence of Proposition 2.4 gives a sufficient condition that guarantees the existence of at least one zero for such a function. Lemma 2.6. If α is a smooth family of closed 1-forms on M and Φ is a Hamiltonian isotopy, then for each t, the function x → Δt (α, Φ)(x) has at least one vanishing point on M . Proof. Assume α is a smooth family of closed 1-forms and Φ is a Hamiltonian isotopy. For each fixed t ∈ [0, 1], we have a smooth function x → Δt (α, Φ)(x) from the compact M into the set of real numbers. Thus, the latter function achieves its bounds; and this implies that 

 ω ≤

min Δt (α, Φ)(x)

x∈M

M

 Δt (α, Φ)ω ≤ max Δt (α, Φ)(x)

n

n

ωn ,

x∈M

M

(2.48)

M

i.e., minx∈M Δt (α, Φ)(x) ≤ 0, and 0 ≤ maxx∈M Δt (α, Φ)(x) since by Proposition 2.4, we have  Δt (α, Φ)ω n = 0. 2 M McDuff–Salamon [12] proved that the orbits of Hamiltonian loops are null-homologous (see Lemma 10.31, [12]). This is equivalent to the following result. Proposition 2.7. Let Ψ be any symplectic isotopy whose flux is non-trivial. Then any loop γ in the homotopy class of an orbit of any Hamiltonian loop (relatively to a fixed base point) trivializes the flux of Ψ, i.e., F lux(Ψ).[γ] = 0.

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Without appealing to Floer’s theory, we provide the following alternative proof of Proposition 2.7 as a consequence of Lemma 2.6. Proof of Proposition 2.7. Let Ψ be a symplectic isotopy whose flux is non-trivial. Let HΨ denote the harmonic representative in the de Rham cohomology class F lux(Ψ). Consider a Hamiltonian loop Φ = {φt } in Symp0 (M, ω), and derive from (2.19) that dΔ1 (HΨ , Φ) = 0, i.e., the function Δ1 (HΨ , Φ) is constant. In fact, we have Δ1 (HΨ , Φ)(x) = 0 for all x ∈ M because Lemma 2.6 implies that the function x → Δ1 (HΨ , Φ)(x) has a vanishing point on M , and M is connected. On the other hand, for each x ∈ M , consider the loop γx,Φ : t → φt (x), and check by the mean of (2.44) that  HΨ = Δ1 (HΨ , Φ)(x) = 0.

F lux(Ψ).[γx,Φ ] =

(2.49)

γx,Φ

In addition, if β is any representative in the homotopic class [γx,Φ ] (relatively to fixed base point), then   Stokes’ theorem implies that γx,Φ HΨ = β HΨ . This completes the proof. 2 Proposition 2.7 tells us that on the 2-dimensional torus T2 equipped with the symplectic structure ω, induced by the standard symplectic form on R2 , no meridian circle of T2 is an orbit of a Hamiltonian loop in Symp0 (T2 , ω) since a meridian of T2 yields a non-trivial homology class: This shows how an orbit of a Hamiltonian loop winds on T2 . 2.7. Reparameterization of symplectic isotopies [5,13] We shall need the following basic formula for the S-generator of a reparameterized symplectic isotopy. If Φ = {φt } is a symplectic isotopy generated by (U, H), and ξ : [0, 1] → [0, 1] is a smooth function, then the reparameterized path t → φξ(t) , denoted Φξ , is generated by the element (U, H)ξ defined as (U, H)ξ := (U ξ , Hξ ),

(2.50)

ξ ˙ where Hξ denotes the smooth family of non-exact 1-forms t → ξ(t)H ξ(t) ; U denotes the smooth family ˙ ˙ of smooth functions (x, t) → ξ(t)U ξ(t) (x); while ξ(t) is the derivative of the function ξ at the point t. The ξ inverse element of (U, H) is given by

 ξ , Φξ ), −Hξ ), (U, H)ξ = (−U ξ ◦ Φξ − Δ(H

(2.51)

ξ(t) ˙ ˙ ˙ Δt (H , Φ ) = ξ(t)H ξ(t) (φu ) ◦ φu du = ξ(t)Δξ(t) (H, Φ),

(2.52)

with ξ

ξ

0

for each t, i.e., Δ(Hξ , Φξ ) = Δξ (H, Φ).

(2.53)

Definition 2.8. ([13]). Given a smooth function ξ : [0, 1] → R, its norm ξham is defined by ˙ L , ξham = ξC 0 + ξ 1 ˙ L = with ξ 1

1 0

˙ |ξ(t)|dt, and ξC 0 = supt |ξ(t)|.

(2.54)

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2.8. Boundary flat symplectic isotopies [5,13] Definition 2.9. Given (U, H) ∈ T(M, ω, S), we say that (U, H) is boundary flat if there exists δ ∈]0, 1[ such that (Ut , Ht ) = (0, 0), for all t in [0, δ[∪]1 − δ, 1]. In other words, a symplectic path {φt } is boundary flat if there exists a constant 0 < δ < 1 such that φt = idM for all 0 ≤ t < δ, and φt = φ1 for all 1 − δ < t ≤ 1. 2.9. The C 0 -topology Let Homeo(M ) denote the group of all homeomorphisms of M equipped with the C 0 - compact-open topology. This is the metric topology induced by the following distance d0 (f, h) = max(dC 0 (f, h), dC 0 (f −1 , h−1 )),

(2.55)

dC 0 (f, h) = sup d(h(x), f (x)).

(2.56)

where

x∈M

On the space of all continuous paths λ : [0, 1] → Homeo(M ) such that λ(0) = idM , we consider the C 0 -topology as the metric topology induced by the following metric ¯ μ) = max d0 (λ(t), μ(t)). d(λ, t∈[0,1]

(2.57)

2.10. Regularization of symplectic isotopies Definition 2.10. A symplectic path {φt } is said to be regular if for every t, the vector field φ˙ t does not vanish identically. First of all, we recall that a regularization method for Hamiltonian paths is due to Polterovich [14]. As far as I know, when H 1 (M, R) = {0}, a general regularization method (in the sense of Definition 2.10) for the whole group of symplectic isotopies is unknown. By regularization method we mean: Given any element φ ∈ Symp0 (M, ω), one can modify any symplectic Φ from the identity to φ until to obtain another symplectic Ψ with time-one map φ, which is regular in the sense of (Definition 2.10). The goal of this subsection is to provide such a result for non-Hamiltonian symplectic paths: Given a symplectic isotopy Φ whose S-generator is (U, H); in view of Proposition 5.2.A–[14], for the above Hamiltonian U , there exists a Hamiltonian loop φ(r,0) which is close to the constant loop identity (in the C ∞ -sense), and in particular, its generating function r is arbitrarily small in the L(1,∞) -version of Hofer’s norm so that osc(−rt + Ut ) = 0,

(2.58)

for all t. Now consider (V, K) to be the product (r, 0) 1S (U, H), which can be written immediately as  (V, K) = (−r ◦ φ(r,0) + U ◦ φ(r,0) + Δ(H, φ(r,0) ), H).

(2.59)

We claim that the symplectic isotopy generated by (V, K) defined in (2.59) is regular in the sense of Definition 2.10 (no matter the choice of the section S). The proof of this claim relies on the splitting of the space of closed 1-forms Z 1 (M ) and Proposition 5.2.A–[14]. Arguing indirectly, we assume that there exists a time s for which the vector field Xs = φ˙ s(V,K) vanishes identically, i.e.,

182

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ι(Xs )ω = dVs + Ks = 0.

(2.60)

d(−rs ◦ φs(r,0) + Us ◦ φs(r,0) + Δs (H, φ(r,0) )) + Hs = 0.

(2.61)

Inserting (2.59) in (2.60) gives

From (2.61), we derive that the 1-form Hs belongs to H1 (M, S) ∩ B1 (M ) = {0}, i.e., the latter 1-form must be trivial. This implies that s Hs (φ˙ u(r,0) ) ◦ φu(r,0) du = 0,

Δs (H, φ(r,0) ) =

(2.62)

0

i.e., the function −rs ◦ φs(r,0) + Us ◦ φs(r,0) is constant since M is connected. The latter argument contradicts (2.58), and the claim follows. 2 As a consequence of the above regularization method, we derive as in [14] that using any regular symplectic path φ(V,K) , we can define a function ζ : [0, 1] → [0, 1] to be the inverse of the map, s s → 01 0

(ν B (dVt ) + λKt L2 )dt (ν B (dVt ) + λKt L2 )dt

,

(2.63)

for some non-negative λ, and a norm ν B on B1 (M ). The derivative of ζ is given explicitly by: 1 

ζ (s) =

(ν B (dVt ) + λKt L2 )dt 0 , ν B (dVζ(s) ) + λKζ(s) L2

(2.64)

for each s. If ζ is only C 1 , then we can approximate ζ in the C 1 -topology by a smooth diffeomorphism κ : [0, 1] → [0, 1], that fixes 0 and 1 [8]. 3. Main results Throughout this section, we introduce the main results of this paper. We start by recalling the definitions of lengths for symplectic isotopies following the ideas of Banyaga [2]. As in [2], given a symplectic isotopy Φ whose S-generator is (U, H), then the L(1,∞) -version and the L∞ -version of Hofer-like lengths of a Φ are defined respectively by (1,∞) lλ,S (Φ)

1 =

 B  ν (dUt ) + λHt L2 dt,

(3.1)

0

and ∞ lλ,S (Φ) = max (ν B (dUt ) + λHt L2 ). t∈[0,1]

(3.2)

In the case that H 1 (M, R) vanishes, the above lengths are called Hofer’s lengths whenever ν B is the oscillation norm. It is clear that if Φ is a Hamiltonian isotopy generated by (U, 0), then its inverse Φ−1 is generated by (−U ◦ Φ, 0); and we have lλ,S (Φ) = lλ,S (Φ−1 ), (1,∞)

(1,∞)

(3.3)

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and ∞ ∞ lλ,S (Φ) = lλ,S (Φ−1 ),

(3.4)

ν B (ρ∗ (dUt )) = ν B (dUt ),

(3.5)

whenever

for all ρ ∈ Symp0 (M, ω) (symmetry). The condition ν B (ρ∗ (dUt )) = ν B (dUt ), is always true whenever ν B is the oscillation norm. On the other hand, for non-Hamiltonian symplectic isotopies we do not know whether the above Hofer-like lengths are symmetric or not (no matter whether ν B is the oscillation norm or not). Indeed, if such a symplectic isotopy Φ has S-generator (U, H), then the formula for inversion of generators  tells us that Φ−1 has S-generator (−U ◦ Φ − Δ(H, Φ), −H), and the gap of symmetry in each of the above  lengths seems to come from the presence of the function Δ(H, Φ) in the S-generator of Φ−1 . This does not −1  mean that the presence of Δ(H, Φ) in the generator of Φ justifies the non-symmetry of the lengths of Φ. In fact, the following example shows the existence of a larger number of non-Hamiltonian symplectic isotopies whose Hofer-like lengths are symmetric. Example 3.1 (Non-Hamiltonian symplectic paths with symmetric lengths). Let α be a non-trivial harmonic 1-form. The symplectic flow {ρt } generated by (0, α) is non-Hamiltonian since its flux is non-trivial. Now, consider Y to be the smooth time-independent vector field such that ι(Y )ω = α, and for each t compute t Δt (α, {ρt })(x) =

ρ∗s (α(ρ˙ s ))(x)ds

t α(Y ) ◦ ρs (x)ds =

=

0

t

0

ω(Y, Y ) ◦ ρs (x)ds = 0,

(3.6)

0

for all x ∈ M because ω(Y, Y ) = 0. That is, the function Δt (α, {ρt }) is constant for each t. Since {ρt }−1 is  generated by (−Δ(α, {ρt }), −α), one derives from (3.6) that (1,∞) lλ,S ({ρt }−1 )

1 =

   t (α, {ρt }) + λαL2 dt = λαL2 = l(1,∞) ({ρt }), ν B −dΔ λ,S

(3.7)

0

and

  ∞  t (α, {ρt }) + λαL2 = λαL2 = l∞ ({ρt }). lλ,S ({ρt }−1 ) = max ν B −dΔ λ,S t∈[0,1]

(3.8)

Remark 3.2. Given a symplectic isotopy Φ and a smooth function ξ : [0, 1] → [0, 1] that fixes 0 and 1, one can derive from Subsection 2.7 that (1,∞)

(1,∞)

lλ,S (Φξ ) = lλ,S (Φ),

(3.9)

i.e., the L(1,∞) -length is invariant under reparameterization via smooth curves ξ that fix 0 and 1. To put the above lengths into further perspective, let assume that ν B is the oscillation norm. In this case, Hofer–Zehnder [10] showed that one can use the Hofer lengths to investigate the Hamiltonian nature of the C 0 -limit of a sequence of Hamiltonian diffeomorphisms (see Theorem 6, [10]). Since the above Hofer-like lengths generalize the Hofer lengths, it is natural to investigate whether one can use the Hofer-like lengths to elaborate the symplectic analogues of Theorem 6 – [10] or not. Of course, by analogy with the Hamiltonian case, in presence of positive symplectic displacement energies, such symplectic analogues can be provided

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(see [4], in the case of Hodge’s decomposition). The worst imaginable scenario is to think of a symplectic analogue of Theorem 6 – [10] in the lack of a positive symplectic displacement energy. However, when ν B is equivalent to the oscillation norm, our proof of the following main result shows that, one can provide various symplectic analogues of Theorem 6 – [10] without appealing to the positivity result of any symplectic displacement energy (no matter the choice of the section S). Theorem 3.3. Let (M, ω) be a closed symplectic manifold. Let {φti } be a sequence of symplectic isotopies, let {ψ t } be another symplectic isotopy, and let φ : M → M be a map such that • (φ1i ) converges uniformly to φ, and ∞ • lλ,S ({ψ t }−1 ◦ {φti }) → 0, i → ∞. Then we must have φ = ψ 1 . A key ingredient in the proof of Theorem 3.3 is the fact that: if a sequence {Hi } ⊂ PH1 (M, S) converges uniformly to H ∈ PH1 (M, S), then the sequence of S-isotopies generated by (0, Hi ) converges in both C 0 -metric and L∞ -metric to the symplectic path generated by (0, H). This is technically supported by the following facts. Lemma 3.4 (Naturality of uniform sup norms). Let {ρti } be a sequence of S-isotopies generated by (0, Hi ) and let {ρt } be another S-isotopy generated by (0, H) such that maxt∈[0,1] Hti − Ht L2 → 0, i → ∞. Then, the following properties hold ∞ ({ρt }−1 ◦ {ρti }) → 0, i → ∞, and (1) lλ,S (2) {ρti } converges uniformly to {ρt }.

Proof. This follows as a simultaneous application of the standard continuity theorem of ODE for Lipschitz vector fields together with a result found in Subsection 2.5. For (2), we define a sequence (Zti) of smooth families of vector fields such that ι(Zti )ω = Hti , for each i and for all t. Likewise, we define another smooth family of vector fields (Zt ) such that ι(Zt )ω = Ht , for all t. Since by assumption the sequence (Hi ) converges uniformly to H, it turns out that the sequence (Zti ) converges uniformly to (Zt ). Therefore, it follows from the standard continuity theorem of ODE for Lipschitz vector fields that the sequence of paths generated by (Zti ) converges uniformly to the path generated by (Zt ), i.e., {ρti } converges uniformly to {ρt }. For (1), compute  i − H, {ρt }), Hi − H), (0, H) 1S (0, Hi ) = (Δ(H

(3.10)

for each i, and derive from (2.26) that max osc(Δt (Hi − H, {ρt })) ≤ 2(1 + )diam(M )(1 + sup |Dρt (γy0 (s))|) max |Ht − Hti |,

t∈[0,1]

t,s

t∈[0,1]

(3.11)

for all > 0, where Dρt stands for the tangent map of ρt , y0 ∈ M , and γy0 is a certain curve (see Subsection 2.5). In particular, for small, the right hand side in (3.11) tends to zero when i goes at infinity since the quantity (1 +supt,s |Dρt (γy0 (s))|) is bounded, and by assumption we have maxt∈[0,1] Hti −Ht L2 → 0, i → ∞. This completes the proof. 2 Remark 3.5. It is clear from Lemma 3.4 that if {ρti } is a sequence of S-isotopies generated by (0, Hi ) and ∞ {ρt } is another S-isotopy generated by (0, H), then the convergence lλ,S ({ρt }−1 ◦ {ρti }) → 0, i → ∞, is equivalent to the convergence maxt∈[0,1] Hti − Ht L2 → 0, i → ∞.

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The following fact is a consequence of Lemma 3.4. Corollary 3.6. Let Φi be a sequence of symplectic isotopies and let Φ be another symplectic isotopy such that ∞ lλ,S (Φ−1 ◦ Φi ) → 0, i → ∞. If for each i, μi is the Hamiltonian isotopy in the S-decomposition of Φi , and ∞ μ is the Hamiltonian isotopy in the S-decomposition of Φ, then we have lλ,S (μ−1 ◦ μi ) → 0, i → ∞. Proof. Let Φ be a symplectic isotopy generated by (U, H), and Φi be a sequence of symplectic isotopies generated by (U i , Hi ). Then, by Corollary 2.3, the Hamiltonian part μ in the S-decomposition of Φ is generated by (U ◦ φ(0,H) , 0), and for each i, the Hamiltonian part μi in the S-decomposition of Φi is generated by (U i ◦ φ(0,Hi ) , 0). For each i, compute osc(Uti ◦ φt(0,Hi ) − Ut ◦ φt(0,H) ) ≤ osc(Uti − Ut ) + osc(Ut ◦ φt(0,Hi ) − Ut ◦ φt(0,H) ),

(3.12)

for all t, and derive from the uniform continuity of the map (t, x) → Ut (x) together with Lemma 3.4 that maxt (osc(Ut ◦ φt(0,Hi ) − Ut ◦ φt(0,H) )) → 0, i → ∞, while by assumption we have maxt (osc(Uti − Ut )) → 0, i → ∞. Hence, max(osc(Uti ◦ φt(0,Hi ) − Ut ◦ φt(0,H) )) → 0, i → ∞. t

(3.13)

This completes the proof. 2 Proof of Theorem 3.3. Firstly, note that we do not use the positivity result of any symplectic displacement energy; and this renders the proof rather delicate. So, we shall proceed in several steps. • Step (1). (Convergence of symplectic isotopies). For each i, let {ρti } (resp. {ρt }) be the S-isotopy arising ∞ in the S-decomposition of the isotopy {φti } (resp. {ψ t }). From the convergence lλ,S ({ψ t }−1 ◦ {φti }) → 0, i → ∞, we derive by the mean of Lemma 3.4 that ∞ ({ρt }−1 ◦ {ρti }) → 0, i → ∞, and (1) lλ,S (2) {ρti } converges uniformly to {ρt }. • Step (2). (Decomposition of the map φ = limC 0 (φ1i )). For each i, let {μti } (resp. {μt }) denote the Hamiltonian isotopy arising in the S-decompositions of the isotopy {φti } (resp. {ψ t }). One derives from the assumption that the sequence of time-one maps (φ1i ) converges uniformly to φ, and according to step (1) the sequence of time-one maps (ρ1i ) converges uniformly to the time-one map ρ1 . The preceding arguments imply that the sequence of time-one maps (μ1i ) converges uniformly to a continuous map σ : M → M because μ1i = (ρ1i )−1 ◦ φ1i for each i, and each factor of the composition converges in the C 0 -metric. Note that at this level, we have no guarantee that the maps σ and φ are invertible. Since the composition of maps is continuous with respect to the C 0 -metric, it follows from the above C 0 -convergences that the map φ decomposes as follows: φ = ρ1 ◦ σ.

(3.14)

• Step (3). (The Hamiltonian nature of the map σ). To achieve the proof, all we have to show is that σ = μ1 , where μ1 is the time-one map of the Hamiltonian isotopy {μt }. Arguing indirectly, i.e. assuming that σ = μ1 , we derive that there exists a small non-empty open ball B ⊂ M such that B is completely displaced by (μ1 )−1 ◦ σ, i.e., B ∩ [(μ1 )−1 ◦ σ](B) = ∅. Since B is compact and the convergence μ1i → σ is uniform, we must have B ∩ [(μ1 )−1 ◦ μ1i ](B) = ∅, for i large enough. The above arguments tell us that we can apply the energy-capacity inequality theorem [11] to obtain 0 < C(B)/2 ≤ l∞ ({μt }−1 ◦ {μti }),

(3.15)

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for i large enough, where C(B) represents the Gromov area of the ball B. But, Corollary 3.6 implies that the right hand side in (3.15) tends to zero for i large enough, and this contradicts the positivity of the Gromov area C(B). Hence, we have proved that σ = μ1 .

(3.16)

Finally, (3.14) together with (3.16) implies that, φ = ρ1 ◦ σ = ρ1 ◦ μ1 = ψ 1 .

(3.17)

This completes the proof. 2 The following result is an immediate consequence of Theorem 3.3, and it can justify the definition of strong symplectic isotopies in the L∞ -context [15,6]. Corollary 3.7. Let Φi = {φti } be a sequence of symplectic isotopies, Ψ = {ψt } be another symplectic isotopy, and let η : t → ηt be a family of maps ηt : M → M , such that the sequence Φi converges uniformly to η and ∞ lλ,S (Ψ−1 ◦ Φi ) → 0, i → ∞. Then η = Ψ. Proof. Assume the contrary, i.e., assume that Ψ = η. This is equivalent to say that there exists t ∈]0, 1] such that ηt = ψt . Therefore, the sequence of symplectic paths Ξi : s → φst i contradicts Theorem 3.3. This completes the proof. 2 Although the latter studies of Hofer-like lengths give us some informations about some symplectic dynamical systems, they do not directly address a question that turns out to be of paramount interest in the study of Hofer-like geometry: Is Hofer-like geometry independent the choice of the Hofer-like metric in general? The answer of the above question is affirmative whenever H 1 (M, R) vanishes [14]. But, when H 1 (M, R) is not necessary trivial, in order to investigate the above question we shall recall the following definitions. 3.1. Hofer-like norms Let φ ∈ Symp0 (M, ω). Following the ideas of Banyaga [2], we use the lengths defined in (3.1) and (3.2) to define respectively the L(1,∞) -energy and L∞ -energy of φ as follows: (1,∞)

eλ,S (φ) = inf(lλ,S (Φ)),

(3.18)

∞ e∞ λ,S (φ) = inf(lλ,S (Φ)),

(3.19)

and,

where each infimum is taken over the set of all symplectic isotopies Φ with time-one map equal to φ. Therefore, the L(1,∞) -version and the L∞ -version of the Hofer-like norms of φ are respectively defined by, (1,∞)

φλ,S

= (eλ,S (φ) + eλ,S (φ−1 ))/2,

(3.20)

and ∞ ∞ −1 φ∞ ))/2. λ,S = (eλ,S (φ) + eλ,S (φ

(3.21)

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∞ Note that if φ ∈ Symp0 (M, ω) such that φ∞ λ,S = 0, then eλ,S (φ) = 0; and the definition of the Hofer-like energy combined with Theorem 3.3 imply immediatly that φ = idM (the non-degeneracy of various Hoferlike norms). 2

are the symplectic analogues of the Hofer norms for Hamiltonian diffeoThe norms .∞ λ,S and .λ,S ∞ morphisms in the following sense: if H 1 (M, R) vanishes, then the norm .∞ λ,S is called the L -version of (1,∞)

(1,∞)

Hofer’s norm, and the norm .λ,S is called the L(1,∞) -version of Hofer’s norm whenever ν B is the oscillation norm [9]. When ν B is the oscillation norm, Polterovich [14] proved that the two versions of Hofer’s (1,∞) norms are equal. That is, the two norms .∞ are equal whenever H 1 (M, R) = {0}, and ν B λ,S and .λ,S is the oscillation norm (see Lemma 5.1.C found in [14]). However, the following main result shows that the (1,∞) two norms .∞ continue to coincide regardless of whether H 1 (M, R) is trivial or not, i.e., the λ,S and .λ,S Hofer-like geometry is independent to the choice of the Hofer-like norm whenever ν B is equivalent to the oscillation norm. Theorem 3.8. Let (M, ω) be a closed symplectic manifold. For each fixed linear section S, and for each positive λ, we have φ∞ λ,S = φλ,S , (1,∞)

(3.22)

for each φ ∈ Symp0 (M, ω). When λ = 1, and the linear section S is chosen so that the S-decomposition is the Hodge decomposition, then Theorem 3.8 was announced in [5]. It yields the symplectic analogues of Lemma 5.1.C–[14]. The main idea behind the proof of Theorem 3.8 is the deformation of any non-trivial symplectic isotopy Φ into a regular symplectic isotopy Ψ so that the L∞ -Hofer-like length of Ψ is bounded from above by the L(1,∞) -Hofer-like length of Φ up to an additive positive , and both paths Φ and Ψ still have the same extremities. This is supported by the following generalized version of a result found in [5]. Lemma 3.9 (Fundamental lemma). Fix a linear section S, and a positive λ. Let Φ be a symplectic isotopy. For any positive real number , there exists a regular symplectic isotopy Ψ with the same extremities than Φ such that ∞ lλ,S (Ψ) < lλ,S (Φ) + . (1,∞)

(3.23)

Proof of Theorem 3.8. It is clear from the definition of the Hofer-like energies that φλ,S ≤ φ∞ λ,S . For the converse, consider φ ∈ Symp0 (M, ω), and derive from the characterization of the infimum that for each positive real number , there exists a symplectic isotopy Φ with time-one map equals to φ such (1,∞) that lλ,S (Φ ) ≤ eλ,S (φ) + . By Lemma 3.9, there exists another symplectic isotopy Ψ with the same (1,∞)

∞ ∞ extremities that of Φ such that lλ,S (Ψ ) < lλ,S (Φ ) + . This yields e∞ λ,S (φ) ≤ lλ,S (Ψ ) < eλ,S (φ) + 2 , i.e. (1,∞)

e∞ λ,S (φ) < eλ,S (φ) + 2 .

(3.24)

In a similar way we use again Lemma 3.9 to deduce that for each positive real number , we have −1 e∞ ) < eλ,S (φ−1 ) + 2 . λ,S (φ

Therefore, adding (3.24) and (3.25) member to member leads to

(3.25)

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∞ −1 −1 φ∞ ) + e∞ ) + 4 )/2 = φλ,S λ,S = (eλ,S (φ λ,S (φ))/2 < (eλ,S (φ) + eλ,S (φ

(1,∞)

+ 2

(3.26)

Since (3.26) holds for any arbitrary positive , we conclude that φ∞ λ,S ≤ φλ,S . This completes the proof. 2 (1,∞)

3.2. Proof of Lemma 3.9 In this subsection, we will always denote by r(g) the injectivity radius of the Riemannian metric g. We shall need the following fact. Lemma 3.10. Let (M, g) be a closed oriented Riemannian manifold, and let H = (Ht ) be a smooth family of closed 1-forms. The following facts hold: (1) If Ψ = {ψt } is an isotopy, and ξj : [0, 1] → [0, 1], j = 1, 2 are two smooth monotonic functions that fix 0, then there exists a constant B2 which depends only on H and Ψ such that 1 osc(Δt (H, Ψξ1 ) − Δt (H, Ψξ2 ))dt ≤ B2 ξ1 − ξ2 ham .

(3.27)

0

¯ Ψ) ≤ r(g)/2, then (2) If Φ = {φt } and Ψ = {ψt } are two isotopies such that d(Φ, ¯ Ψ). max osc(Δt (H, Φ) − Δt (H, Ψ)) ≤ 4 max |Ht |0 d(Φ,

t∈[0,1]

(3.28)

t∈[0,1]

˙ ξj (t) = Proof. For each j = 1, 2, differentiating the reparameterized path Ψξj in the variable t yields Ψ ˙ξj (t)ψ˙ ξ (t) , for all t ∈ [0, 1]. Now, compute j t

t ξ˙j (s)Ht (ψ˙ ξj (s) ) ◦ ψξj (s) ds,

˙ ξj (s)) ◦ Ψξj (s)ds = Ht (Ψ

Δt (H, Ψξj ) = 0

(3.29)

0

for each t, and use a suitable variable change to see that the right hand side in (3.29) can be written as t

ξj (t)

ξ˙j (s)Ht (ψ˙ ξj (s) ) ◦ ψξj (s) ds =

Ht (ψ˙ u ) ◦ ψu du.

0

(3.30)

0

Relation (3.29) together with (3.30) yields 1

1 osc(Δt (H, Ψξ1 ) − Δt (H, Ψξ2 ))dt =

0

⎛ ⎜ osc ⎝

0

max{ξ 1 (t),ξ2 (t)}

⎞

⎟ Ht (ψ˙ u ) ◦ ψu du⎠ dt,

min{ξ1 (t),ξ2 (t)}

≤ 2 sup |Ht (ψ˙ s )(x)|ξ1 − ξ2 C 0 , s,t,x

≤ 2 sup |Ht (ψ˙ s )(x)|ξ1 − ξ2 ham .

(3.31)

s,t,x

Therefore, the last inequality in (3.31) suggests that the desired B2 can be chosen as B2 := 2 sups,t,x |Ht (ψ˙ s )(x)| < ∞. For (2), as in Subsection 2.5, fix a point m ∈ M , and for all x in M , pick any curve γx from m to x, and therefore define a smooth function μ ¯t by

S. Tchuiaga / Differential Geometry and its Applications 61 (2018) 170–196

 μ ¯t (x) :=

(φ∗t (Ht ) − ψt∗ (Ht )) ,

189

(3.32)

γx

for each t. Now, let y0 be any point of M that realizes the supremum of the function x → |¯ μt (x)|, i.e., 

(φ∗t (Ht )

sup |¯ μt (x)| = | x

−

ψt∗ (Ht )) |





=|

Ht −

φt ◦γy0

γy0

Ht |.

(3.33)

ψt ◦γy0

  In the what follows, for each fixed t, we shall express the quantity ( φt ◦γy Ht − ψt ◦γy Ht ) as a difference 0 0 between two integrals of the closed 1-form Ht over two minimizing geodesics, and next we shall combine it with a well-known result from Riemannian geometry to achieve the proof. For this purpose, using the ¯ Ψ) ≤ r(g)/2, we derive that for each fixed t ∈ [0, 1], and for all z ∈ M , topological assumption d(Φ, the points φt (z) and ψt (z) can be connected through a minimizing geodesic χz . This induces a homotopy H t : [0, 1] × [0, 1] → M between the curves s → (φt ◦ γy0 )(s) and s → (ψt ◦ γy0 )(s), i.e., we have H t (0, s) = φt (γy0 (s)) and H t (1, s) = ψt (γy0 (s)) for all s ∈ [0, 1]. We may define H t (u, s) to be the unique minimizing geodesic χ(γy0 (s)) that connects φt (γy0 (s)) to ψt (γy0 (s)) for all s ∈ [0, 1]. On the other hand, put t := {H t (u, s) | (u, s) ∈ [0, 1] × [0, 1]}, and derive from Stokes’ theorem that

 ∂t

Ht = 0, where ∂t represents the boundary of the set t , i.e., 





Ht − φt ◦γy0

(3.34)

Ht =

ψt ◦γy0

 Ht −

χy0

Ht ,

(3.35)

χm

  for each t. Observe that in (3.35), each of the integrals χy Ht and χm Ht is bounded from above by 0 ¯ Ψ) because the speed is constant and equals to the distance between the end points. Finally, we |Ht |0 d(Φ, see that inserting (3.35) in (3.33) leads to ¯ Ψ). max osc(Δt (H, Φ) − Δt (H, Ψ)) ≤ 4 max |Ht |0 d(Φ,

t∈[0,1]

t∈[0,1]

(3.36)

This completes the proof. 2 Proof of Lemma 3.9. This proof is based on the general regularization method introduced in Subsections 2.10, and Lemma 3.10. Let Φ be a symplectic isotopy whose S-generator is (U, H), and let be a positive real number. Consider Ξ to be the isotopy obtained by regularizing the isotopy Φ as explained in Subsection 2.10. It follows from Subsection 2.10 that the isotopy Ξ is generated by the element (V, K) defined in (2.59) so that (1,∞) lλ,S (Ξ)

1 (osc(Vt ) + λKt L2 )dt ≤

=

(1,∞) lλ,S (Φ)

1 +

0

1 osc(rt )dt +

0

osc(Δt (H, φ(r,0) ))dt, 0

1 where φ(r,0) is a Hamiltonian loop such that 0 osc(rt )dt < /2 (see Subsection 2.10). Since Polterovich’s arguments provided in Subsection 2.10 state that the path φ(r,0) is arbitrarily close to the constant path 1 identity (in the C ∞ -topology), we derive from Lemma 3.10 that one can assume 0 osc(Δt (H, φ(r,0) ))dt <

/2. Hence, (1,∞)

(1,∞)

lλ,S (Ξ) < lλ,S (Φ) + .

(3.37)

190

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Now, we use the isotopy Ξ to define a curve ζ as in Subsection 2.10 (see (2.63)), and we let Ξζ to be the path obtained by a reparameterization of Ξ via curve ζ. Next, consider Ωs := ζ  (s)(osc(Vζ(s) ) + λKζ(s) L2 ), (1,∞)

for each s, and derive from (2.64) that Ωs = lλ,S (Ξ), i.e. ∞ lλ,S (Ξζ ) = max Ωs = l(1,∞) (Ξ).

(3.38)

s

Relations (3.37) and (3.38) imply that ∞ lλ,S (Ξζ ) < lλ,S (Φ) + . (1,∞)

Therefore, to complete the proof, it suffices to take Ψ = Ξζ .

2

∞ (.), where lλ,S (.) is invariant under repaRemark 3.11. For each fixed section S, we have lλ,S (.) ≤ lλ,S ∞ rameterization, while lλ,S (.) is far from being invariant. Actually, it is clear from the proof of Lemma 3.9 that any regular symplectic path Φ can be reparameterized to obtain another path Ψ with the same extremities (1,∞) ∞ than Φ so that lλ,S (Ψ) = lλ,S (Ψ). (1,∞)

(1,∞)

Remark 3.12. Observe that: If ν B has the following properties: (1) ν B (ρ∗ (df )) ≤ ν B (df ), for all smooth function f , and for all ρ ∈ Symp0 (M, ω), and (2) there exists a positive finite constant θ(S, λ) such that ν B (dΔt (H, Φ)) ≤ θ(S, λ)osc(Δt (H, Φ)), for all t, for each H ∈ H1 (M, S), and for each Hamiltonian isotopy Φ sufficiently closed to the constant path identity in the C ∞ -sense, then Theorem 3.8 will continue to hold with respect to the norm ν B on B 1 (M ). 4. Some auxiliary results The goal of this section is to elaborate various symplectic analogues of some approximation lemmas found in [13]. In what follows, we shall assume that S is given and λ is a fixed non-negative constant. Definition 4.1. The L(1,∞) -version of Hofer-like topology on the group Iso(M, ω) is the topology induced by the following metric: 1 Dλ,S ((U, H), (V, K)) =

D0λ,S ((U, H), (V, K)) + D0λ,S ((U, H), (V, K)) , 2

(4.1)

where 1 D0λ,S ((U, H), (V, K)) = 0

We will need the following lemma.



 ν B (dUt − dVt ) + λHt − Kt L2 dt.

(4.2)

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191

Lemma 4.2 (Reparameterization lemma). Let (M, g) be a closed oriented Riemannian manifold and let > 0. Let H ∈ PH1 (M, S), and let Φ = {φt } ∈ Iso(M, ω). If ξj : [0, 1] → [0, 1], j = 1, 2 are two smooth functions such that ξ1 is monotonic, then there exists a constant B1 ( ) which depends on H, Φ, the metric g and such that 1 osc(Δt (Hξ1 , Φ) − Δt (Hξ2 , Φ))dt ≤ B1 ( )ξ1 − ξ2 ham . 0

Proof. From the equality, Δ(Hξ2 , Φ) − Δ(Hξ1 , Φ) = Δt (Hξ2 − Hξ1 , Φ), we derive by the mean of (2.20) that 1

1 Htξ1 − Htξ2 L2 dt,

osc(Δt (H , Φ) − Δt (H , Φ))dt ≤ 2(1 + )diam(M )(1 + sup |Dφt (γy0 (s))|)K(g) ξ2

ξ1

t,s

0

0

(4.3) where Dφt is the tangent map of φt , and γy0 is a certain curve (see Subsection 2.5). Since we always have Htξ1 − Htξ2 L2 ≤ ξ˙1 (t)Hξ1 (t) − ξ˙1 (t)Hξ2 (t) L2 + ξ˙1 (t)Hξ2 (t) − ξ˙2 (t)Hξ2 (t) L2 , we use the Lipschitz nature of the map t → Ht to derive the existence of a constant c0 > 0 (which depends on H) such that, Htξ1 − Htξ2 L2 ≤ maxHt L2 |ξ˙1 (t) − ξ˙2 (t)| + c0 ξ1 − ξ2 C 0 |ξ˙1 (t)|, t

(4.4)

for each t. Therefore, integrating (4.4) between 0 and 1 with respect to t gives 1

1 Htξ1

−

Htξ2 L2 dt

|ξ˙1 (t) − ξ˙2 (t)|dt + c0 ξ1 − ξ2 C 0

≤ maxHt L2 t

0

0

≤ 2 max(c0 , maxHt L2 )ξ1 − ξ2 ham . t

Finally, (4.3) together with (4.4) yields 1 osc(Δt (Hξ2 , Φ) − Δt (Hξ1 , Φ))dt ≤ B1 ξ1 − ξ2 ham , 0

where B1 ( ) = 4(1 + )K(g)diam(M ) max(c0 , maxHt L2 )(1 + sup |Dφt (γy0 (s))|) < ∞. t

This completes the proof. 2

t,s

(4.5)

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192

Lemma 4.3. If (U, H) ∈ T(M, ω, S), and ξj : [0, 1] → [0, 1], j = 1, 2 are two smooth monotonic functions, then there exists a constant C(S, λ) which depends on (U, H), S and λ such that, 1 Dλ,S ((U, H)ξ1 , (U, H)ξ2 ) ≤ C(S, λ)ξ1 − ξ2 ham .

We shall give a complete proof of Lemma 4.3 later on. The following result is an immediate consequence of Lemma 4.3. 1 , and ξl : [0, 1] → [0, 1], l = 1, 2 be Lemma 4.4. Let (U i , Hi ) ⊂ T(M, ω, S) be a Cauchy sequence in Dλ,S two monotonic smooth functions. Given > 0, there exists a positive constant δ = δ((U i , Hi )), and a larger positive integer j0 = j0 ((U i , Hi )) such that if the inequality ξ1 − ξ2 ham < δ holds, then 1 Dλ,S ((U i , Hi )ξ1 , (U i , Hi )ξ2 ) < ,

for all i ≥ j0 . 1 1 , one can choose an integer j0 large enough such that Dλ,S ((U i , Hi ), Proof. Since (U i , Hi ) is Cauchy in Dλ,S j0 j0 j0 j0 (U , H )) < /3 for all i ≥ j0 . Assume this is done. Now we apply Lemma 4.3 with (U , H ), ξ1 , and ξ2 to derive that there exists a constant C(S, λ) which depends on (U j0 , Hj0 ) such that, 1 Dλ,S ((U j0 , Hj0 )ξ1 , (U j0 , Hj0 )ξ2 ) ≤ C(S, λ)ξ1 − ξ2 ham .

(4.6)

Taking δ = /3C(S, λ), we compute 1 1 Dλ,S ((U i , Hi )ξ1 , (U i , Hi )ξ2 ) ≤ Dλ,S ((U i , Hi )ξ1 , (U j0 , Hj0 )ξ1 ) 1 1 ((U j0 , Hj0 )ξ1 , (U j0 , Hj0 )ξ2 ) + Dλ,S ((U j0 , Hj0 )ξ2 , (U i , Hi )ξ2 ) + Dλ,S

≤ /3 + /3 + /3, as long as ξ1 − ξ2 ham < δ, and i ≥ j0 . This completes the proof. 2 The following result yields various symplectic analogues of a slight variation of the L(1,∞)- approximation lemma found in [13]. It implies that a symplectic isotopy Φ can be approximated in each of the metrics 1 Dλ,S and d¯ by a boundary flat symplectic path with the same extremities than Φ. Lemma 4.5. Let Φ be a symplectic isotopy generated by (U, H), and let be a positive real number. Then, there exists a boundary flat symplectic isotopy Ψ = ψ(V,K) with the same extremities than Φ such that 1 ¯ Φ) < . Dλ,S ((U, H), (V, K)) < , and d(Ψ, Proof. Let be a positive real number, and consider ξ : [0, 1] → [0, 1] to be any smooth and increasing function which is the trivial map 0 on [0, δ] and the constant map 1 on [1 − δ, 1] where 0 < δ < 1/13. Therefore, define (V, K) to be the element (U, H)ξ obtained by re-parameterizing (U, H) via the curve ξ as explained in Subsection 2.7. It follows from the definition of the curve ξ that the symplectic isotopy Ψ generated by (V, K) is boundary flat and it has the same extremities than Φ. Applying Lemma 4.3 with ξ1 = id[0,1] and ξ2 = ξ, we derive from the above arguments that 1 Dλ,S ((U, H), (V, K)) ≤ C(S, λ)ξ − id[0,1] ham ,

where C(S, λ) is the constant in Lemma 4.3 which only depends on (U, H). On the other hand, since the maps (t, x) → φt(U,H) (x) and (t, x) → φ−t (U,H) (x) are Lipschitz continuous, it turns out that there exists a constant l0 > 0 which depends only on (U, H) such that

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193

¯ (U,H) , ψ(V,K) ) ≤ l0 ξ − id[0,1] C 0 < l0 ξ − id[0,1] ham . d(φ Finally, to conclude, it suffices to choose the function ξ so that ξ − id[0,1] ham ≤ min{ /C(S, λ); /l0 ; }. This completes the proof. 2 Remark 4.6. In the proof of Lemma 4.5 the choice of δ less than 1/13 has no particular meaning than to satisfy the condition [0, δ] ∩ [1 − δ, 1] = ∅. Proof of Lemma 4.3. Let Φ be a symplectic isotopy generated by (U, H). • Step (1). Consider the normalized function V = U ξ1 − U ξ2 , and compute |Vt | = |ξ˙1 (t)Uξ1 (t) − ξ˙2 (t)Uξ2 (t) | ≤ |ξ˙1 (t)||Uξ1 (t) − Uξ2 (t) | + |ξ˙1 (t) − ξ˙2 (t)||Uξ2 (t) |,

(4.7)

for each t. Since the map (t, x) → Ut (x) is smooth on a compact set, it is Lipschitz, i.e., there exists a positive constant k0 which depends on U such that maxx∈M |Ut (x) − Us (x)| ≤ k0 |t − s| for all t, s ∈ [0, 1]. The latter Lipschitz inequality together with (4.7) yields 0 ≤ max Vt (x) ≤ k0 |ξ˙1 (t)||ξ1 (t) − ξ2 (t)| + max{max(Uξ2 (t) (x)), − min(Uξ2 (t) (x))}|ξ˙1 (t) − ξ˙2 (t)|. x

x∈M

x

(4.8)

Similarly, one can check that 0 ≤ − min Vt (x) ≤ k0 |ξ˙1 (t)||ξ1 (t) − ξ2 (t)| + max{max(Uξ2 (t) (x)), − min(Uξ2 (t) (x))}|ξ˙1 (t) − ξ˙2 (t)|. x

x∈M

x

(4.9)

Adding (4.8) and (4.9) member to member, and integrating the resulting inequality in the variable t gives 1

1 |ξ˙1 (t) − ξ˙2 (t)|dt.

osc(Vt )dt ≤ 2k0 max |ξ1 (t) − ξ2 (t)| + 2 max(osc(Ut )) t

t

0

(4.10)

0

• Step (2). Put K = Hξ1 − Hξ2 , and compute Kt L2 = ξ˙1 (t)Hξ1 (t) − ξ˙2 (t)Hξ2 (t) L2 ≤ Hξ1 (t) − Hξ2 (t) L2 |ξ˙1 (t)| + |ξ˙1 (t) − ξ˙2 (t)|Hξ2 (t) L2 ,

(4.11)

for each t. The Lipschitz nature of the map t → Ht implies that there exists a positive constant c0 such that Ht − Hs L2 ≤ c0 |t − s| for all s, t ∈ [0, 1]. This Lipschitz inequality together with (4.11) implies Kt L2 ≤ c0 |ξ1 (t) − ξ2 (t)||ξ˙1 (t)| + |ξ˙1 (t) − ξ˙2 (t)|Hξ2 (t) L2 .

(4.12)

Integrating (4.12) in the variable t yields, 1

1 |ξ˙1 (t) − ξ˙2 (t)|dt.

Kt L2 dt ≤ c0 max |ξ1 (t) − ξ2 (t)| + maxHt L2 t

t

0

(4.13)

0

Adding (4.10) and (4.13) member to member gives D0λ,S ((U, H)ξ1 , (U, H)ξ2 ) ≤ B3 (λ)ξ1 − ξ2 ham , where B3 (S, λ) = 2 max{2k0 + λc0 , λ maxHt L2 + 2 max osc(Ut )}. t

t

(4.14)

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194

• Step (3). On the other hand, for each j = 1, 2, we compute  ξj , Φξj ), −Hξj ), (U, H)ξj = (−U ξj ◦ Φξj − Δ(H with  t (Hξj , Φξj ) = ξ˙j (t)Δ  ξ (t) (H, Φ), Δ j for all t, and derive from the definition of D0λ,S that 1 D0λ,S ((U, H)ξ1 , (U, H)ξ2 )

≤

  t (Hξ1 , Φξ1 ) − Δ  t (Hξ2 , Φξ2 ) dt osc Δ

0

1 osc(Utξ2 − Utξ1 ) + |Htξ1 − Htξ2 |dt

+ 0

1 +

 osc Utξ1 ◦ Φξ2 (t) − Utξ1 ◦ Φξ1 (t) dt,

0

1 ≤

  t (Hξ1 , Φξ1 ) − Δ  t (Hξ2 , Φξ2 ) dt osc Δ

0

+ D0λ,S ((U, H)ξ1 , (U, H)ξ2 ) + k1 ξ1 − ξ2 ham .

(4.15)

Note that in (4.15), to obtain the quantity k1 ξ1 − ξ2 ham we have used the Lipschitz natures of the maps (x, t) → Ut (x), (x, t) → Φ−1 (t)(x) and (x, t) → Φ(t)(x) to derive the existence of a positive constant k1 which depends on Φ such that 1

  osc ξ˙1 (t)Uξ1 (t) ◦ Φξ1 (t) − ξ˙1 (t)Uξ1 (t) ◦ Φξ2 (t) dt ≤ k1 ξ1 − ξ2 ham .

0

Since the map (x, t) → Δt (H, Φ)(x) is smooth and M is compact, we derive as in step (1) that there exists a Lipschitz constant k2 which depends on Φ such that 1

 ξ (t) (H, Φ) − ξ˙2 (t)Δ  ξ (t) (H, Φ))dt ≤ 2k2 max |ξ1 (t) − ξ2 (t)| osc(ξ˙1 (t)Δ 1 2 t

0

1 |ξ˙1 (t) − ξ˙2 (t)|dt.

+ max(osc(Δt (H, Φ))) t

(4.16)

0

Observe that maxt (osc(Δt (H, Φ))) ≤ B2 (H, Φ), where B2 is the constant in Lemma 3.10, and derive from (4.16) that 1 0

 ξ (t) (H, Φ) − ξ˙2 (t)Δ  ξ (t) (H, Φ))dt ≤ 2 max{2k2 , B2 }ξ1 − ξ2 ham . osc(ξ˙1 (t)Δ 1 2

(4.17)

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195

• Step (4). Since,  1 Dλ,S ((U, H)ξ1 , (U, H)ξ2 ) = D0λ,S ((U, H)ξ1 , (U, H)ξ2 ) + D0λ,S ((U, H)ξ1 , (U, H)ξ2 ) /2, it follows from step (2) and step (3) that 1 Dλ,S ((U, H)ξ1 , (U, H)ξ2 ) ≤ C(S, λ)ξ1 − ξ2 ham ,

where C(S, λ) := B3 (S, λ) + 0.5k1 + max{2k2 , B2 } < ∞. This completes the proof. 2 Example 4.7. Consider the torus T2l with coordinates (θ1 , . . . , θ2l ) and equip it with the flat Riemannian metric g0 . Assume that λ = 1. Consider S : H 1 (T2l , R) → Z 1 (T2l ), to be a linear section of the natural projection P : Z 1 (T2l ) → H 1 (T2l , R), such that Z 1 (M ) = H1 (M, S) ⊕S B1 (M ) stands for the Hodge decomposition. Note that all the 1-forms dθi , i = 1, . . . , 2l are harmonic. Take the 1-forms dθi for i = 1, . . . , 2l l as basis for the space of harmonic 1-forms and consider the symplectic form ω = i=1 dθi ∧ dθi+l . Given v = (a1 , . . . , al , b1 , . . . , bl ) ∈ R2l , the translation x → x + v on R2l induces a rotation Rv on T2l , which is a symplectic diffeomorphism. Therefore, the smooth mapping {Rvt } : t → Rtv defines a symplectic l isotopy generated by (0, H) with H = i=1 (ai dθi+l − bi dθi ). For each integer j > 0, consider the function t(j + 1) j f sin( ), and denote by {Rvj } the smooth path t → Rvj (t) where vj (t) = fj : [0, 1] → [0, 1], t → 1+j j f

d¯

fj (t)v. Since the sequence (fj ) converges uniformly to f : t → sin(t), we derive that {Rvj } − → {Rvf }. For each t, we have t Δt (H, {Rvf })

f˙(s)H(R˙ vf (s) )

=

◦

Rvf (s) ds

f (t)  = H(R˙ vu ) ◦ Rvu du = Δf (t) (H, {Rvt }),

0

0

t

fj (t)

f˙j (s)H(R˙ vfj (s) ) ◦ Rvfj (s) ds =

Δt (H, {Rvfj }) =

(4.18)

0

H(R˙ vu ) ◦ Rvu du = Δfj (t) (H, {Rvt }),

(4.19)

0

for each j. Therefore, we derive from Example 3.1 that for each t, we have Δt (H, {Rvf }) = 0, and f Δt (H,  {Rvj }) = 0 because {Rvt } is the symplectic flow generated by the harmonic 1-form H. In particular, 1 fj f osc Δt (H, {Rv }) − Δt (H, {Rv }) dt → 0, j → ∞. The latter convergence can follow from Lemma 3.10 0 d¯

f

since {Rvj } − → {Rvf }. On the other hand, using the product rule in the group T(T2l , ω, S), we check can that  fj , {Rf }) − Δ(H  f , {Rf }), Hfj − Hf ) (0, Hf ) 1 (0, Hfj ) = (Δ(H v v   {Rvf }) − f  Δ(H, {Rvf }), Hfj − Hf ) = (fj Δ(H, = (0, fj H − f  H), because Δ(H, {Rvf }) = 0. Thus, we derive from the above equalities that

l

(1,∞)

({Rvf }−1

1 ◦

{Rvfj })

HL2 |fj (t) − f (t)|dt ≤ HL2 fj − f C 0 → 0, j → ∞.

= 0

Therefore, Corollary 3.7 tells us that the isotopy {Rvf } is symplectic. This is true from the definition of the isotopy {Rvf }. 2

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Remark 4.8. It is not too hard to see that: If the norm ν B has the following properties: (1) ν B (ρ∗ (df )) ≤ ν B (df ), for all smooth function f , and all ρ ∈ Symp0 (M, ω), and (2) there exists a positive finite constant θ(S, λ) such that ν B (df ) ≤ θ(S, λ)osc(f ), for all smooth function f . Then, the results in the last Section of this paper will continuous to hold with respect to the norm ν B on B 1 (M ). Acknowledgements I would like to thank the referees for carefully reading earlier versions of this paper and for raising questions. References [1] A. Banyaga, Sur la structure de difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978) 174–2227. [2] A. Banyaga, A Hofer-like metric on the group of symplectic diffeomorphisms, in: Symplectic Topology and Measure Preserving Dynamical Systems, 2010, pp. 1–23. [3] A. Banyaga, On fixed points of symplectic maps, Invent. Math. 56 (1980) 215–229. [4] A. Banyaga, E. Hurtubise, P. Spaeth, The symplectic displacement energy, J. Symplectic Geom. 16 (2018) 69–83. [5] A. Banyaga, S. Tchuiaga, The group of strong symplectic homeomorphisms in L∞ -metric, Adv. Geom. 14 (2014) 523–539. [6] A. Banyaga, S. Tchuiaga, Uniqueness of generators of strong symplectic isotopies, IMHOTEP-Math. J. 1 (2016) 7–23. [7] G. Bus, R. Leclercq, Pseudo-distance on symplectomorphisms groups and application to the flux theory, Math. Z. 272 (2012) 1001–1022. [8] M. Hirsch, Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer Verlag, New York–Heidelberg, 1976, corrected reprint (1994). [9] H. Hofer, On the topological properties of symplectic maps, Proc. R. Soc. Edinb. 115A (1990) 25–38. [10] H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts, Birkhäuser Verlag, 1994. [11] F. Lalonde, D. McDuff, The geometry of symplectic energy, Ann. Math. 141 (1995) 711–727. [12] D. McDuff, D. Salamon, Introduction to Symplectic Topology, second ed., Oxford Mathematical Monographs, Oxford University Press, New York, 1998. [13] Y-G. Oh, S. Müller, The group of Hamiltonian homeomorphisms and C 0 -symplectic topology, J. Symplectic Geom. 5 (2007) 167–225. [14] L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphism, Lecture in Mathematics ETH Zürich, Birkhäuser Verlag, Basel-Boston, 2001. [15] S. Tchuiaga, Some structures of the group of strong symplectic homeomorphisms, Glob. J. Adv. Res. Class. Mod. Geom. 2 (1) (2013) 36–49. [16] S. Tchuiaga, Hofer-like geometry and flux theory, arXiv:1609.07925, 2016. [17] F. Warner, Foundation of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, vol. 94, SpringerVerlag, New York, 1983.