Boris Kolev, Marie-Christine Peroueme
An orientation-preserving recurrent homeomorphism of the two-sphere which is not the identity is shown to admit exactly two fixed points. A recurrent homeomorphism of a compact surface with negative Euler characteristic is periodic.
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In this paper we show that the group of automorphisms of a non-recurrent tent map inverse limit is very simple by demonstrating that every homeomorphism of such a space is isotopic to a power of the induced shift homeomorphism.
Let $X$ be a compact metric space and let $f:X\rightarrow X$ be a homeomorphism on $X$. We show that if $f$ is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some symbolic system. Moreover, if $f$ is pointwise positively recurrent, then the subshift is semisimple; a counterexample is given to show the necessity of positive recurrence to ensure the semisimilicity.
Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points
本文给出平面保向同胚 Brouwer 不动点定理的简洁新证明，并引入非游荡点的推广概念 BP-链回归点，证明存在该点即可推出不动点。
We prove that the genus two surface admits a cw-expansive homeomorphism with a fixed point whose local stable set is not locally connected.
We exploit the techniques developed in [Le] to study N-expansive homeomorphisms on surfaces. We prove that when f is a 2-expansive homeomorphism defined on a compact boundaryless surface M without wandering points then f is expansive. This condition on the wandering set cannot be relaxed: we present an example of a 2-expansive homeomorphisms on the bitorus with wandering points that is not expansive.
In this article, we describe all the group morphisms from the group of orientation-preserving homeomorphisms of the circle to the group of homeomorphisms of the annulus or of the torus.
In this article we characterize monotone extensions of cw-expansive homeomorphisms of compact metric spaces. We study the topology of its quotient space in the case of a compact surface. These results are applied to prove that there are cw-expansive homeomorphisms of compact surfaces with infinitely many fixed points and empty wandering set. The examples are quotients of topological perturbations of pseudo-Anosov diffeomorphisms. We also show that there is a cw-expansive homeomorphism with the shadowing property of the 2-sphere.
Let S be a compact, oriented surface with negative Euler characteristic and let f be a homeomorphism of S that is isotopic to the identity. If there exists a periodic orbit with a non-zero rotation vector, then there exists a simple braid with the same rotation vector.
N-expansive homeomorphisms with the shadowing property
本文研究紧度量空间上具有阴影性质的n-扩张同胚，构造了非(n-1)-扩张且有无穷多链回归类的例子，并利用局部稳定集证明极限阴影性质，为拓扑混合n-扩张同胚的存在性问题提供方向。
We classify minimal sets of (closed and oriented) hyperbolic surface homeomorphisms by studying the connected components of their complement. This extends the classification given by F. Kwakkel, T.Jäger and A. Passeggi in the torus. The given classification is studied in the non-wandering setting and in light of the Nielsen-Thurston Theory.
We prove that if a homeomorphism of a closed orientable surface S has no wandering points and leaves invariant a compact, connected set K which contains no periodic points, then either K=S and S is a torus, or $K$ is the intersection of a decreasing sequence of annuli. A version for non-orientable surfaces is given.
We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple…
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.
We study the dynamics of Topologically Anosov homeomorphisms of non compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f:S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \R ^ 2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in a previous work.
Let $M$ be an orientable connected closed surface and $f$ be an $R$-closed homeomorphism on $M$ which is isotopic to identity. Then the suspension of $f$ satisfies one of the following condition: 1) the closure of each element of it is minimal and toral. 2) there is a minimal set which is not locally connected. Moreover, we show that any positive iteration of an $R$-closed homeomorphism on a compact metrizable space is $R$-closed.
Continuum-wise hyperbolic homeomorphisms on surfaces
本文研究曲面连续统双曲同胚的动力学，证明含有限多个脊的 $cw_F$-双曲同胚为 $cw_2$-双曲，并证明 $cw_3$-双曲性蕴含 $cw_2$-双曲性，同时揭示脊的孤立性。
Let S be a compact orientable surface and f be an element of the group Homeo_{0}(S) of homeomorphisms of S isotopic to the identity. Denote by F a lift of f to the universal cover of S. In this article, the following result is proved: if there exists a fundamental domain D of the universal cover of S such that the sequence (d_{n}log(d_{n})/n) converges to 0 where d_{n} is the diameter of F^{n}(D), then the homeomorphism f is a distortion element of the group Homeo_{0}(S).
Anomalous cw-expansive homeomorphisms on compact surfaces of higher genus
本文构造了高亏格紧致表面上的异常cw-扩张同胚，解决了Artigue提出的问题，并给出了非扩张同胚的例子。
The study of the dynamics of a surface homeomorphism in the neighbourhood of an isolated fixed point leads us to the following results. If the fixed point index is greater than 1, a family of attractive and repulsive petals is constructed, generalizing the Leau-Fatou flower theorem in complex dynamics. If the index is less than 1, we get a family of stable and unstable branches, generalizing the stable manifold theorem in hyperbolic dynamics.
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…
Given an orientation-preserving and area-preserving homeomorphism $f$ of the sphere, we prove that every point which is in the common boundary of three pairwise disjoint invariant open topological disks must be a fixed point. As an application, if $K$ is an invariant Wada type continuum, then $f^n|_K$ is the identity for some $n>0$. Another application is an elementary proof of the fact that invariant disks for a nonwandering homeomorphisms homotopic to the identity in an arbitrary surface are homotopically bounded if the fixed point set is inessential. The main results in this article are sel…
This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel's classification of zero entropy maps of $S^2$ for non-wandering homeomorphisms; we show that if $f$ is a Hamiltonian homeomorphism of the annulus, then the rotation set of $f$ is either a singleton or it co…
Generic surface homeomorphisms are almost continuum-wise expansive
本文证明了无边紧致曲面上cw-膨胀同胚在 $C^0$ 拓扑下的稠密性，并进一步证明一般同胚可被半共轭且满足连通性、不分隔性与直径约束的cw-膨胀同胚逼近。
In 1912 Brouwer published his translation theorem, which implies, for example, that an orientation preserving homeomorphism of the plane having a periodic point also has a fixed point. This theorem has given rise to a number of developments, leading among other things to Le Calvez's proof of the existence of a Brouwer foliation for surface homeomorphisms homotopic to identity. Recently, Le Calvez and Tal used this foliation to construct a forcing theory intrinsically topological which, like Brouwer's theorem, allows to deduce the existence of new orbits from certain dynamic properties of homeo…
Let S be a compact connected surface and let f be an element of the group Homeo\_0(S) of homeomorphisms of S isotopic to the identity. Denote by \tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be non-spreading if the sequence (d\_{n}/n) converges to 0, where d\_{n} is the diameter of \tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugate…
Surface homeomorphisms with big rotation set
本文研究了高亏格表面同胚映射在大旋转集情况下的结构定理及性质，包括旋转集的凸性、不变开放集的存在性以及对Boyland猜想的解答。
Let p be a saddle fixed point for an orientation-preserving surface diffeomorphism f admitting a homoclinic point q. Let V be an open 2-cell bounded by a simple loop formed by two arcs joining p to q lying respectively in the stable and unstable curves at p. It is shown that f|V has fixed point index 1 or 2 depending only on the geometry of V near p.
We study the dynamics of surface homeomorphisms around isolated fixed points whose Poincaré-Lefschetz index is not equal to 1. We construct a new conjugacy invariant, which is a cyclic word on the alphabet $\{\ua, \ra, \da, \la\}$. This invariant is a refinement of the P.-L. index. It can be seen as a canonical decomposition of the dynamics into a finite number of sectors of hyperbolic, elliptic or indifferent type. The contribution of each type of sector to the P.-L. index is respectively -1/2, $+1/2$ and 0. The construction of the invariant implies the existence of some canonical dynamical s…
A rotational hyperbolic theory for surface homeomorphisms
本文提出了一种针对表面同胚的旋转双曲理论，通过旋转同宿类和异宿连接的分析，揭示了同胚的旋转行为结构及其应用。
Let $f$ be an $R$-closed homeomorphism on a connected orientable closed surface $M$. In this paper, we show that If $M$ has genus more than one, then each minimal set is either a periodic orbit or an extension of a Cantor set. If $M = \mathbb{T}^2$ and $f$ is neither minimal nor periodic, then either each minimal set is finite disjoint union of essential circloids or there is a minimal set which is an extension of a Cantor set. If $M = \mathbb{S}^2$ and $f$ is not periodic but orientation-preserving (resp. reversing), then the minimal sets of $f$ (resp. $f^2$) are exactly two fixed points and …
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…
In this work we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy and we obtain several applications. For homeomorphisms of the open annulus $\mathbb{A}$ with zero topological entropy, we show that rotatio…
Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_0$ with Lefschetz index one. Le Roux conjectured that $z_0$ is accumulated by periodic orbits. In this article, we will approach Le Roux's conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_0$ and if the area of $M$ is finite, then $z_0$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_0$ is the limit in weak-star topology of a sequence of invaria…
A note on weak conjugacy for homeomorphisms of surfaces
本文研究曲面上同位于恒等映射的同胚群中的弱共轭关系，并给出相关结论。
A generalization of the Brouwer plane translation theorem
该研究通过证明含拓扑链回归点的平面保向同胚必有不动点，推广了Brouwer平面平移定理。
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…
We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set is totally disconnected, then f is topologically conjugate to a conformal transformation.
In this article, the main goal is to give a dynamical point of view of Floer homology barcodes for Hamiltonian homeomorphisms of surfaces. More specifically, we describe a way to construct barcodes for Hamiltonian homeomorphisms of surfaces from graphs. We will define graphes associated to maximal isotopies of a Hamiltonian homeomorphism using Le Calvez's positively transverse foliation theory and to those graphs we will associate barcodes. In particular, we will prove that for the simplest cases, our constructions coincide with the Floer Homology barcodes.
One of the main problems of the theory of dynamical systems is the determination of the existence of periodic orbits of a self-map and more generally, the structure of the set of periods. Define the minimum period of a class os self-maps of a fixed set as the minimum of the positive integers such that each map in the class has a periodic point whose period is at most this number. The problem of the determination of the minimum period of the classes of homeomorphisms of closed surfaces was completely solved, in successive steps, from 1910 to 1996. The aim of our work is, for each compact, conne…
Generalized foliations for homeomorphisms isotopic to a pseudo-Anosov homeomorphism: a geometric realization of a result by Fathi
本研究给出Fathi一个结果的新几何证明，证明闭曲面上同痕于伪Anosov同胚的同胚都存在与伪Anosov同胚叶状结构性质相似的稳定、不稳定不变划分。
We show that any pseudo-Anosov map that is a lift of pseudo-Anosov homeomorphism of a nonorientable surface has vanishing SAF invariant. We also provide a criterion to certify that a pseudo-Anosov map is not such a lift.
We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surface. For genus g=2 to 5, the mimimum dilatation is the smallest Salem number for polynomials of degree 2g.
We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surface of genus g tends to zero at rate 1/g (as g goes to infinity). We also show that our uniform lower bound sqrt{2} is sharp. More precisely the least dilatation of a pseudo-Anosov on a genus g>1 translation surface in a hyperelliptic component belongs to the interval ]sqrt{2},sqr…
We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of non-parabolic elements in the periodic field of certain translation surfaces.
Dold coefficients of quasi-unipotent homeomorphisms of orientable surfaces
本文根据 Dold 系数描述了具有有界 Lefschetz 数的曲面同胚（拟单映射），探讨了最小化曲面亏格以实现给定代数周期集合的问题，并计算列出了小亏格可定向曲面的所有可能 Dold 系数与代数周期。
On homeomorphisms of three-dimensional manifolds with pseudo-Anosov attractors and repellers
本文研究带有限曲面吸引/排斥子的三维流形保定向同胚，证明一类伪阿诺索夫型同胚的环境Ω-共轭性及其中中心化子的构成性质。
In this work we study homeomorphisms of closed orientable surfaces homotopic to the identity, focusing on the existence of non-contractible periodic orbits. We show that, if $g$ is such a homeomorphism, and if $\hat g$ is its lift to the universal covering of $S$ that commutes with the deck transformations, then one of the following three conditions must be satisfied: (1) The set of fixed points for $\hat g$ projects to a closed subset $F$ which contains an essential continuum, (2) $g$ has non-contratible periodic points of every sufficiently large period, or (3) there exists an uniform bound …
We give a new definition for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then, we apply this concept to prove that there exists a C0-dense subset of the set of the area-preserving homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero new Lyapunov exponents for Lebesgue almost every point. Finally, we prove that the function that associates an area-preserving h…
Let $BS(1,n)= <a,b : a b a ^{-1} = b ^n>$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces with (pseudo-)Anosov elements. That is, we consider a closed surface $S$, and homeomorphisms $f, h: S \to S$ such that $h f h^{-1} = f^n$, for some $ n\geq 2$. It is known that $f$ (or some power of $f$) must be homotopic to the identity. Suppose that $h$ is pseudo-Anosov with stretch factor $λ>1$. We show that $<f,h>$ is not a faithful representation of $BS(1, n)$ if $λ> n$. Moreover, we show that there are no faithful …
Essential factors for surface homeomorphisms
针对亏格至少为1的闭可定向曲面上同痕于伪阿诺索夫自同胚的自同胚，通过坍缩万有覆盖中迭代直径一致有界的极大连续统构造单调因子，该因子具强混沌特征且可保面积，其极大熵测度为全支撑伯努利测度。
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 …
Suppose that a closed surface $S \subseteq \mathbb{R}^3$ is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points $W$ is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. Using this result and a modification of the classical construction of a wild sphere due to Antoine we show that there exist uncountably many different $2$--spheres in $\mathbb{R}^3$ none of which can be realized as an attractor for a homeomorphism.
We exhibit low-dilatation families of surface homeomorphisms among monodromies of Lorenz knots.
We consider a self-homeomorphism h of some surface S. A subset F of the fixed point set of h is said to be unlinked if there is an isotopy from the identity to h that fixes every point of F. With Le Calvez' transverse foliations theory in mind, we prove the existence of unlinked sets that are maximal with respect to inclusion. As a byproduct, we prove the arcwise connectedness of the space of homeomorphisms of the two dimensional sphere that preserves the orientation and pointwise fix some given closed connected set F.
Let $M$ be a closed surface. By $\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says that for any closed surface $M$ of genus $\g \ge 2$, there is no homomorphic section $\E:\MC(M) \to \Homeo(M)$ of the standard projection map $\Proj:\Homeo(M) \to \MC(M)$.
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。
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。
Periodic points of rational area-preserving homeomorphisms
本文证明了亏格至少为二的紧曲面上具有有理旋转方向的保面积同胚要么是恒等映射，要么具有无界最小周期的周期点，回答了Seyfaddini的问题，并讨论了多种变体。
A filling curve $γ$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(γ)$ of $S$ via the process of "point-pushing along $γ$." We consider the relationship between the self-intersection number $i(γ)$ of $γ$ and the dilatation of $P(γ)$; our main result is that the dilatation is bounded between $(i(γ)+1)^{1/5}$ and $9^{i(γ)}$. We also bound the least dilatation of any pseudo-Anosov in the point-pushing subgroup of a closed surface and prove that this number tends to infinity with genus. Lastly, we investigate the minimal entropy of any pseudo-Anosov homeomorphism obtained by pu…
We study the problem of existence of a periodic point in the boundary of an invariant domain for a surface homeomorphism. In the area-preserving setting, a complete classification is given in terms of rationality of Carathéordory's prime ends rotation number, similar to Poincaré's theory for circle homeomorphisms. In particular, we prove the converse of a classic result of Cartwright and Littlewood. This has a number of consequences for generic area preserving surface diffeomorphisms. For instance, we extend previous results of J. Mather on the boundary of invariant open sets for $C^r$-generic…
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