Flowability of plane homeomorphisms

具有 BP-链回归点的平面保向同胚

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。

There is no minimal action of Z^2 on the plane

In this paper it is proved that there is no minimal action (i.e. every orbit is dense) of Z^2 on the plane. The proof uses the non-existence of minimal homeomorphisms on the infinite annulus (Le Calvez-Yoccoz's theorem), and the theory of Brouwer homeomorphisms.

Rotation numbers for planar attractors of equivariant homeomorphisms

Given an integer m>1 we consider Zm-equivariant and orientation preserving homeomorphisms in the plane with an asymptotically stable fixed point at the origin. We present examples without periodic points and having some complicated dynamical features. The key is a preliminary construction of Zm-equivariant Denjoy maps of the circle.

Existence de points fixes enlacés à une orbite périodique d'un homéomorphisme du plan

Let f be an orientation-preserving homeomorphism of the plane such that f-Id is contracting. Under these hypotheses, we establish the existence, for every periodic orbit, of a fixed point which has nonzero linking number with this periodic orbit.

Generic homeomorphisms with shadowing of one-dimensional continua

In this article we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.

平面同胚的链接轨道与Gambaudo-Kolev定理

Linked orbits of homeomorphisms of the plane and Gambaudo-Kolev Theorem

本文证明对任意保定向平面同胚，其任一有界轨道都与某个不动点在Gambaudo意义下链接，从而支持Gambaudo-Kolev定理所描述的轨道与不动点之间的拓扑纠缠关系。

Bifurcations, robustness and shape theory of attractors of discrete dynamical systems

We study in this paper global properties, mainly of topological nature, of attractors of discrete dynamical systems. We consider the Andronov-Hopf bifurcation for homeomorphisms of the plane and establish some robustness properties for attractors of such homeomorphisms. We also give relations between attractors of flows and quasi-attractors of homeomorphisms in Rn. Finally, we give a result on the shape (in the sense of Borsuk) of invariant sets of IFSs on the plane, and make some remarks about the recent theory of Conley attractors for IFS.

布劳威尔平面平移定理的一个推广

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。

环面上无周期点 distal 同胚的结构

The structure of periodic point free distal homeomorphisms on the annulus

本文证明了环面上保持边界分支且无周期点的 distal 同胚可连续分解为不变圆周，且各圆周上的限制共享同一无理旋转数并按包含关系线性有序。

Stably non-synchronizable maps of the plane

Pecora and Carroll presented a notion of synchronization where an (n-1)-dimensional nonautonomous system is constructed from a given $n$-dimensional dynamical system by imposing the evolution of one coordinate. They noticed that the resulting dynamics may be contracting even if the original dynamics are not. It is easy to construct flows or maps such that no coordinate has synchronizing properties, but this cannot be done in an open set of linear maps or flows in $\R^n$, $n\geq 2$. In this paper we give examples of real analytic homeomorphisms of $\R^2$ such that the non-synchronizability is s…

有界轨道猜想反例的一种新构造

A new construction of counterexamples to the bounded orbit conjecture

本文针对有界轨道猜想，通过在正方形上构造具有特定 $\omega$ 极限集和 $\alpha$ 极限集的反向同胚，并借助半共轭将其诱导为平面同胚，从而给出该猜想更易理解的反例构造。

Simplicity of the group of compactly supported area preserving homeomorphisms of the open disc and fragmentation of symplectic diffeomorphisms

In 1980, Albert Fathi asked whether the group of area-preserving homeomorphisms of the 2-disc that are the identity near the boundary is a simple group. In this paper, we show that the simplicity of this group is equivalent to the following fragmentation property in the group of compactly supported, area preserving diffeomorphisms of the plane: there exists a constant m such that every element supported on a disc D is the product of at most m elements supported on topological discs whose area are half the area of D.

Pruning fronts and the formation of horseshoes

Let f:E -> E be a homeomorphism of the plane E. We define open sets P, called {\em pruning fronts} after the work of Cvitanović, for which it is possible to construct an isotopy H: E x [0,1] -> E with open support contained in the union of f^{n}(P), such that H(*,0)=f(*) and H(*,1)=f_P(*), where f_P is a homeomorphism under which every point of P is wandering. Applying this construction with f being Smale's horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behaviour. This family is a 2-dim…

Bounded orbits and global fixed points for groups acting on the plane

Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g, h in G, and the action of G satisfies a nonwandering hypothesis, then the action has a global fixed point. In particular, any group of measure-preserving orientation preserving homeomorphisms of the plane with uniformly bounded orbits has a global fixed point. The constant k/\sqrt{3} is sharp. We also show that a group acting on the plane with orbits bounded as above is left orde…

关于同伦布劳威尔理论的分层观点

A foliated viewpoint on homotopy Brouwer theory

本文提出一个统一框架，使Le Calvez的分层方法能够恢复并改进Handel理论的经典结果。

Fixed points for nilpotent actions on the plane and the Cartwright-Littlewood theorem

The goal of this paper is proving the existence and then localizing global fixed points for nilpotent groups generated by homeomorphisms of the plane satisfying a certain Lipschitz condition. The condition is inspired in a classical result of Bonatti for commuting diffeomorphisms of the 2-sphere and in particular it is satisfied by diffeomorphisms, not necessarily of class $C^{1}$, whose linear part at every point is uniformly close to the identity. In this same setting we prove a version of the Cartwright-Littlewood theorem, obtaining fixed points in any continuum preserved by a nilpotent act…

Extension of boundary homeomorphisms to mappings of finite distortion

We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half-plane or the disk, respectively. Moreover, we can ensure that the quasiconformal dilatation of the extension satisfies certain integrability conditions, such as $p$-integrability or exponential integrability. Mappings satisfying the latter integrability condition are also known as David homeomorphisms. Our extension operator is the same as the one used by Beurling and Ahlfors in their celebrated work. We prove an optimal bound for th…

One some planar Baumslag-Solitar actions

Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group for $n \geq 2$. We study representations of $BS(1, n)$ on the plane by orientation preserving homeomorphisms, assuming that $a$ acts as a linear map and $b$ as a map with bounded displacement. We find that the possibilities for a faithful action depend greatly on the Jordan canonical form of the map $h$ defined by the action of $a$. In case $h$ is diagonalizable over $\mathbb R$, we shall give examples or prove rigidity theorems depending on the eigenvalues. We also show some rigidity in the cases whe…

Conjugacy of free mappings embedded in a flow

In this paper we study free mappings of the plane, that is orientation preserving fixed point free homeomorphisms of $\mathbb{R}^2$. We provide a necessary and sufficient condition under which two free mappings of the plane that are embedded in flows are conjugate to one another using Haefliger-Reeb theory of plane foliations.

平面映射的拟中间值定理与外翼弧定理

Quasi-intermediate value theorem and outflanking arc theorem for plane maps

本文通过圆盘边界上的限制条件提出平面映射的拟中间值定理，并据此建立外翼弧不动点定理，推广了 Brouwer 关于保向同胚周期点蕴含不动点的经典结果。

On the struture of homeomorphims of the open annulus

Let $h$ be a without fixed point lift to the plane of a homeomorphism of the open annulus isotopic to the identity and without wandering point. We show that $h$ admits a $h$-invariant dense open set $O$ on which it is conjugate to a translation and we study the action of $h$ on the compactly connected components of the closed and without interior set ${\bf R}^2 \setminus O$.

Free planar actions of the Klein bottle group

We describe the structure of the free actions of the Klein bottle group by orientation preserving homeomorphisms of the plane. This group is generated by two elements $a,b$, where the conjugate of $b$ by $a$ equals the inverse of $b$. The main result is that $a$ must act properly discontinuously, while $b$ cannot act properly discontinuously. As a corollary, we describe some torsion free groups that cannot act freely on the plane. We also find some properties which are reminiscent of Brouwer theory for the infinite cyclic group $Z$, in particular that every free action is virtually wandering.

Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles

This paper studies the behavior under iteration of the maps T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^s that map rays from the origin into rays from the origin. This paper shows the existence of special parameter values where T_{ab} has every nonzero orbit an invariant circle with irrational rotation …

Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers

This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of piecewise-linear maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^2 that map rays from the origin into rays from the origin. The action on rays defines a map S_{ab} of the circle, which has a well-defined rotation number. This paper characterizes …

Total rigidity of generic quadratic vector fields

We consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on $\mathbb C P^2$ are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the leaves and in $\mathbb C P^2$ and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a…

Action on the circle at infinity of foliations of ${\mathbb R}^2 $

This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on ${\mathbb R}^2 $ preserving one foliation or two transverse foliations. In particular the action on the circle at infinit…

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)- y, x) of the plane R^2, in which F_{ab}(x)= ax if x>0 and bx if x<0. These maps are area-preserving homeomorphisms of the plane that map rays from the origin into rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This difference equation can be written in an eigenvalue form for a nonlinear difference operator of Schrodinger type, in which μ= 1/2(a-b) is viewed as fixed and the energy E=2- 1/2(a+b). The paper…

Circular groups, planar groups, and the Euler class

We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the Euler class of C^infty diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus >1 admits a C^infty action with arbitrary Euler class. On the other hand, we show that Z oplus Z actions satisfy a homological rigidity property: every orientatio…