On two remarkable groups of area-preserving homeomorphisms

Topological extension of Calabi homomorphism and non-simpleness of the area-preserving homeomorphism group of two-disc

The paper contains a silly mistake in the usage of the equation (5.4) in the formula (6.5). Due to this mistake, the effect of the transfer map $phi$ is nullified. I am deeply sorry for not being very careful in my submission and making this mistake.

Extension of Calabi homomorphism and non-simpleness of the area preserving homeomorphism group of $D^2$

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.

具有有理旋转方向的曲面保面积同胚

Area preserving homeomorphisms of surfaces with rational rotational direction

本文证明亏格至少为2的闭曲面上，若保测度同胚的旋转向量是有理方向，则其具有无穷多个周期轨道，且这些轨道的旋转向量可任意逼近任一固定遍历测度的旋转向量。

Reduction of symplectic homeomorphisms

In a previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity. To prove our main…

Continuous Hamiltonian dynamics and area-preserving homeomorphism group of $D^2$

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…

Some properties of Hamiltonian homeomorphisms on closed aspherical surfaces

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…

Proof of the simplicity conjecture

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…

纽结谱不变量的次领头渐近行为与曲面的同胚群

Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

本文研究亏格为零的曲面上纽结谱不变量的次领头渐近行为，证明其对光滑含时哈密顿量有界、对有限临界值自治圆盘映射可恢复Ruelle不变量，并推导了Calabi同态的延拓性质与hameomorphism群上Calabi同态核的非单性。

The group of Hamiltonian homeomorphisms and continuous Hamiltonian flows

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 construct…

穿孔曲面上保面积映射周期轨道的通有稠密性

Generic density of periodic orbits of area-preserving maps on punctured surfaces

本文在有限穿孔闭曲面上证明保面积微分同胚的C∞闭合引理，并由此得到通有周期点稠密性，进而在穿孔2-球面上建立周期轨道均匀分布的定量结果。

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} $Hameo(M,ω)$ 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…

The displaced disks problem via symplectic topology

We prove that a $C^0$--small area preserving homeomorphism of a closed surface with vanishing mass flow can not displace a topological disk of large area. This resolves the displaced disks problem posed by F. Béguin, S. Crovisier, and F. Le Roux.

Fragmentation norm and relative quasimorphisms

We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure.

莫比乌斯带上保密度微分同胚群的拟态射

Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band

本文首次研究非可定向流形上保密度微分同胚群，证明莫比乌斯带上的该群存在可数多个线性无关的无界拟态射，并证明紧致连通带边非可定向曲面上的保密度微分同胚群弱可缩。