CUBO A Mathematical Journal Vol.19, No¯ 02, (49–71). June 2017 On topological symplectic dynamical systems S. Tchuiaga1, M. Koivogui2, F. Balibuno3 and V. Mbazumutima3 1Department of Mathematics The University of Buea, South West Region, Cameroon. tchuiagas@gmail.com 2Ecole Supérieure Africaine des Technologies de l’Information et de Communication, Côte d’ Ivoire. moussa.koivogui@esatic.ci 3Institut de Mathématiques et des Sciences Physiques Bénin. balibuno.lugando@imsp-uac.org, mbazumutima.vianney@aims-cameroon.org ABSTRACT This paper contributes to the study of topological symplectic dynamical systems, and hence to the extension of smooth symplectic dynamical systems. Using the positivity result of symplectic displacement energy [4], we prove that any generator of a strong symplectic isotopy uniquely determine the latter. This yields a symplectic analogue of a result proved by Oh [12], and the converse of the main theorem found in [6]. Also, tools for defining and for studying the topological symplectic dynamical systems are provided: We construct a right-invariant metric on the group of strong symplectic homeomorphisms whose restriction to the group of all Hamiltonian homeomorphism is equivalent to Oh’s metric [12], define the topological analogues of the usual symplectic displacement energy for non-empty open sets, and we prove that the latter is positive. Several open conjectures are elaborated. 50 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) RESUMEN Este art́ıculo contribuye al estudio de los sistemas dinámicos simplécticos topológicos, y por tanto a la extensión de los sistemas dinámicos simplécticos suaves. Usando el resultado de la positividad de la enerǵıa de desplazamiento simpléctico [4], demostramos que cualquier generador de una isotoṕıa simpléctica determina esta última. Esto entrega un análogo simpléctico de un resultado demostrado por Oh [12], y el inverso del teorema principal encontrado en [6]. También entregamos herramientas para definir y estudiar los sistemas dinámicos simplécticos topológicos: construimos una métrica invariante por derecha en el grupo de homeomorfismos fuertemente simplécticos cuya restricción al grupo de homeomorfismos Hamiltonianos es equivalente a la métrica de Oh [12], definimos los análogos topológicos de la enerǵıa de desplazamiento simpléctico usual para conjuntos no-vaćıos, y demostramos que esta última es positiva. Planteamos varios problemas abiertos. Keywords and Phrases: Isotopies, Diffeomorphisms, Homeomorphisms, Displacement energy, Hofer-like norms, Mass flow, Riemannian metric, Lefschetz type manifolds, Flux geometry. 2010 AMS Mathematics Subject Classification: 53D05, 53D35, 57R52, 53C21. CUBO 19, 2 (2017) On topological symplectic dynamical systems 51 1 Introduction Gromov [10] showed that on a symplectic manifold, the C0−closure of the group of symplectic diffeomorphisms in the group of diffeomorphisms is either the group of symplectic diffeomorphisms itself, or the group of volume preserving diffeomorphisms. Eliashberg [9] proved that the symplec- tic nature of a sequence of symplectic diffeomorphisms survives topological limits. This result is known as the ”celebrated rigidity” result of Eliashberg, which motivated various remarkable stud- ies of continuum phenomena in the field of symplectic geometry. Especially, based on this rigidity result, Oh-Müller [13] defined the group of symplectic homeomorphisms as the C0−closure of the group of symplectomorphisms in the group of homeomorphisms. They also defined both versions of C0−Hamiltonian topologies on the space of Hamiltonians paths and used them to define the group of Hamiltonian homeomorphisms. This group is at the center of the study of topological Hamiltonian dynamical systems (see Viterbo [19], Buhovsky and Seyfaddini [8, 12]). More recently, motivated again by the celebrated rigidity result of Eliashberg, Banyaga [2, 3] defined two contexts of symplectic topologies on the space of symplectic isotopies that generalize the C0−Hamiltonian topologies. These topology had been used to define a new class of symplectic homeomorphisms named the ”group strong symplectic homeomorphisms” (see Banyaga [3]), which had been studied in Banyaga-Tchuiaga [6, 5, 14]. This group could be the right topological analogue of the identity component in the group of symplectomorphisms. However, for that to be possible, we have to define, and then study what will be the equivalents (or analogues) of some well known smooth symplectic objects in the world (or context) of strong symplectic homeomorphisms. Therefore, one purpose of this paper is to point out further studies of generators for strong sym- plectic isotopies [6], and then use them to construct a framework in which the flux homomorphism and the Hofer-like geometry can be extended to some category of continuous maps (see [18]). We organize the present paper as follows: In Sections 2 and Section 3, we recall some funda- mental tools needed in the definition of strong symplectic homeomorphisms: The description of symplectic isotopies that was introduced in [5], and the displacement energy. Section 4 deals with the definitions of the C0−compact open topology, the origin of strong symplectic homeomorphisms, and the definition of strong symplectic isotopies with their generators. These tools are used to show a bijective correspondence between the group of strong symplectic isotopies and the group of their generators (Theorem 4.4). The Hamiltonian version of this result is well known. This section also includes Lemma 4.6 which shows that any strong symplectic isotopy which is a 1−parameter group decomposes as composition of a smooth harmonic flow and a continuous Hamiltonian flow in the sense of Oh-Müller. In Section 5, we use the results of Section 4 to introduce a topological version of Hofer-like ge- ometry: We construct a topological counterpart of the Hofer-like metric for strong symplectic homeomorphisms, and we prove that its restriction to the group of Hamiltonian homeomorphisms is equivalent to Oh’s metric [12]. Therefore, the definition of a topological analogue of the usual symplectic displacement for non-empty sets is given, and we prove that it is positive. Finally, in 52 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) Section 6, we elaborate some conjectures and some examples are given. 2 Preliminaries Let M be an 2n−dimensional manifold of class C∞. A differential 2−form ω on M is called a symplectic form if it is closed and non-degenerate. The nondegeneracy of ω, implies that ωn is a volume form on M. A symplectic manifold is an even dimensional smooth manifold M that admits a symplectic form ω. From now on, we assume that M is an 2n−dimensional closed symplectic manifold with a symplectic form ω. We also equip M with a Riemannian metric g. Note that for technical reasons, we will sometimes assume that the symplectic manifold (M,ω) is of Lefschetz type. That is, the mapping ωn−1 : H1(M, R) → H2n−1(M, R),α !→ α ∧ ωn−1, is an isomorphism. The category of Lefschetz manifolds includes all Kähler manifolds, such has oriented surfaces and even dimensional tori. 2.1 Harmonics 1−forms Let H1(M, R) denote the first de Rham cohomology group (with real coefficients) of M, and let H1(M, g) denote the space of harmonic 1−forms on M with respect to the Riemannian metric g. The set H1(M, g) forms a finite dimensional vector space over R which is isomorphic to H1(M, R), and whose dimension is denoted b1(M), and called the first Betti number of the manifold M [20]. Taking (hi)1≤i≤b1(M) as a basis of the vector space H 1(M, g), we equip H1(M, g) with the Euclidean norm |.| defined as follows: for all H ∈ H1(M, g) with H = b1(M)∑ i=1 λihi, its norm is defined as |H| := b1(M)∑ i=1 |λi|. (2.1) We denote by PH1(M, g) the space of all smooth mappings H : [0, 1] → H1(M, g). 3 On the classical symplectic dynamical systems 3.1 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). CUBO 19, 2 (2017) On topological symplectic dynamical systems 53 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 field φ̇t := dφt dt ◦ φ−1 t is symplectic for each t. In particular, a symplectic isotopy {ψt} is a Hamiltonian isotopy if for each t, the vector field ψ̇t := dψt dt ◦ ψ−1 t is Hamiltonian, i.e. there exists a smooth function F : [0, 1] × M → R, called generating Hamil- tonian such that ι(ψ̇t)ω = dFt, 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 Ftω n = 0 for all t. Let N([0, 1] × M , R) denote the vector space of all smooth normalized Hamiltonians. We denote by Iso(M,ω) the group of all symplectic isotopies of (M,ω), and by Symp0(M,ω), the group of time−1 maps of all symplectic isotopies. 3.2 Description of the classical symplectic isotopies We now recall the description of symplectic isotopies introduced in [5]. Given any symplectic isotopy Φ = {φt}, one derives from Hodge’s theory that the closed 1−form ι(φ̇t)ω decomposes in a unique way as the sum of an exact 1−form dUΦt and a harmonic 1−form H Φ t [20]. Denote by U the normalized Hamiltonian of UΦ = (UΦ t ), and by H the smooth family of harmonic 1−forms HΦ = (HΦ t ). In [5], the Cartesian product N([0, 1] × M, R) × PH1(M, g) is denoted T(M,ω, g), and equipped with a group structure which makes the bijection A : Iso(M,ω) → T(M,ω, g),Φ !→ (U, H) (3.1) a group isomorphism. Under this identification, any symplectic isotopy Φ is denoted by φ(U,H) to mean that A 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 a harmonic isotopy, and a symplectic isotopy φ(U,0), is a Hamiltonian isotopy. 3.3 Group structure on T(M,ω, g) The product rule in T(M,ω, g) is given by, (U, H) ✶ (V, K) = (U + V ◦ φ−1 (U,H) + !∆(K,φ−1 (U,H) ), H + K). (3.2) The inverse of (U, H), say (U, H) is given by (U, H) = (−U ◦ φ(U,H) − !∆(H,φ(U,H)), −H). (3.3) In (3.2) and (3.3) the quantity !∆ is defined as follows: for any symplectic isotopy Ψ = {ψt}, and for any smooth family of closed 1−forms α = (αt), we have !∆t(α,Ψ) = ∆t(α,Ψ) − ∫ M ∆t(α,Ψ)ω n ∫ M ωn , 54 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) where ∆t(α,Ψ) := ∫t 0 αt(ψ̇ s) ◦ ψsds, for all t (see [5]). Also, it is proved in [16, 18] that ∆(α,Ψ) is a 1−cocycle. 3.4 Metric structures on T(M,ω, g) For all (U, H), (V, K) ∈ T(M,ω, g), set D (1,∞) 0 ((U, H), (V, K)) = ∫ 1 0 [|Ht − Kt| + osc(Ut − Vt)] dt, (3.4) and D∞ 0 ((U, H), (V, K)) = max t [|Ht − Kt| + osc(Ut − Vt)] , (3.5) where osc(f) = max x f(x) − min x f(x), for all f ∈ C∞(M, R). Therefore, the L∞−Hofer-like metric and the L(1,∞)−Hofer-like on T(M,ω, g) are defined respectively as follows: D(1,∞)((U, H), (V, K)) = D (1,∞) 0 ((U, H), (V, K)) + D(1,∞) 0 ((U, H), (V, K)) 2 , (3.6) and D∞((U, H), (V, K)) = D∞ 0 ((U, H), (V, K)) + D∞ 0 ((U, H), (V, K)) 2 . (3.7) (see [2, 5]). 3.5 Displacement energy Definition 3.1. ([4]) The symplectic displacement energy eS(A) of a non empty set A ⊂ M is: eS(A) = inf{∥φ∥HL|φ ∈ Symp(M,ω)0,φ(A) ∩ A = ∅}. Theorem 3.2. ([4]) For any non empty open set A ⊂ M, eS(A) is a strict positive number. Note that in the definition of the displacement energy, the quantity ∥.∥HL stands for the usual Hofer-like norm defined in [2]. 4 On topological symplectic dynamical systems 4.1 The C0−topology Let Homeo(M) be the group of all homeomorphisms of M equipped with the C0− compact-open topology. This is the metric topology induced by the following distance d0(f, h) = max(dC0(f, h), dC0(f −1, h−1)), CUBO 19, 2 (2017) On topological symplectic dynamical systems 55 where dC0(f, h) = supx∈M d(h(x), f(x)). 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 metric d̄(λ, µ) = max t∈[0,1] d0(λ(t), µ(t)). 4.2 The origin of strong symplectic isotopies A result found in [15] (Corollary 3.7-[15], or more generally a result found in [16]) states that: Let Φi = {φ t i } be a sequence of symplectic isotopies, Ψ = {ψt} be another symplectic isotopy, and η : t !→ ηt be a family of maps ηt : M → M, such that the sequence Φi converges uniformly to η and l∞(Ψ−1 ◦ Φi) → 0, i → ∞, then η = Ψ. Note that if, Ψ generated by (U, H) and Φi generated by (Ui, Hi), then replacing the condition l∞(Ψ−1 ◦ Φi) → 0, i → ∞, by the condition D∞((U, H), (Ui, Hi)) → 0, i → ∞, does not break the result of Corollary 3.7-[15, 16]. Therefore, we can then ask the following ques- tion: If in Corollary 3.7-[15, 16], the convergence D∞((U, H), (Ui, Hi)) → 0, i → ∞, is replaced by the condition D∞((Ui+1, Hi+1), (Ui, Hi)) → 0, i → ∞, then what can we say about the geometries and the structures of the space of all such paths η? The seek of a possible answer to the above question motivated the following definition: Definition 4.1. ([6]) A continuous map ξ : [0, 1] → Homeo(M) with ξ(0) = idM, is called strong symplectic isotopy if there exists a D∞−Cauchy sequence {(Fi,λi)} ⊂ T(M,ω, g) such that d̄(φ(Fi,λi),ξ) → 0, i → ∞. We denote by PSSympeo(M,ω) the space of all strong symplectic isotopies. It is proved in [6, 14] that PSSympeo(M,ω) is a group. If the manifold is simply connected, then the group PSSympeo(M,ω) reduces to the group of continuous Hamiltonian flows [12]. The set of time−1 maps of all strong symplectic isotopies coincides with the group of all strong symplectic homeo- morphisms, denoted here by SSympeo(M,ω) (see [5, 3]). 56 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) 4.3 Generators of ssympeotopies Let N 0([0, 1]×M , R) denotes the completion of the metric space N([0, 1]×M , R) with respect to the L∞−Hofer norm, and let PH1(M, g)0 denotes the completion of the metric space PH1(M, g) with respect to the uniform sup norm. Consider the map J0(M,ω, g) := N 0([0, 1] × M , R) × PH1(M, g)0, and the inclusion map i0 : T(M,ω, g) → J 0(M,ω, g). This map is uniformly continuous with respect of the topology induced by the metric D∞ on the space T(M,ω, g), and the natural topology of the complete metric space J0(M,ω, g). Now, let L(M,ω, g) denotes the image of T(M,ω, g) under i0, and T(M,ω, g)0 be the closure of L(M,ω, g) inside the complete metric space J0(M,ω, g). That is, T(M,ω, g)0 consists of pairs (U, H) where the mappings (t, x) !→ Ut(x) and t !→ Ht are continuous, and for each t, Ht lies in H1(M, g) such that there exists a D∞−Cauchy sequence (Ui, Hi) ⊂ T(M,ω, g) that converges to (U, H) ∈ J0(M,ω, g). Note that the sequence (Fj,λj) in definition (4.1) converges necessarily in the complete metric space T(M,ω, g)0. The latter limit is called the ”generator” of strong symplectic isotopy (see [6]). We will often write (Fi,λi) L ∞ −−→ (F,λ) to mean that the sequence (Fi,λi) converges to (F,λ) in the space J0(M,ω, g). Definition 4.2. ([6]) The set GSSympeo(M,ω, g) is defined as the space of all the pairs (ξ, (U, H)) where ξ is a strong symplectic isotopy generated (U, H). Group structure on the space GSSympeo(M,ω, g) For all (ξ, (F,λ)), (µ, (V,θ)) ∈ GSSympeo(M,ω, g), their product is given by, (ξ, (F,λ)) ∗ (µ, (V,θ)) = (ξ ◦ µ, (F + V ◦ ξ−1 + ∆0(θ,ξ−1),λ + θ)), and the inverse of the element (ξ, (F,λ)) is given by, (ξ, (F,λ)) = (ξ−1, (−F ◦ ξ − ∆0(λ,ξ), −λ)), with ∆0(θ,ξ−1) := lim L∞ (!∆(θi,φ−1 (Fi,λi) ), (4.1) ∆0(λ,ξ) := lim L∞ (!∆(λi,φ(Fi,λi)), (4.2) where (Fi,λi), and (Vi,θi) are two arbitrary sequences in T(M,ω, g) such that (Fi,λi) L ∞ −−→ (F,λ), φ(Fi,λi) d̄ −→ ξ, and (Vi,θi) L ∞ −−→ (V,θ), φ(Vi,θi) d̄ −→ µ, CUBO 19, 2 (2017) On topological symplectic dynamical systems 57 with !∆t(λi,φ−1(Fi,λi)) the normalized function of ∆t(λ i,φ−1 (Fi,λi) ). This set is known as a topological group with respect to the symplectic topology [6]: The sym- plectic topology on the space GSSympeo(M,ω, g) is defined to be the subspace topology induced by the inclusion of the latter in the complete topological space P(Homeo(M), id) × T(M,ω, g)0. Question (a) Let (Mi,ωi) be two closed symplectic manifolds equipped with two Riemannian metrics gi, for i = 1, 2. If the group T(M1,ω1, g1)0 is isomorphic to the group T(M2,ω2, g2)0, then what can we say about: The manifolds M1 and M2? The symplectic structures ω1 and ω2? The Riemannian structures g1 and g2? The following uniqueness results show that there is a bijective correspondence between the group of strong symplectic isotopies and that of their generators. Theorem 4.3. ([6]) Let (M,ω) be a Lefschetz closed symplectic manifold. Any strong symplectic isotopy determines a unique generator. In the presence of a positive symplectic displacement energy from Banyaga-Hurtubise-Spaeth [4], we point out the following converse of Theorem 4.3, which in the same time gives the symplectic analogue of a result prove by Oh [12]. Theorem 4.4. Any generator corresponds to a unique strong symplectic isotopy, i.e. if (γ, (U, H)), (ξ, (U, H)) ∈ GSSympeo(M,ω, g), then we must have γ = ξ. Proof. Let (γ, (U, H)) and (ξ, (U, H)) be two elements of GSSympeo(M,ω, g). By definition of the group GSSympeo(M,ω, g), there exist two sequences of symplectic isotopies φ(Ui,Hi) and φ(Vi,Ki) such that: φ(Ui,Hi) d̄ −→ ξ, (Ui, Hi) L ∞ −−→ (U, H), and φ(Vi,Ki) d̄ −→ γ, (Vi, Ki) L ∞ −−→ (U, H). Assume that γ ≠ ξ, i.e. there exists s0 ∈]0, 1] such that γ(s0) ≠ ξ(s0). Since the map γ −1(s0)◦ξ(s0) belongs to Homeo(M), then we derive from the identity γ−1(s0) ◦ ξ(s0) ≠ id, that there exists of a closed ball B which is entirely moved by γ−1(s0) ◦ ξ(s0). From the compactness of B, and the uniform convergence of the sequence φ−1 (Ui,Hi) ◦ φ(Vi,Ki) to γ −1 ◦ ξ, we derive that (φ−s0 (Ui,Hi) ◦ φs0 (Vi,Ki) )(B) ∩ (B) = ∅, (4.3) 58 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) for all sufficiently large i. Relation (4.3) implies that, 00, there exists a strong symplectic isotopy η with the same extremities that β ∗r γ, such that l∞(η)0 (we refer to [4] for the definition of symplectic displacement energy). For such a ball B, we set δ = eS(B). The characterization of the infimum tells us that one can find a strong symplectic isotopy γδ (U,H) with γδ (U,H) (1) = h, such that ∥h∥SHL>l∞(γ δ (U,H)) − δ 4 . On the other hand, it follows from the definition of strong symplectic isotopies that there exists a sequence (φ(Fi,λi)) that converges to γ(U,H) with respect to the (C 0 + L∞)−topology. So, we can find a larger integer i0 for which the path φ(Fi 0 ,λi 0 ) is sufficiently close to γ(U,H) [resp. φ −1 (Fi 0 ,λi 0 ) sufficiently close to γ−1 (U,H) ] with respect to the (C0 + L∞)−topology, and so that l∞(γ δ (U,H))>l∞(φ(Fi 0 ,λi 0 )) − δ 4 , where φi0 = φ 1 (Fi 0 ,λi 0 ) displaces B. It follows from the definition of Banyaga’s Hofer-like norm ∥, ∥HL (see [2]) that l∞(φ(Fi 0 ,λi 0 )) ≥ ∥φi0∥HL, i.e. l∞(φ(Fi 0 ,λi 0 )) − 2δ 4 ≥ ∥φi0∥HL − δ 2 . Then, we derive from the definition of symplectic displacement energy eS (see [4]) that ∥φi0∥HL − δ 2 ≥ eS(B) − δ 2 = δ − δ 2 = δ 2 >0. Summarizing the above statements together gives, ∥h∥SHL>l∞(γ δ (U,H)) − δ 4 >l∞(φ(Fi 0 ,λi 0 )) − 2δ 4 ≥ ∥φi0∥HL − δ 2 = δ 2 >0. For (3), let h and f be two strong symplectic homeomorphisms, pick γ ∈ "(h) and β ∈ "(f), and derive from Remark 5.1 that for all ϵ>0, there exists a strong symplectic isotopy η with the same extremities as the right concatenation γ ∗r β such that ē(h ◦ f) ≤ l∞(η)0, such that ∥φ∥SHL ≤ ∥φ∥Oh ≤ κ∥φ∥SHL, for all Hamiltonian homeomorphism φ. This result is motivated in party by a conjecture which can be found in [2]. The latter conjec- ture first was proved by Buss-Leclercq [7], and an alternate proof of the same conjecture is given 66 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) in [17]. This conjecture stated that the restriction of Banyaga’s Hofer-like norm to the group of Hamiltonian diffeomorphisms is equivalent to Hofer’s norm. Proof of Theorem 5.5. By construction, we always have ∥.∥SHL ≤ ∥.∥Oh. To complete the proof, we need to show that there exists a positive finite constant κ such that ∥.∥Oh ≤ κ∥.∥SHL, (5.15) or equivalently, via the sequential criterion, it suffices to prove that any sequence of Hamiltonian homeomorphisms converging to the constant map identity with respect to the norm ∥.∥SHL, con- verges to the constant map identity with respect to Oh’s norm. The proof that we give here heavily relies the ideas that Oh [12] and Banyaga [2] used in the proof of the nondegeneracy of their norms. Let ψi be a sequence of Hamiltonian homeomorphisms that converges to the identity with respect to the norm ∥.∥SHL. For each i, and any ϵ>0 there exists a strong symplectic isotopy ψ(Ui,ϵ,Hi,ϵ) such that ψ1 (Ui,ϵ,Hi,ϵ) = ψi, and l∞(ψ(Ui,ϵ,Hi,ϵ))<∥ψ i ∥SHL + ϵ. (5.16) On the other hand, for a fixed i, there exists a sequence of symplectic isotopies φ(Vi,j,Ki,j) such that ψ(Ui,ϵ,Hi,ϵ) = lim C0+L∞ (φ(Vi,j,Ki,j)). (5.17) In particular, one can find a sufficiently large integer j0 such that φ(Vi,j 0 ,Ki,j 0 ) is sufficiently close to ψ(Ui,ϵ,Hi,ϵ) with respect to the (C 0 + L∞)−topology, and so that l∞(ψ(Ui,ϵ,Hi,ϵ))>l∞(φ(Vi,j 0 ,Ki,j 0 )) − ϵ 4 . (5.18) Since ∥ψi∥SHL → 0, i → ∞, then the Hofer-like length of the isotopy φ(Vi,j 0 ,Ki,j 0 ) can be con- sidered as being sufficiently small for i sufficiently large. This implies that the flux of the path φ(Vi,j 0 ,Ki,j 0 ) can be considered as arbitrarily small for all i sufficiently large. Hence, it follows from Banyaga [2] that, for all i sufficiently large, the time−1 map of φ(Vi,j 0 ,Ki,j 0 ) is a Hamiltonian diffeomorphism. So, we can assume (without breaking the generality) that φ1 (Vi,j 0 ,Ki,j 0 ) is Hamil- tonian for i ≤ j0, and i sufficiently large. Therefore, the above statements together with formula (5.18) imply that l∞(ψ(Ui,ϵ,Hi,ϵ))>∥φ 1 (Vi,j 0 ,Ki,j 0 )∥HL − ϵ 4 . (5.19) For i ≤ j0, and i sufficiently large, since the diffeomorphism φ 1 (Vi,j 0 ,Ki,j 0 ) is Hamiltonian, we derive from a result found in [7, 16] that there exists a positive finite constant D which does not depend on neither i, nor j0, such that 1 D ∥φ1(Vi,j 0 ,Ki,j 0 )∥H ≤ ∥φ 1 (Vi,j 0 ,Ki,j 0 )∥SHL, (5.20) CUBO 19, 2 (2017) On topological symplectic dynamical systems 67 where ∥.∥H represents the Hofer norm of Hamiltonian diffeomorphisms. This implies that ∥ψi∥SHL + ϵ> 1 D ∥φ1(Vi,j 0 ,Ki,j 0 )∥H − ϵ 4 , (5.21) for i ≤ j0, and i sufficiently large. At this level, we use the fact that Oh’s norm restricted to the group of Hamiltonian diffeomorphisms is bounded from above by Hofer’s norm to get ∥ψi∥SHL + ϵ> 1 D ∥φ1(Vi,j 0 ,Ki,j 0 )∥Oh − ϵ 4 , (5.22) for i ≤ j0, and i sufficiently large. Passing to the limit in the latter estimate, yields lim i→∞ ∥ψi∥SHL + 5ϵ 4 ≥ 1 D lim j0≥i,i→∞ ∥φ1(Vi,j 0 ,Ki,j 0 )∥Oh = 1 D lim i→∞ ∥ψi∥Oh, (5.23) for all ϵ. Finally, we have proved that for all positive real number δ (replacing ϵ by 4δ 5D ), we have δ ≥ lim i→∞ ∥ψi∥Oh. (5.24) This completes the proof.! 5.3 Topological symplectic displacement energy In this Section, we extend the symplectic displacement energy to the world of strong symplectic homeomorphisms. This is motivated by the uniqueness result from [5] and the uniqueness of Banyaga’s Hofer-like geometry [15]. Definition 5.6. The strong symplectic displacement energy e0,∞ S (B) of a non empty compact subset B ⊂ M is : e0,∞S (B) = inf{∥h∥SHL|h ∈ SSympeo(M,ω), h(B) ∩ B = ∅}. Lemma 5.7. For any non empty compact subset B ⊂ M, e0,∞ S (B) is a strict positive number. Proof. Let ϵ>0, by definition of e0,∞ S (B), there exists a strong symplectic isotopy ψ(Fϵ,λϵ) such that ψ1 (Fϵ,λϵ) = h, and e0,∞S (B) + ϵ 2 >l∞(ψ(Fϵ,λϵ)). (5.25) On the other hand, there exists a sequence of symplectic isotopies (φ(Fi,λi)) that converges to ψ(Fϵ,λϵ) with respect to the (C 0 + L∞)−topology. So, one can choose integer j0 large enough such that φ1 (Fi,λi) displaces B, and l∞(ψ(Fϵ,λϵ))>l∞(φ(Fi,λi)) − ϵ 4 . (5.26) 68 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) for all i>j0. It follows from the definition of symplectic displacement energy that ∥φ1(Fi,λi)∥HL ≥ eS(B)>0, (5.27) for all i>j0. Relations (5.25), (5.26) and (5.27) implies that e0,∞ S (B) + ϵ 2 + ϵ 4 >l∞(φ(Fi,λi)) ≥ ∥φ 1 (Fi,λi) ∥HL ≥ eS(B)>0, (5.28) for i sufficiently large. Therefore, e0,∞S (B) + ϵ>eS(B)>0, for all positive ϵ. This completes the proof. ✷ 6 Conjectures and Examples 6.1 Conjectures: Conjecture (A): For any h ∈ SSympeo(M,ω), we have ∥h∥SHL = ∥h∥ (1,∞) SHL . ⋆ Conjecture−(A) is supported by the uniqueness result of Hofer-like geometry from [15] or more generally in [16]. Conjecture (B): If h ∈ SSympeo(M,ω), then the norm φ !→ ∥h ◦φ◦ h−1∥SHL is equivalent to the norm φ !→ ∥φ∥SHL.♣ Conjecture−(B) is supported by a result found in [4] (Theorem 7). Conjecture (C): Let σ be the canonical volume form on the unit circle S1 given by the orientation of the circle. If λ is a strong symplectic isotopy generated by (U, H), then the Fathi’s mass flow of λ is exactly one of the following quantities: ± 1 (n − 1)! ∫ M ""∫1 0 Htdt # ∧ ωn−1 ∧ f∗(σ) # , for any mapping f : M → S1.♥ In particular, any continuous Hamiltonian flow has a trivial mass flow. Therefore, is any strong symplectic isotopy with trivial Fathi’s mass flow homotopic (relatively to fixed endpoints) to a continuous Hamiltonian flow? CUBO 19, 2 (2017) On topological symplectic dynamical systems 69 Conjecture (D): If λ is a strong symplectic isotopy generated by (U, H), then the mapping, λ !→ $∫1 0 [Ht]dt % ∈ H1(M, R), is a well defined group homomorphism which only depend on the homotopic class of λ relatively to fixed ends, where [, ] stands for the de Rham cohomology class.$ 6.2 Examples 6.2.1 Example A harmonic 1−parameter group is an isotopy β = {βt} generated by the vector field X defined by ι(X)ω = K, where K is a harmonic 1−form, and let φ be a non-smooth Hamiltonian homeomor- phism [13]. By definition of φ, there exists a sequence of Hamiltonian isotopies Φj which is Cauchy in (C0 + L∞), and φj := Φj(1) → φ, uniformly. The isotopy Ψj : t !→ φ −1 j ◦ βt ◦ φj, has time−1 map φ−1 j ◦β1 ◦φj, and it is generated by ( ∫ 1 0 K(Φ̇j(s)) ◦Φj(s)ds, K). In fact, the smooth function x !→ ∫ 1 0 K(Φ̇j(s)) ◦ Φj(s)ds(x), does not depend on the choice of any isotopy with time−1 map φj := Φj(1) [5, 16, 18]. As it can be checked, the sequence of generators ( ∫ 1 0 K(Φ̇j(s))◦Φj(s)ds, K) is Cauchy in D∞ if and only if the sequence of functions x !→ ( ∫1 0 K(Φ̇j(s)) ◦ Φj(s)ds)(x) is Cauchy in the L∞−Hofer norm. But, Lemma 3.9 from [15, 16] shows that the sequence of func- tions x !→ ( ∫ 1 0 K(Φ̇j(s)) ◦ Φj(s)ds)(x), is Cauchy in the L∞−Hofer norm provided the sequence Φj is Cauchy in the metric d̄; which is the case. Thus, the latter converges in the complete met- ric space N 0([0, 1] × M , R) to a time-independent continuous function that we denote F0. The strong symplectic isotopy Φ : t !→ φ ◦ βt ◦ φ −1, is generated by (F0, K), and its time−1 map x !→ (φ◦β1 ◦φ −1)(x) is not necessary C1, but continuous. Hence, we have constructed separately an example of strong symplectic isotopy which is a 1−parameter subgroup and whose generator is time independent. The Hofer-like length of Φ are given by l∞(Φ) = osc(F0) + |K| = l(1,∞)(Φ), (6.1) and we also have, ē0(φ ◦ β1 ◦ φ −1) ≤ ē(φ ◦ β1 ◦ φ −1) ≤ osc(F0) + |K|<∞. (6.2) △ 6.2.2 Example Consider the torus T 2l with coordinates (θ1, . . . ,θ2l) and equip it with the flat Riemannian metric g0. Note that all the 1−forms dθi, i = 1, . . . , 2l are harmonic. Take the 1-forms dθi for i = 1, . . . , 2l as basis for the space of harmonic 1-forms and consider the symplectic form ω = ∑l i=1 dθi ∧dθi+l. Given v = (a1, . . . , al, b1, . . . , bl) ∈ R 2l , the translation x !→ x+v on R2l induces a rotation Rv on T 2l , which is a symplectic diffeomorphism. Therefore, the smooth mapping {Rt v } : t !→ Rtv defines a symplectic isotopy generated by (0, H) with H = ∑ l i=1 (aidθi+l − bidθi). Now, consider the 70 S. Tchuiaga, M. Koivogui, F. Balibuno & V. Mbazumutima CUBO 19, 2 (2017) torus T 2 as the square: ✷ := {(p, q) | 0 ≤ p ≤ 1, 0 ≤ q ≤ 1} ⊂ R2, with opposite sides identified. Then, the action of the unit circle S 1 on T 2 : ρ : S1 × T2 → T2, (α, (θ1,θ2)) !→ (θ1 + α,θ2 + α), induces a non-Hamiltonian diffeomorphism ρα : T 2 → T2, because the latter has no fixed point for α small and non-trivial. Assume this done. Let D2 ⊂ R2 be the 2−disk of radius τ ∈]0, 1/8[ centered at A = (a, 0) with 7/8 ≤ a<1, and let Λ2(τ) be the corresponding subset in T2. For any ν<1/4, consider the nonempty open subset B(ν) = {(x, y) | 0