Patrice Le Calvez
The goal of the article is to characterize the conservative homeomorphisms of a closed orientable surface $S$ of genus $\geq 2$, that have finitely many periodic points. By conservative, we mean a map with no wandering point. As a particular case, when $S$ is furnished with a symplectic form, we characterize the symplectic diffeomorphisms of $S$ with finitely many periodic points.
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Linearisation of conservative toral homeomorphisms
本文给出了两环面上保守同胚的无理旋转的半共轭存在性的一个等价刻画。这导致了一个类似的庞加莱的分类圆同胚保守toral同胚具有唯一的旋转向量和一定的有界平均运动属性。对于极小toral同胚，结果推广到任意维数。进一步，我们给出了同伦于恒等式的非游荡环面同胚动力学的一个基本分类。
We prove the genericity of the shadowing and periodic shadowing properties for both conservative and dissipative homeomorphisms on a compact connected manifold. Our proof is valid for topological manifolds and still holds in the dissipative case. As a consequence of this result, we establish the genericity of the specification property, the average shadowing property and the asymptotic average shadowing property in the conservative case.
Irrational rotation factors for conservative torus homeomorphisms
我们提供了一个等价的表征的保守环面同胚，最终不是环形的一维无理旋转因子的存在。它指出，一个面积保持的非环形环面同胚$f$是共轭的一个无理旋转$R_α$的圆当且仅当存在一个明确的速度旋转在某个合理的方向上的环面，和偏差的常数旋转在这个方向上是一致有界的。通过一个反例，我们还证明了一个类似的特征不成立最终环形环面同胚。
Typical conservative homeomorphisms have total metric mean dimension
该研究针对带局部正非原子测度的高维紧致光滑无边流形，证明典型保测同胚的上度量平均维数等于流形维数，且该测度是对应变分原理下具有最大度量平均维数的测度。
Fully chaotic conservative models for some torus homeomorphisms
研究同伦于恒等映射且旋转集有非空内部的环面同胚，证明其单调半共轭于保勒贝格测度且旋转集相同的同胚，且商映射具有拓扑混合、周期点稠密和连续统扩张等混沌性质。
Dynamical properties of spatial discretizations of a generic homeomorphism
本文讨论动力系统的动力学行为与其数值模拟的动力学行为之间的联系。在这里，我们的模型数值截断作为一个空间离散化的系统。以前的一些工作，精心挑选的例子表明，动力系统的动力学行为和他们的离散化可以是完全不同的。本文主要研究紧致流形的通有同胚。因此，我们的目标是解决以下问题：可以检测的动力学性质的一般同胚的空间离散化这个同胚？我们研究保守和耗散同胚（即有或没有假设同胚保持一个给定的措施）。我们将证明，一个通用的保守同胚的一个单一的离散化的动力学不依赖于同胚本身，而是在用于离散化的网格。因此，一个给定的一般保守同胚的动力学性质不能使用单一的离散化检测。然而，我们也将证明，一个一般的保守同胚的一些动力学特征（如所有周期点的周期集），可以读取一个序列的精细和精细的离散。耗散情形在某种程度上是相反的：我们将看到，一般耗散同胚的动力学被这个同胚的任何足够精细的离散化的动力学很好地近似。
The goal of this article is to exhibit two new families of finitely generated simple groups of homeomorphisms of $\mathbf{R}$. These families are strikingly different from existing families owing to the nature of their actions on $\mathbf{R}$, and exhibit surprising algebraic and dynamical features. In particular, one construction provides the first examples of finitely generated simple groups of homeomorphisms of the real line which also admit a minimal action by homeomorphisms on the circle. This provides new examples of finitely generated simple groups with infinite commutator width, and th…
This memoir is concerned with the generic dynamical properties of conservative homeomorphisms of compact manifolds. Several important techniques allowing to prove genericity results are presented: we emphasize the important role played by periodic approximations of homeomorphisms, and by the embedding of the space of homeomorphisms in the space of bi-measurable automorphisms.
On real extensions of distal minimal homeomorphisms
本文证明了远端极小紧度量$\Z$ -流的拓扑保守实斜积扩张的一个结构定理。主要结果表明，每一个这样的扩展可以表示为一个扰动的Rokhlin斜积。此外，我们给出了一些反例，指出该结构的所有组成部分实际上都是必然的。
The concept of bounded mean motion for toral homeomorphisms
两环面的保守无理伪旋转与无理旋转半共轭当且仅当它具有有界平均运动的性质[10]。(Here“无理伪旋转”表示具有唯一定义的完全无理旋转向量的环面同胚。本说明的目的是进一步探讨这一概念。例如，我们提供了一个例子，表明前面的陈述在非保守情况下不成立。此外，我们收集了一些意见的情况下，有界平均运动属性失败。特别地，我们证明了具有无界平均运动的双环面的非游荡无理伪旋转对初始条件具有敏感的依赖性。
Burnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms
如果对于任何$g\in G$，存在具有$g^n=id$的正整数$n$，则称群$G$是周期性的。证明了2-环面上保持测度$μ$的同胚周期群是有限的。此外，如果群包含与单位元同素的同胚，则它是阿贝尔群，并且自由作用于$\mathbb{T}^2$。在附录中，我们证明了每一个生成的2-群的toral同胚是有限的。
We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$, $d\ge 2$. In the conservative setting, we prove that there exists a Baire residual subset of the set $\text{Homeo}_{0, λ}(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$, and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$-generic conservative homeomorphisms on $\mathbb T^d$ is …
Burnside problem for groups of homeomorphisms of compact surfaces
如果对于$Γ$中的任何$g$，存在具有$g^n=id$的正整数$n$，则称组$Γ$是周期性的。我们首先证明了由具有有限轨道的$C^1$ -环同态作用于2-球面$\SS^2$的环生成周期群是有限的并且共轭于$\mathrm{O}(3,\R)$的子群，并利用它证明了由具有偶数有界阶的球面环同态生成的周期群是有限的.最后，我们证明了除了2-球面或2-环面（这是作者以前的一篇论文的目的）之外的任何可定向紧致曲面的同胚的一个可生成的周期群是有限的。
Maximal Entropy Measures for Piecewise Affine Surface Homeomorphisms
从熵的角度研究了分段仿射曲面同胚的动力学性质。在正拓扑熵的假设下，建立了极大熵的多个遍历不变概率测度的存在性，并证明了周期点个数的一个乘性下界.这是为了作为一个步骤的理解表面的仿射。我们继续建立一个跳跃变换，不使用第一个回报，但仔细选择的“好”回报，以免除马尔可夫分割。我们通过一些熵和遍历参数来控制这些良好的回报。
To every dynamical system $(X,φ)$ over a totally disconnected compact space, we associate a left-orderable group $T(φ)$. It is defined as a group of homeomorphisms of the suspension of $(X,φ)$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple, and if it is a subshift then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that i…
On groups of homeomorphisms of the interval with finitely many fixed points
本文证明含两生成元自由半群的Diff₊(I)子群非C₀-离散，并借此推广Hölder定理，完成对区间同胚群中每个非单位元至多有N个不动点的子群的分类。
We prove that every finitely-generated group of homeomorphisms of the 2-dimensional sphere all of whose elements have a finite order which is a power of 2 and so that there exists a uniform bound for the order of group elements is finite. We prove a similar result for groups of area-preserving homeomorphisms without the hypothesis that the orders of group elements are powers of 2 provided there is an element of even order.
The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal identity isotopy such that we can give a fine descriptions of the dynamics of its transverse foliation. We will define a kind of identity isotopies: torsion-low isotopies. In particular, when $f$ is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy $I$ of $f$ is torsion-low if and only if for every point $z$ fixed along the isotopy, the (real) rotation number $ρ(I,z)$, which is well defin…
The group of almost-periodic homeomorphisms of the real line
研究了实直线的概周期同胚群。我们的主要结果是：实直线上无整体不动点的有限生成群的作用共轭于无几乎不动点的几乎周期作用。这就等于说，实直线上的作用可以紧化为紧空间的一维层上的作用，而没有全局不动点。作为应用，我们给出了Witte定理的另一种证明：顺从左序群是局部可指示的。
This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system $(X,T)$ can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.
We show that, in many situations, a homeomorphism $f$ of a manifold $M$ may be recovered from the (marked) isomorphism class of a finitely generated group of homeomorphisms containing $f$. As an application, we relate the notions of {\em critical regularity} and of {\em differentiable rigidity}, give examples of groups of diffeomorphisms of 1-manifolds with strong differential rigidity, and in so doing give an independent, short proof of a recent result of Kim and Koberda that there exist finitely generated groups of $C^α$ diffeomorphisms of a 1-manifold $M$, not embeddable into $\mathrm{Diff}…
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。
Translation numbers define generators of $F_k^+\to {\text{\rm Homeo}_+}(\mathbb{S}^1)$
考虑了由圆的同胚生成的半群的极小作用，证明了单个元素的平移数的集合完全决定了生成元的集合（直到一个共同的连续坐标变换）.证明中使用的主要工具之一是圆同胚随机动力学的同步性质：安东诺夫定理及其推论。
Symmetric random walks on Homeo+(R)
研究了实直线上保向同胚的有限生成群上的对称随机游动。我们建立了线上诱导马尔可夫链的一个振动性质，该性质隐含弱常返形式。除少数特殊情况外，我们证明了“有限距离上的全局稳定性”的一个性质：粗略地说，存在一个紧区间，使得当任意两条轨道中的一条返回到该紧区间时，任意两条轨道越来越近.这里采用的概率技术导致有趣的结果，在线上的群作用的研究。例如，我们表明，在适当的坐标变化，每个点的漂移变为零，只要行动是最小的。作为一个副产品，我们恢复的事实，即每一个有限生成群同胚的实线是拓扑共轭的一组（全球）Lipschitz同胚。此外，我们表明，这样的共轭可以选择在这样一种方式，每个元素的位移是一致有界的。
Pseudo-rotations of the open annulus
本文研究了开环的伪旋转，开环的\emph{i.e.}保守同胚，其旋转集被约化为一个无理数（伪旋转角）.特别证明了对于角$ρ$的每一个伪旋转$h$，角$ρ$的刚性旋转都在$h$的共轭类的闭包中.我们还证明了伪旋转是不持久的$C^r$拓扑的任何$r\geq 0$。
We study homeomorphisms of a Cantor set with $k$ ($k < +\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and their certain orbit-cut sub-C*-algebras. In the case that $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank zero if in addition $(X, σ)$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli-Vershik-Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of th…
We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of finitely-generated groups with undecidable conjugacy problems, whose descriptions arguably do not involve any type of computation. For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every finitely-generated subgroup containing both ha…
Conservative Coexpanding on Average Diffeomorphisms
本文证明保守IID随机系统在平均余维1平面扩张时，其生成元在Sobolev空间上本质谱半径小于1，从而具有有限遍历分量，并在单分量条件下满足指数混合与中心极限定理，且这些性质对小扰动稳定。
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。
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。
We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson's group $F$. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a canonical quotient which is also a chain group, and which has a simple commutator subgroup. On the other hand, every finitely generated subgroup of $\operatorname{Homeo}^+(I)$ can be realized as a subgr…
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