Hamiltonian Homeomorphisms Research Articles

C0-transport of flux geometry

The goal of this paper is to study, in a large scale point of view, the flux geometry of a closed symplectic manifold (M,ω): namely, the topological counterpart of the flux homomorphism. Using metrics arising from the decomposition of closed 1-forms with respect to an arbitrary linear section S, we generalize the construction of the group of strong symplectic homeomorphisms. The flux homomorphism for symplectomorphisms is extended to a surjective group homomorphism Sω0 on the group of S-homeomorphisms. We prove that the kernel of Sω0 is path connected, coincides with the subgroup Hameo(M,ω) of all Hamiltonian homeomorphisms and investigate the discreteness of the corresponding flux group SΓω. Later on, without appealing to any lifting map, we give an alternative proof of a result from the classical flux geometry saying that any smooth symplectic isotopy in Ham(M,ω) is a Hamiltonian isotopy. Furthermore under some hypothesis, we prove that any S-topological isotopy in Hameo(M,ω) is a continuous Hamiltonian isotopy. We also proved that any S-topological isotopy with trivial flux is homotopic to a continuous Hamiltonian isotopy, relatively to fixed endpoints.

An Arnold-type principle for non-smooth objects

In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnold-type principle continues to hold in \(C^0\) settings: suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite. We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: \(C^0\) Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to 0-section. An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles.

Quantitative Heegaard Floer cohomology and the Calabi invariant

Abstract We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).

Barcodes and area-preserving homeomorphisms

In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.

The action spectrum and $$C^0$$ symplectic topology

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.

Some results of Hamiltonian homeomorphisms on closed aspherical surfaces

On closed symplectically aspherical manifolds, by using Floer homology, Schwarz proved a classical result, i.e., that the action function of a nontrivial Hamiltonian diffeomorphism is not constant. In this article, we generalize Schwarz's theorem to the C0-case on closed aspherical surfaces. Our methods are purely topological and involve the theory of transverse foliations for dynamical systems of surfaces and the recent progress inspired by Le Calvez. As an application, we prove that the contractible fixed points set (and consequently the fixed points set) of a nontrivial Hamiltonian homeomorphism is not connected. We also get a similar result for an orientation preserving nonwandering point homeomorphism of the two-sphere. In the end, we give further applications based on the C0-Schwarz Theorem.

On variants of Arnold conjecture

In this note we discuss the collection of statements known as Arnold conjecture for Hamiltonian diffeomorphisms of closed symplectic manifolds. We provide an overview of the homological, stable and strong versions of Arnold conjecture for non-degenerate Hamiltonian systems, a few versions of Arnold conjecture for possibly degenerate Hamiltonian systems, the degenerate version of Arnold conjecture for Hamiltonian homeomorphisms and Sandon’s version of Arnold conjecture for contactomorphisms.

C0–Symplectic Geometry Under Displacements

ABSTRACTThis paper continues the study of the group Hameo(M, ω), of all Hamiltonian homeomorphisms of a closed symplectic manifold (M, ω). After given a direct proof of the positivity result of the symplectic displacement energy, we show that the uniqueness theorem of generators of strong symplectic isotopies extends to any closed symplectic manifold: An explicit formula for the mass flow of any strong symplectic isotopy with respect to its generator is given. We show that Hameo(M, ω) inherits under the C0 -Hamiltonian topology, the fragmentation property, the algebraic perfectness, and coincides with the commutator sub-group of the group of all strong symplectic homeomorphisms. This solves a Banyaga's conjecture, and some other conjectures are also formulated.

A $$C^0$$ C 0 counterexample to the Arnold conjecture

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold must have at least as many fixed points as the minimal number of critical points of a smooth function on the manifold. It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher. More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point.

Generalizations of the Action Function in Symplectic Geometry

In symplectic geometry, the action function is a classical object defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. On closed aspherical surfaces, we give a dynamical interpretation of this function, which permits us to generalize it to the case of a diffeomorphism that is isotopic to identity and preserves a Borel finite measure of rotation vector zero. We define a boundedness property on the contractible fixed points set of the time-one map of an identity isotopy. We generalize the classical action function to any Hamiltonian homeomorphism, provided that the proposed boundedness condition is satisfied. We prove that the generalized action function only depends on the time-one map but not on the isotopy. Finally, we define the action spectrum and show that it is invariant under conjugation by an orientation and measure preserving homeomorphism.

On topological symplectic dynamical systems

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.

C 0-Limits of Hamiltonian Paths and the Oh–Schwarz Spectral Invariants

In this article we study the behavior of the Oh-Schwarz spectral invariants under C^0-small perturbations of the Hamiltonian flow. We obtain an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C^0-distance of its flow from the identity. Using the mentioned estimate we show that, unlike the Hofer norm, the spectral norm is C^0-continuous on surfaces. We also present applications of the above results to the theory of Calabi quasimorphisms and improve a result of Entov, Polterovich and Py. In the final section of the paper we use our results to answer a question of Y.-G. Oh about spectral Hamiltonian homeomorphisms.

Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems

AbstractWe compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results of Gambaudo and Ghys [Enlacements asymptotiques. Topology 36(6) (1997), 1355–1379] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the isotopy. Based on these results, we provide positive answers to two questions posed by Arnold in [The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5(4) (1986), 327–345]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume-preserving homeomorphisms, and is invariant under conjugation by volume-preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed by Banyaga and Spaeth [On the uniqueness of generating Hamiltonians for topological strictly contact isotopies.Preprint, 2012], Buhovsky and Seyfaddini [Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symplectic Geom. to appear, arXiv:1003.2612v2], Müller [The group of Hamiltonian homeomorphisms in the$L^\infty $-norm. J. Korean Math. Soc.45(6) (2008), 1769–1784], Müller and Oh [The group of Hamiltonian homeomorphisms and$C^0$-symplectic topology. J. Symplectic Geom. 5(2) (2007), 167–219], Müller and Spaeth [Topological contact dynamics I: symplectization and applications of the energy-capacity inequality.Preprint, 2011, arXiv:1110.6705v2] and Viterbo [On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. (2006), 34028; Erratum,Int. Math. Res. Not.(2006), 38748]. Moreover, we generalize an example of Furstenberg [Strict ergodicity and transformation of the torus. Amer. J. Math. 83(1961), 573–601] on topologically conjugate but not$C^1$-conjugate area-preserving diffeomorphisms of the two-torus to trivial$T^2$-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not$C^1$-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.

Computations of Hamiltonian Homeomorphisms under Symplectic Reduction in the New Sense

This paper studies the extended Hofer norm of hamiltonian homeomorphisms on the symplectic quotient by a compact Lie group.The hamiltonian property is proved for the maps under the symplectic reduction, and the relation between the hamiltonian homeomorphism’s extended Hofer norm and the induced one is also shown in the new sense

On the group of symplectic homeomorphisms

Let ( M , ω ) be a closed symplectic manifold. We define a Hofer-like metric d on the identity component Sym ( M , ω ) 0 in the group Symp ( M , ω ) of all symplectic diffeomorphisms of ( M , ω ) . Unlike the Hofer metric on the group Ham ( M , ω ) of Hamiltonian diffeomorphisms, the metric d is not bi-invariant. We show that the metric topology τ defined by d is natural (i.e. independent of the choice involved in its definition). We define the symplectic topology as a blend of the Hofer-like topology τ and the C 0 -topology. We use it to construct a subgroup SSympeo ( M , ω ) of the group Sympeo ( M , ω ) of all symplectic homeomorphisms, containing the group Hameo ( M , ω ) of Hamiltonian homeomorphisms (introduced by Oh and Muller). If M is simply connected SSympeo ( M , ω ) coincides with Hameo ( M , ω ) . Moreover its commutator subgroup [ SSympeo ( M , ω ) , SSympeo ( M , ω ) ] is contained in Hameo ( M , ω ) . To cite this article: A. Banyaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

THE GROUP OF HAMILTONIAN HOMEOMORPHISMS IN THE L∞-NORM

The group Hameo (M, <TEX>$\omega$</TEX>) of Hamiltonian homeomorphisms of a connected symplectic manifold (M, <TEX>$\omega$</TEX>) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the <TEX>$L^{(1,{\infty})}$</TEX>-Hofer norm (and not the <TEX>$L^{\infty}$</TEX>-Hofer norm) on the space of Hamiltonian paths (see below for the definitions). A justification for this choice was given in [7]. In this article we study the <TEX>$L^{\infty}$</TEX>-case. In view of the fact that the Hofer norm on the group Ham (M, <TEX>$\omega$</TEX>) of Hamiltonian diffeomorphisms does not depend on the choice of the <TEX>$L^{(1,{\infty})}$</TEX>-norm vs. the <TEX>$L^{\infty}$</TEX>-norm [9], Y.-G. Oh and D. McDuff (private communications) asked whether the two notions of Hamiltonian homeomorphisms arising from the different norms coincide. We will give an affirmative answer to this question in this paper.

Periodic orbits of Hamiltonian homeomorphisms of surfaces

John Franks and Michael Handel [FH2] have recently proved that any nontrivial Hamiltonian diffeomorphism of a closed surface of genus at least one has periodic orbits of arbitrarily large period. They proved a similar result for a nontrivial area-preserving diffeomorphism of a sphere with at least three fixed points. We extend these results to the case of the homeomorphisms. When the genus is at least one, we prove, moreover, that the periodic orbits may be chosen contractible if the set of contractible fixed points is contained in a disk. When the surface is a sphere, we extend the result to the case of a nontrivial homeomorphism with no wandering points. The proofs make use of an equivariant foliated version of Brouwer's plane translation theorem (see [B, Proposition 2.1]) and some properties of the linking number of fixed points

Une version feuilletée équivariante du théorème de translation de Brouwer

The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a Brouwer line: a proper topological embedding C of R, disjoint from its image and separating f(C) and f–1(C). Suppose that f commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines. We apply this result to give simple proofs of previous results about area-preserving homeomorphisms of surfaces and to prove the following theorem: any Hamiltonian homeomorphism of a closed surface of genus g ≥ 1 has infinitely many contractible periodic points.

The group of Hamiltonian homeomorphisms and C^0-symplectic topology

The main purpose of this paper is to carry out some of the foundational study of $C^0$-Hamiltonian geometry and $C^0$-symplectic topology. We introduce the notions of the strong and the weak {\it Hamiltonian topology} on the space of Hamiltonian paths, and on the group of Hamiltonian diffeomorphisms. We then define the {\it group} $Hameo(M,\omega)$ and the space $Hameo^w(M,\omega)$ of {\it Hamiltonian homeomorphisms} such that $$ Ham(M,\omega) \subsetneq Hameo(M,\omega) \subset Hameo^w(M,\omega) \subset Sympeo(M,\omega) $$ where $Sympeo(M,\omega)$ is the group of symplectic homeomorphisms. We prove that $Hameo(M,\omega)$ is a {\it normal subgroup} of $Sympeo(M,\omega)$ and contains all the time-one maps of Hamiltonian vector fields of $C^{1,1}$-functions. We prove that $Hameo(M,\omega)$ is path connected and so contained in the identity component $Sympeo_0(M,\omega)$ of $Sympeo(M,\omega)$. In the case of an orientable surface, we prove that the {\it mass flow} of any element from $Hameo(M,\omega)$ vanishes, which in turn implies that $Hameo(M,\omega)$ is strictly smaller than the identity component of the group of area preserving homeomorphisms when $M \neq S^2$. For the case of $S^2$, we conjecture that $Hameo(S^2,\omega)$ is still a proper subgroup of $Homeo^\omega_0(S^2) = Sympeo_0(S^2,\omega)$.