The group of Hamiltonian homeomorphisms and continuous Hamiltonian flows
Yong-Geun OH
备注：
28 pages; All main theorems are re-stated in $C^0$ setting due to lack of the proof of uniqueness of Hamiltonian in $L^{(1,\infty)}$ setting, a section containing a discussion of a wild homeomorphism is added; (v6) the final version to appear, English improved, presentation improved by rearranging some sections and adding explanations
In this paper, we study the dynamical aspects of the \emph{Hamiltonian homeomorphism group} $Hameo(M,ω)$ which was introduced by Müller and the author. We introduce the notion of autonomous continuous Hamiltonian flows and extend the well-known conservation of energy to such flows. The definitions of the Hofer length and of the spectral invariants $ρ_a$ are extended to continuous Hamiltonian paths, and the Hofer norm and the spectral norm $γ:Ham(M,ω) \to \R_+$ are generalized to the corresponding intrinsic norms on $Hameo(M,ω)$ respectively. Using these extensions, we also extend the construction of Entov-Polterovich's Calabi quasi-morphism on $S^2$ to the space of continuous Hamiltonian paths. We also discuss a conjecture concerning extendability of Entov-Polterovich's quasi-morphism and its relation to the extendability of Calabi homomorphism on the disc to $Hameo(D^2,\del D^2)$, and their implication towards the simpleness question on the area preserving homeomorphism groups of the disc $D^2$ and of the sphere $S^2$.
In this article, the main goal is to give a dynamical point of view of Floer homology barcodes for Hamiltonian homeomorphisms of surfaces. More specifically, we describe a way to construct barcodes for Hamiltonian homeomorphisms of surfaces from graphs. We will define graphes associated to maximal isotopies of a Hamiltonian homeomorphism using Le Calvez's positively transverse foliation theory and to those graphs we will associate barcodes. In particular, we will prove that for the simplest cases, our constructions coincide with the Floer Homology barcodes.
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 po…
The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $Homeo^Ω(D^2,\partial D^2)$ of area preserving homeomorphisms of the 2-disc $D^2$. We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism ${\rm Cal}:Diff^Ω(D^1,\partial D^2) \to \mathbb R$ to a homomorphism $\overline{\rm Cal}:Hameo(D^2,\partial D^2) \to \mathbb R$ to that of the vanishing of the basic phase function $f_{\underline{\mathbb F}}$, a Floer theoretic graph se…
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} $Hameo(M,ω)$ and the space $Hameo^w(M,ω)$ of {\it Hamiltonian homeomorphisms} such that $$ Ham(M,ω) \subsetneq Hameo(M,ω) \subset Hameo^w(M,ω) \subset Sympeo(M,ω) $$ where $Sympeo(M,ω)$ is the group of symplectic homeomorphisms. We prove that $Hameo(M,ω)$ is a {\it n…
We define boundedness properties on the contractible fixed points set of the time-one map of an identity isotopy on a closed oriented surface with genus $g\geq1$. In symplectic geometry, a classical object is the notion of action function, defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. We give a dynamical interpretation of this function that permits us to generalize it in the case of a homeomorphism isotopic to identity that preserves a Borel finite measure of rotation vector zero, provided that a boundedness condition is satisfied. We give some p…
We prove that on a symplectic sphere, the group of Hamiltonian homeomorphisms in the sense of Oh and Müller is a proper normal subgroup of the group of finite energy Hamiltonian homeomorphisms. Moreover we detect infinite-dimensional flats inside the quotient of these groups endowed with the natural Hofer pseudo-metric.
The group Hameo (M,ω) of Hamiltonian homeomorphisms of a connected symplectic manifold (M,ω) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the L^{(1,\infty)}-Hofer norm (and not the L^\infty-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 L^\infty-case. In view of the fact that the Hofer norm on the group Ham (M,ω) of Hamiltonian diffeomorphisms does not depend on the choice of the L^{(1,\infty)}-norm vs. the L^\infty-norm [9], Y.-G. …
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 Hamilt…
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 homeomorp…
On closed symplectically aspherical manifolds, Schwarz proved a classical result that the action function of a nontrivial Hamiltonian diffeomorphism is not constant by using Floer homology. In this article, we generalize Schwarz's theorem to the $C^0$-case on closed aspherical surfaces. Our methods involve the theory of transverse foliations for dynamical systems of surfaces inspired by Le Calvez and its recent progresses. 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. Furthe…
We show that the Calabi homomorphism extends to some groups of homeomorphisms on exact symplectic manifolds. The construction is based on the uniqueness of the generating Hamiltonian (proved by Viterbo) of continuous Hamiltonian isotopies (introduced by Mueller and Oh).
Elementary sub-Riemannian geometry on the Heisenberg group H(n) provides a compact picture of symplectic geometry. Any Hamiltonian diffeomorphism on $R^{2n}$ lifts to a volume preserving bi-Lipschitz homeomorphisms of H(n), with the use of its generating function. Any curve of a flow of such homeomorphisms deviates from horizontality by the Hamiltonian of the flow. From the metric point of view this means that any such curve has Hausdorff dimension 2 and the $\mathcal{H}^{2}$ (area) density equal to the Hamiltonian. The non-degeneracy of the Hofer distance is a direct consequence of this fact.
We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group $SSympeo(M,ω)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,ω)$ of hamiltonian homeomorphisms introduced by Oh and Muller. The group $SSympeo(M,ω)$ is arcwise connected, is contained in the identity component of $Sympeo(M,ω)$; it contains $Hameo(M,ω)$ as a normal subgroup and coincides with it when $M$ is simply connected. Finally its commutator subgroup $[SSympeo(M,ω),SSympeo(M,ω)]$ is contained…
Stefan M$\ddot{\mathrm{u}}$ller posed the problem "Do Hofer's metrics on the group of Hamiltonian diffeomorphism and the one of Hamiltonian homeomorphisms (Hameomorphisms) correspond?". Let $(M,ω)$ be a compact exact symplectic manifold. We prove that the group of Hamiltonian homeomorphisms is not a simple group if the positive answer of M$\ddot{\mathrm{u}}$ller's question holds . To prove this results, we introduce fragmented Hofer's norms on a normal subgroup of the group of Hamiltonian homeomorphisms.
The content of this paper has no mathematical flaw except that the proof of the main theorem relies on the homotopy invariance of spectral invariants of topological Hamiltonian paths. Since the latter is still up in the air, the main result of the paper is the reduction of the extension problem of the Calabi homomorphism to the group of hameomorphism is still conjectural.
This sequel to our previous paper [MS11b] continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. The article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is a…
This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, δ)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic hom…
In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n \ge 3$, asked if the same statement holds in dimension $2$. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles what is known as the "simplicity conjecture" in the affirmative. In fact, we prove the a priori stronger statement that this group is not perfect. An important step in our proof involves verifying for certain smooth twist maps a conjecture of Hutchings concerning recovering the C…
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 an…
We show that if a sequence of Hamiltonian flows has a $C^0$ limit, and if the generating Hamiltonians of the sequence have a limit, then this limit is uniquely determned by the limiting $C^0$ flow. This answers a question by Y.G. Oh.
