Conservative surface homeomorphisms with finitely many periodic points

Two new families of finitely generated simple groups of homeomorphisms of the real line

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…

Propriétés dynamiques génériques des homéomorphismes conservatifs

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

We prove a structure theorem for topologically conservative real skew product extensions of distal minimal compact metric $\Z$-flows. The main result states that every such extension can be represented by a perturbation of a Rokhlin skew product. Moreover, we give certain counterexamples to point out that all components of the construction are in fact inevitable.

The concept of bounded mean motion for toral homeomorphisms

A conservative irrational pseudo-rotation of the two-torus is semi-conjugate to the irrational rotation if and only if it has the property of bounded mean motion [10]. (Here 'irrational pseudo-rotation' means a toral homeomorphism with uniquely defined, totally irrational rotation vector.) The aim of this note is to explore this concept some further. For instance, we provide an example which shows that the preceding statement does not hold in the non-conservative case. Further, we collect a number of observations concerning the case where the bounded mean motion property fails. In particular, …

Burnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms

A group $G$ is said to be periodic if for any $g\in G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $μ$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.

On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$

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

A group $Γ$ is said to be periodic if for any $g$ in $Γ$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $\mathrm{O}(3,\R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose …

Maximal Entropy Measures for Piecewise Affine Surface Homeomorphisms

We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected "good" returns to dispense with Markov partitions. We control these good returns throu…

Groups of piecewise linear homeomorphisms of flows

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…

关于区间上具有有限多个不动点的同胚群

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。

On periodic groups of homeomorphisms of the 2-dimensional sphere

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.

Existence of torsion-low maximal identity isotopies for area preserving surface homeomorphisms

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

We study the group of almost-periodic homeomorphisms of the real line. Our main result states that an action of a finitely generated group on the real line without global fixed point is conjugated to an almost-periodic action without almost fixed point. This is equivalent to saying that the action on the real line can be compactified to an action on a 1-dimensional lamination of a compact space, without global fixed point. As an application we give an alternative proof of Witte's theorem: an amenable left orderable group is locally indicable.

Invariant measures for Cantor dynamical systems

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.

Reconstructing maps out of groups

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}…

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。

Translation numbers define generators of $F_k^+\to {\text{\rm Homeo}_+}(\mathbb{S}^1)$

We consider a minimal action of a finitely generated semigroup by homeomorphisms of a circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov's theorem and its corollaries.

Symmetric random walks on Homeo+(R)

We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of "global stability at a finite distance": roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for t…

Pseudo-rotations of the open annulus

In this paper, we study pseudo-rotations of the open annulus, \emph{i.e.} conservative homeomorphisms of the open annulus whose rotation set is reduced to a single irrational number (the angle of the pseudo-rotation). We prove in particular that, for every pseudo-rotation $h$ of angle $ρ$, the rigid rotation of angle $ρ$ is in the closure of the conjugacy class of $h$. We also prove that pseudo-rotations are not persistent in $C^r$ topology for any $r\geq 0$.

C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

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…

Conjugacy of reversible cellular automata and one-head machines

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…

保守的平均扩张微分同胚的余扩张

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。

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。

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。

Chain groups of homeomorphisms of the interval

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…