Katrin Gelfert, Örjan Stenflo
We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.
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Pivoting technique for the circle homeomorphism group
本文把枢轴技术推广到圆同胚群，并据此为概率性Tits二择性和指数同步提供了新证明。
Singular measures of circle homeomorphisms with two break points
设$T_{f}$是具有两个断点$a_{b},c_{b}$和无理旋转数$\varrho_{f}$的圆同胚.设其提升$f$的导数$Df$在集合$S^{1}\backslash\{a_{b},c_{b}\}$的每一个连通区间上绝对连续，$DlogDf \in L^{1}$与$ Df $在断点处的跳跃比的乘积非平凡，即$\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\neq1$ .证明了唯一的$T_{f}$不变概率测度$μ_{f}$关于$S^{1}$上的Lebesgue测度$l$是奇异的.
Actions of the group of homeomorphisms of the circle on surfaces
在这篇文章中，我们描述了所有的群态射，从保持方向的同胚群的圆的环或环面的同胚群。
Invariant measures for quasiperiodically forced circle homeomorphisms
研究了拟周期强迫圆同胚的不变遍历测度，得到了要么系统是唯一遍历的，要么任何这样的测度都与某个不变重图相关联。
Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.
Reversibility in the group of homeomorphisms of the circle
我们确定那些元素的同胚群的圆，可以表示为一个复合的两个对合。
We provide a classification of random orientation-preserving homeomorphisms of $\mathbb{S}^1$, up to topological conjugacy of the random dynamical systems generated by i.i.d. iterates of the random homeomorphism. This classification covers all random circle homeomorphisms for which the noise space is a connected Polish space and an additional extremely weak condition is satisfied.
Polynomial entropy of induced maps of circle and interval homeomorphisms
本文计算了具有有限多个非游荡点的区间或圆周同胚在超空间上诱导映射的多项式熵。
Orbit structure for groups of homeomorphisms of the circle
在本文中，我们将改进Sacksteder的定理的情况下，有一个有限的轨道集的情况下，保持方向同胚的圆群。我们将给出一个分类的拓扑可能性的轨道的点的圆，随着例子。
We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction expansion, existence of a continuous solution of the corresponding cohomological equation and displacement sequence along the orbit.
Minimal sets for torus homeomorphisms with an irrational circle factor
本文研究具有无理圆周因子的环面同胚，在纤维为Jordan曲线等条件下证明极小集唯一性，并构造具有多个极小Cantor集的例子。
On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms
本文证明具有无理旋转数和零平均非线性的充分正则分段光滑圆同胚的唯一不变测度通常具有零Hausdorff维数，并借助广义区间交换变换的组合旋转数概念编码该一般性条件。
Rigidity for piecewise smooth homeomorphisms on the circle
我们找到了圆的两个分段C^{2+ν}同胚f和g是C^1共轭的条件。除了对映射的组合学的限制（我们假设映射有界组合学）和对f和g不可微的点的单侧导数的必要条件之外，我们还假设f和g的零平均非线性。
Renormalization for piecewise smooth homeomorphisms on the circle
本文研究了作用在圆上分段光滑同胚上的重整化算子，其实质是研究亏格为1的广义区间交换映射的Rauzy-Veech重整化.特别是，我们证明了零平均非线性和满足一定的光滑性和组合假设的重正化的映射收敛到一组分段仿射区间交换映射。
Polynomial entropy for the circle homeomorphisms and for $C^1$ nonvanishing vector fields on $\T^2$
证明了当圆的保向同胚不共轭于旋转时，其多项式熵等于1，否则为0。在第二部分中，我们证明了与$C^1$非零向量场相关联的二维环面上的流的多项式熵小于$1$。此外，我们证明，当流具有周期轨道的多项式熵等于1，除非它是共轭的旋转（在最后一种情况下，多项式熵为零）。
We study renormalizations of piecewise smooth homeomorphisms on the circle, by considering such maps as generalized interval exchange maps of genus one. Suppose that $Df$ is absolutely continuous on each interval of continuity and $D\ln{Df}\in \mathbb{L}_{p}$ for some $p>1$. We prove, that under certain combinatorial assumptions on $f_{1}$ and $f_{2}$, corresponding renormalizations approach to each other in $C^{1+L_{1}}$-norm.
The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+α}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-ν}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from th…
We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic piecewise affine circle homeomorphism with two break points -with respect to the Lebesgue measure- is Morse-Smale.
Periodicity and Rotation Number for Random Circle Homeomorphisms
研究随机动力系统中圆同胚的周期性和旋转数，证明随机周期点存在与旋转数有理性的关系，并指出确定性与随机性下的固定点与旋转数整数性的差异。
Let $f$ be an orientation preserving homeomorphisms on the circle with several break points, that is, its derivative $Df$ has jump discontinuities at these points. We study Rauzy-Veech renormalizations of piecewise smooth circle homeomorphisms, by considering such maps as generalized interval exchange maps with genus one. Suppose that $Df$ is absolutely continuous on the each interval of continuity and $D\ln{Df}\in \mathbb{L}_{p}$ for some $p>1$. We prove that, under certain combinatorial assumptions on $f$, renormalizations $R^{n}(f)$ are approximated by piecewise Möbus functions in $C^{1+L_{…
In this work we study the following realization problem: given a piecewise homeomorphism $Φ: S^1 \rightarrow S^1$, which geometrical and dynamical conditions on $Φ$ are sufficient to realize it as a Bowen-Series-like map associated to a surface group, seen as a discrete subgroup of ${\rm Homeo}^+ (S^1)$?
Anderson Localization for Schrödinger Operators with Monotone Potentials over Circle Homeomorphisms
本文在旋转数条件弱于丢番图条件的保向圆同胚上，针对单调双李普希茨位势的薛定谔算子，建立了获得纯点谱的方案，并证明在大耦合常数下局域化具有一致性。
Displacement sequence of an orientation preserving circle homeomorphism
我们给出了一个完整的描述的位移序列的行为$η_n(z)=Φ^n(x) - Φ^{n-1}(x) \ \rmod \ 1$，$z=\exp(2π\rmi x)$，沿轨迹$\{φ^{n}(z)\}$，其中$φ$是一个方向保持圆同胚和$Φ:\mathbb{R} \to \mathbb{R}$其升降机。如果旋转数$\varrho(φ)=\frac{p}{q}$是有理数，则$η_n(z)$是半周期$q$的渐近周期。如果我们承认某些点是向后迭代的，而不是对所有$z$只进行向前迭代，那么$z$中收敛到周期序列是一致的。如果$\varrho(φ) \notin \mathbb{Q}$，则$η_n(z)$的值在一个集合中是稠密的，该集合依赖于将$φ$与$\varrho(φ)$的旋转进行（半）共轭的映射$γ$，并且该映射是位移分布的支撑。我们提供了一个有效的公式的位移分布，如果$φ$是$C^1$ -同胚，并显示近似的位移分布的样本位移测量沿任何其他圆同胚的轨迹是足够接近的初始同胚$φ$。最后，我们证明了即使对于无理旋转数$\varrho$，位移序列也具有一定的正则性。
Towards a classification for quasiperiodically forced circle homeomorphisms
庞加莱的分类动态同胚的循环是一个最早的，但仍然是一个最优雅的，分类结果在动力系统。在这里，我们推广到拟周期强迫圆同胚，这一直是近年来相当感兴趣的主题。赫尔曼在20年前就已经证明，在准周期强迫的情况下，所有的轨道都存在唯一的旋转数。然而，与非强迫的情况下，不存在先验界的平均旋转的偏差。这在尝试的分类中起着重要的作用，事实上，如果这种偏差是有界的，则我们将系统定义为正则的，否则将其定义为不规则的。对于经常的情况下，我们证明了一个密切的类似庞加莱的结果：如果旋转数是合理的有关的旋转速度的基础上，然后存在一个不变的带（一个适当的模拟固定或周期点在这种情况下），否则系统是半共轭的无理平移的环面。在不规则的情况下，这两种选择都不能发生，我们表明，动力学总是拓扑传递的。
David extension of circle homeomorphisms, welding, mating, and removability
本文通过圆周同胚的David扩张结果，在抛物与双曲周期点之间建立共轭联系，并据此证明新型焊接同胚存在性、反有理映射与反射群间的动力联系、相关Julia集与极限集的共形可移除性，以及反多项式与项链反射群的配对和Suffridge多项式的存在性。
Parameterized family of annular homeomorphisms with pseudo-circle attractors
本文构造了环面同胚的参数族，其吸引子均为伪圆，外部素数端点旋转数随参数在[0,1/2]连续变化，并证明保持Lebesgue测度的圆映射的逆极限在一般情形下为Bing伪圆。
On conjugations of circle homeomorphisms with two break points
设$f_i\in C^{2+α}(S^1\setminus \{a_i,b_i\}), α>0, i=1,2$是具有两个断点$a_i,b_i$的圆同胚，即导数$f_i$中的不连续性，具有相同的无理旋转数$rho$和$μ_1([a_1,b_1])= μ_2([a_2,b_2])$，其中$μ_i$是$f_i$的不变测度。假设$Df_1$和$Df_2$的跳跃比的乘积不重合，即$\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times \frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times \frac{Df_2(b_2-0)}{Df_2(b_2+0)}$。则共轭$f_1$和$f_2$的映射$ψ$是奇异函数，即它在$S^1$上是连续的，但$Dψ= 0$ a. e.关于勒贝格测度
Minimal sets of non-resonant torus homeomorphisms
如H所知。Poincare证明了无周期点的保向圆同胚要么是极小的，要么没有稠密轨道，且每个轨道都在唯一极小集上聚集。在第一种情况下，最小集是圆，在后一种情况下，是康托尔集。在本文中，我们研究一个二维模拟这个经典的结果：我们分类的最小集的非共振环面同胚，也就是说，环面同胚同位素的身份旋转集是一个点，合理独立的非理性坐标。
We show that a generic quasiperiodically forced circle homeomorphism is mode-locked: the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system. As a consequence, this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition, the graph of the rotation number as a function of the parameter is a devil's staircase.
We prove that if $Γ$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line.
Random actions of homeomorphisms of Cantor sets embedded in a line and Tits alternative
该研究类比Margulis关于圆周同胚群的Tits二择性结论，证明了实数集紧子集的局部单调同胚群也满足类似的二择性质，拓展了同胚群领域的相关理论结果。
Let f be a class P -homeomorphism of the circle. We prove that there exists a piecewise analytic homeomorphism that conjugate f to a one-class P with prescribed break points lying on pairwise distinct orbits. As a consequence, we give a sharp estimate for the smoothness of a conjugation of class P -homeomorphism f of the circle satisfying the (D)-property (i.e. the product of f-jumps in the break points contained in a same orbit is trivial), to diffeomorphism. When f does not satisfy the (D)-property the conjugating homeomorphism is never piecewise C^1 and even more it is not absolutely contin…
Classical results by Poincaré and Denjoy show that two orientation-preserving $C^2$ diffeomorphisms of the circle are topologically conjugate if and only if they have the same rotation number. We show that there is no possibility of getting such a complete numerical Borel invariant for the conjugacy relation of orientation-preserving circle diffeomorphisms by homeomorphisms with higher degree of regularity. For instance, we consider conjugacy by Hölder homeomorphisms or by $C^k$-diffeomorphisms with $k\in \mathbb{Z}^+ \cup \{\infty\}$. The proof combines techniques from Descriptive Set Theory …
Hyperbolicity of renormalization of circle maps with a break-type singularity
研究了具有断点的圆同胚的重正化算子，证明了它具有双曲马蹄吸引子。
Polynomial entropy of induced homeomorphisms on $C_k(\mathbb{S}^1)$ and $C_k([0,1])$
本文研究了圆同胚诱导映射在$C(\mathbb{S}^1)$上的多项式熵计算问题。
We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
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。
Let $f$ and $g$ be two class $P$-homeomorphisms of the circle $S^{1}$ with break points singularities. Assume that the derivatives $\textrm{Df}$ and $\textrm{Dg}$ are absolutely continuous on every continuity interval of $\textrm{Df}$ and $\textrm{Dg}$ respectively. Denote by $C(f)$ the set of break points of $f$. For $c\in S^{1}$, denote by $π_{s, O_{f}(c)}(f)$ the product of $f-$ jumps in break points lying to the $f-$ orbit of $c$ and by $\textrm{SO}(f) = \{O_{f}(c):~c \in C(f)~\textrm{and}~π_{s, O_{f}(c)}(f)\neq 1\}$, called the set of singular $f$-orbits. The maps $f$ and $g$ are called b…
Structure theorems for subgroups of homeomorphisms groups
在这篇半临时性的论文中，我们研究了不允许非交换自由子群的圆S^1的保向同胚群的集合A。我们利用圆同胚的经典结果和初等动力学方法推导出关于A中群的各种新老结果。在已知的结果中，我们包括了Beklaryan和Malylyan的一族结果，并给出了Margulis定理的一个新证明。我们的主要新结果包括R的可解子群的详细分类。Thompson群T .
Periodic point free homeomorphisms and irrational rotation factors
本文完整刻画了二维环面上以无理旋转为因子的无周期点同胚，指出环状性与非游荡集几何是唯一障碍，并推广了Jäger的结果。
$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval
研究平坦段边界奇点次数不同的圆周映射类，证明其拓扑共轭若为双Lipschitz同胚则必为$C^1$微分同胚，从而两类等价。
Dynamical models for some torus homeomorphisms
众所周知，由H.庞加莱允许从拓扑的角度完全理解这样一个地图的动态。本文收集了与$\T^2$同胚的旋转集相关的拓扑动力学模型存在性的一些结果。
Singular measures in circle dynamics
临界圆同胚有一个关于勒贝格测度的不变测度全奇异.证明了不变测度的奇点是Holder型的。不变测度的豪斯多夫维数小于1但大于0。
In this paper we present a generalization of Poincaré's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements, \emph{à la} Poincaré and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between Rotation Theory and entropy for thes…
We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.
Let $f_{i},i=1,2$ be orientation preserving circle homeomorphisms with a finite number of break points, at which the first derivatives $Df_{i}$ have jumps, and with identical irrational rotation number $ρ=ρ_{f_{1}}=ρ_{f_{2}}.$ The jump ratio of $f_{i}$ at the break point $b$ is denoted by $σ_{f_{i}}(b)$, i.e. $σ_{f_{i}}(b):=\frac{Df_{i}(b-0)}{Df_{i}(b+0)}$. Denote by $σ_{f_{i}}, i=1,2,$ the total jump ratio given by the product over all break points $b$ of the jump ratios $σ_{f_{i}}(b)$ of $f_{i}$. We prove, that for circle homeomorphisms $f_{i}, i=1,2$, which are $C^{2+\varepsilon}, \varepsil…
Circle maps and C*-algebras
本文考虑由圆周上的连续分段单调映射构造C* 代数-代数，它推广了同胚的交叉积构造，更一般地推广了Renault，Deaconu和Anantharaman-Delaroche关于局部同胚的构造.假设映射是满射而不是局部内射，给出了C* 代数单的充要条件，并证明了它是Kirchberg代数.我们提供了计算K-理论群的工具，并将其转化为马尔可夫映射的算法方法。
Modifying Hall's idea in "A C^{\infty} Denjoy counterexample" we construct an example of homeomorphism of the circle which is a Denjoy counterexample (i.e. it is not conjugated to a rotation) and which is a C^{\infty}-diffeomorphism everywhere except in a flat half-critical point.
Denjoy constructions for fibred homeomorphisms of the torus
我们构造了不同类型的具有传递但非极小动力学的拟周期强迫圆同胚。关于最近的庞加莱分类这类映射的Jaeger-斯塔克，我们证明了传递，但非最小的行为可以发生在每一个不同的情况下。这填补了拓扑分类中的最后一个空白。实际上，我们可以得到一些具有相当复杂极小集的传递拟周期强迫圆同胚。例如，我们表明，在我们构造的一些例子中，唯一的最小集是一个康托尔集，它与每个垂直纤维的交集是不可数的，无处密集（但可能包含孤立点）。我们还证明了最小集的后一种不可能发生的动力学是由一个准周期SL（2，R）-上圈的投射作用。更精确地说，我们表明，准周期SL（2，R）-上圈，任何最小严格子集的环面要么是一个联盟的1000多个连续曲线，或包含最多两个点上的一般纤维。
A method via the KAM technique is introduced to study the existence of invariant tori and quasiperiodic solutions for impulsive Duffing-type equations with time period 1. Basing on several planar symplectic homeomorphisms and some estimates of impulsive perturbations under each symplectic homeomorphisms, we prove via the Moser's twist theorem the boundedness (Lagrange stability) and the existence of an invariant circle for the equation with area-preserving impulsive terms. And this invariant circle having any rotation number $ω>ω_0$ with some $ω_0>0$, so we obtain also that the solutions start…
Common Limits of Fibonacci Circle Maps
我们证明了临界指数趋于\infty的极限存在于黄金平均旋转数和Fibonacci圆覆盖的临界圆同胚中。而且，它们是相同的。极限映射在平坦的临界点不是解析的，但具有非平凡的复杂动力学。
Universal minimal flows of the homeomorphism groups of pseudo-solenoids are non-metrizable
本文证明所有伪螺线管（包括伪圆）的同胚群的泛最小流均不可度量化。
We show the chess billiard map, which was introduced in [HM] in order to study a generalization of the $n$-Queens problem in chess, is a circle homeomorphism. We give a survey of some of the known results on circle homeomorphisms, and apply them to this map. We prove a number of new results which give answers to some of the open questions posed in [HM].
Piecewise linear circle maps and conjugation to rigid rational rotations
本文研究分段线性圆周同胚与有理旋转数刚性旋转的共轭准则，通过两类映射验证结论，揭示其与无理旋转数情形的对应性及自然映射族中无模锁定的线性标度特性。
Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose ro…
Bounded vertical deviations and irrational circle factors in Dehn Twist classes
本文证明与具有无理有界垂直偏差的Dehn扭转同伦的环面同胚必有无理圆周因子，去掉小游荡域假设，建立刚性性质。
A piecewise linear homeomorphism of the circle which is periodic under renormalization
本研究构造了一个斜率为2/3幂次、旋转数为√2-1且重整化后周期的圆周分段线性同胚，否定了Calegari的相关问题，证明F_{2/3}无法嵌入F，并给出其变体问题的肯定性证据与计算观测结果。
We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the circle. We provide new examples of iterated functions systems where this behavior persists under perturbation of the generators.
Complex a priori bounds for multicritical circle maps with bounded type rotation number
本文研究了具有多个临界点和有界型旋转数的圆同胚，证明了其复先验界，并得出具有相同有界型旋转数的双三次圆映射具有 $C^{1+α}$ 刚性的结论。
Toral or non locally connected minimal sets for suspensions of $R$-closed surface homeomorphisms
设$M$是可定向连通闭曲面，$f$是$M$上的$R$ -闭同胚，它与恒等式是同构的.则$f$的悬置满足下列条件之一：1）它的每个元素的闭包是极小的且是环面的。2)存在一个非局部连通的极小集。证明了紧致可度量化空间上的$R$ -闭同胚的任何正迭代都是$R$ -闭的.
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