On periodic homeomorphisms of spheres

自相似2维流形的同胚群

Homeomorphism groups of self-similar 2-manifolds

该研究围绕自相似2维流形的同胚群展开，通过综述新旧成果、研究稳定集结构，证明了三项强化/推广型定理，为无限亏格2维流形同胚群的Steinhaus性质提供了首批实例。

圆盘同胚的辫子与连锁周期轨道

Braids and linked periodic orbits of disc homeomorphisms

本文以诱导辫子刻画二维圆盘保向同胚周期轨道的连锁关系，并借助辫子强迫关系提出寻找连锁轨道的方法，构造了 pseudo-Anosov 型 3-周期轨道对应的周期至多 4 的新例子。

Actions of homeomorphism groups of manifolds admitting a nontrivial finite free action

In this paper, we study the action of $\text{Homeo}_0(M)$, the identity component of the group of homeomorphisms of an $n$-dimensional manifold $M$ with an $\mathbb{F}_p$-free action, on another manifold $N$ of dimension $n+k<2n$. We prove that if $M$ is not an $\mathbb{F}_p$-homology sphere, then $N\cong M\times K$ for a homology manifold $K$ such that the action of $\text{Homeo}_0(M)$ on $M$ is standard and on $K$ is trivial. In particular, for $M=S^n$ a sphere, any nontrivial action is a generalization of the "coning-off" construction.

Local product structure for expansive homeomorphisms

Let $f\colon M\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus.

Dynamics of shear homeomorphisms of tori and the Bestvina-Handel algorithm

Sharkovskii proved that the existence of a periodic orbit in a one-dimensional dynamical system implies existence of infinitely many periodic orbits. We obtain an analog of Sharkovskii's theorem for periodic orbits of shear homeomorphisms of the torus. This is done by obtaining a dynamical order relation on the set of simple orbits and simple pairs. We then use this order relation for a global analysis for a quantum chaotic physical system called the kicked accelerated particle.

Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$

We investigate a known problem whether a Sobolev homeomorphism between domains in $\mathbb{R}^n$ can change sign of the Jacobian. The only case that remains open is when $f\in W^{1,[n/2]}$, $n\geq 4$. We prove that if $n\geq 4$, and a sense-preserving homeomorphism $f$ satisfies $f\in W^{1,[n/2]}$, $f^{-1}\in W^{1,n-[n/2]-1}$ and either $f$ is Hölder continuous on almost all spheres of dimension $[n/2]$, or $f^{-1}$ is Hölder continuous on almost all spheres of dimensions $n-[n/2]-1$, then the Jacobian of $f$ is non-negative, $J_f\geq 0$, almost everywhere. This result is a consequence of a mo…

Homogeneous Transformation Groups of the Sphere

In this paper, we study the structure of homogeneous subgroups of the homeomorphism group of the sphere, which are defined as closed groups of homeomorphisms of the sphere that contain the rotation group. We prove two structure theorems about the behaviour and properties of such groups and present a diagram of the structure of these groups partly on the basis of these results. In addition, we prove a number of explicit relations between the groups in the diagram.

Periodic Surface Homeomorphisms and Contact Structures

Periodic surface homemorphisms (diffeomorphisms) play a significant role in the the Nielsen-Thurston classification of surface homeomorphisms. Periodic surface homeomorphisms can be described (up to conjugacy) by using data sets which are combinatorial objects. In this article, we start by associating a rational open book to a slight modification of a given data set, called marked data set. It is known that every rational open book supports a contact structure. Thus, we can associate a contact structure to a periodic map and study the properties of it in terms combinatorial conditions on marke…

末端周期映射环面的体积

Volumes of end-periodic mapping tori

本文概述了两个近期结果，将无限型曲面的末端周期同胚动力学与其紧化映射环面几何联系起来，是Brock定理在有限型伪阿诺夫同胚映射环面情形的类似结果。

端周期伸缩因子的下界

A lower bound on end-periodic stretch factors

本文针对端周期同胚，以其核特征这一拓扑复杂度度量为变量，给出了Handel--Miller伸缩因子的下界，并证明了该下界增长率的精确性。

映射类群$\boldsymbol{\rm Mod}(S_g)$在$(2g-1)$维球面上的忠实作用

A Faithful Action of the Mapping Class Group $\mathrm{Mod}(S_g)$ on the $(2g-1)$-Sphere

该研究证明当$g\boldsymbol{\rm Mod}(S_g)$时，映射类群$\boldsymbol{\rm Mod}(S_g)$可通过等变爆破与托雷利群双序性，在$(2g-1)$维球面上实现保定向同胚的忠实作用。

Injections of Artin groups

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers. We also give a generating set for the automorphism group of the pure braid group on at least 4 strands. The technique, following Ivanov, is to prove that every superinjective map of the complex of curves of a sphere with at least 5 punctures is induced by a homeomorphism.

端周期映射环面体积的下界

A lower bound on volumes of end-periodic mapping tori

本文给出强不可约端周期同胚f的紧化映射环面体积下界，结合已有工作表明其体积与f在裤子图分支上的平移长度可比，扩展了Brock在有限型伪阿诺索夫映射环面体积的研究。

同调球的Alexander技巧

The Alexander trick for homology spheres

本文证明在维度至少为6时，固定边界的紧致可缩流形同胚群可缩，并由此推导出单侧$h$-配边的强唯一性。

作用于边界球面的双曲群的稳定性

Stability for hyperbolic groups acting on boundary spheres

本文证明，当双曲群的 Gromov 边界为拓扑 n-球面时，其边界同胚作用具有动力学拓扑稳定性，任意邻近作用均与标准作用半共轭。

每一个弱佩罗数都是一个端周期拉伸因子

Every Weak Perron Number is an End-Periodic Stretch Factor

论文构造了一个端周期的homeomorphism，其Handel-Miller拉伸因子等于给定的弱佩罗数，研究了无限型表面的拉伸因子性质。

3-manifold invariants and periodicity of homology spheres

We show how the periodicity of a homology sphere is reflected in the Reshetikhin-Turaev-Witten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.

穿孔圆盘上的强Nielsen等价性

Strong Nielsen equivalence on the punctured disc

本文研究穿孔圆盘上保持定向同胚的周期点强Nielsen等价性，在辫子理论中给出两个周期点强Nielsen等价的充要条件，并应用于Jiang--Zheng迹公式，证明强制周期轨道分属不同强Nielsen类。