正则同胚的承载流形的拓扑研究

非紧空间中具有拓扑影子性质的同胚映射全局吸引子的拓扑结构

Topology of global attractors for homeomorphisms with the topological shadowing property in $\mathbb{R}^m$

研究非紧空间中动力系统的行为，探讨全局吸引子和影子性质，证明在特定动力系统中，全局吸引子的拓扑影子性质当且仅当吸引子是平凡的。

具有全支撑的不变测度与在$C^0$-Gromov--Hausdorff拓扑下由零熵系统逼近

Invariant measures with full support and approximation by zero-entropy systems in the $C^0$-Gromov--Hausdorff topology

本文证明紧度量空间上连续映射在$C^0$-Gromov--Hausdorff拓扑下可由零熵映射逼近，依赖遍历分解定理和正轨道点稠密性，揭示拓扑熵不稳定性及周期点稠密性，最终得出全支撑不变测度动态落入零熵系统逼近范围。

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 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 homeomorp…

Fully essential dynamics for area-preserving surface homeomorphisms

We study the interplay between the dynamics of area-preserving surface homeomorphisms homotopic to the identity and the topology of the surface. We define fully essential dynamics and generalize the results previously obtained on strictly toral dynamics to surfaces of higher genus. Non-fully essential dynamics are, in a way, reducible to surfaces of lower genus, while in the fully essential case the dynamics is decomposed into a disjoint union of periodic bounded disks and a complementary invariant externally transitive continuum $C$. When the Misiurewicz-Ziemian rotation set has non-empty int…

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.