Infinite genus surfaces and irrational polygonal billiards
Ferran Valdez
我们证明了与无理多边形桌子上的台球游戏相关联的自然不变曲面与尼斯湖水怪同胚，即唯一可定向的具有恰好一端的无限亏格拓扑实曲面。
We prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one end.
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Polygonal billiards with one side scattering
研究了正方形台球桌上单面垂直镜面散射的多边形台球。我们将这些台球的轨迹与双旋转相关联，并研究轨道行为和复杂性问题。
A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards
讨论了C.卡尔特：每一个多边形外台球有一个周期性的轨迹。
Entropy and Complexity of Polygonal Billiards with Spy Mirrors
证明了带单面镜的多边形台球的拓扑熵为零。在某些情况下，我们显示子指数和其他多项式估计的复杂性。
It is well-known that billiards in polygons cannot be chaotic (hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonal billiard is zero. We consider physical polygonal billiards where a moving particle is a hard disc rather than a point (mathematical) particle and show that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov- Sinai entropy for any positive radius of the moving particle (provided that the particle is not so big that it cannot move within a polygon). This happens because a typical physical…
Lower bounds on directional complexity for irrational triangle billiards
我们给出了一类无理三角形台球在典型方向上复杂性增长的显式下估计
We show that in a typical polygon the billiard map as well as its associated subshift obtained by coding orbits by the sequence of sides they visit are topologically weakly mixing.
Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards
引入了分段凸变换类，并研究了其复杂性。我们将结果应用于多边形台球的复杂性，常曲率的表面。
Does a billiard orbit determine its (polygonal) table?
在多边形台球集合上引入了一个新的等价关系。我们说，两个台球（或多边形）是顺序等价的，如果每个台球有一个轨道，其足点是密集的边界和这些轨道的足点的两个序列具有相同的组合顺序。我们研究这个等价关系与额外的正则性条件的轨道。
Hyperbolic polygonal billiards with finitely many ergodic SRB measures
我们研究多边形台球与反射法收缩的反射角朝向正常。证明了如果一个多边形没有平行边，则相应的台球映射有2个遍历SRB测度，其盆域覆盖一个完整的勒贝格测度集.
SRB measures for hyperbolic polygonal billiards
我们证明了多边形台球与收缩反射法表现出双曲吸引子与可数许多遍历SRB措施。这些措施是强大的反射法的小扰动下，和他们存在的表形成一个脱殊集在所有多边形的空间。具体的多边形表进行了详细研究。
Complexity and growth for polygonal billiards
我们建立了一个多边形台球的字的复杂性和广义对角线的数量之间的关系。我们得出结论，在合理的情况下，复杂性函数有立方的上界和下界。在平铺的情况下，复杂性有立方渐近增长。
Splitting time for irrational triangle billiards
我们找到了一个上估计分裂时间的薄平行梁无理三角形台球的一些数论函数的角度。对于某些角度类别，我们提供了该函数的上限估计。
We consider the set of polygons all of whose sides are vertical or horizontal with fixed combinatorics (for example all the figure "L"s). We show that there is a dense G $δ$ subset of such polygons such that for each polygon in this G $δ$ set the billiard flow is weakly-mixing in almost every direction.
Polygonal Refractive Outer Billiards
该研究将Gutkin等、Tabachnikov等关于常规外台球的两个经典结果，推广到了凸多边形对应的折射外台球系统中。
Code & order in polygonal billiards
两个多边形$P,Q$是代码等价的，如果有台球轨道$u,v$击中相同序列的边，使得轨道的投影在边界上是密集的$\partial P, \partial Q$。我们的主要结果表明，当代码等效多边形有相同的角度，分别。相似，分别。亲密相似
A formal definition of a (mathematical) polygonal Andreev billiard and a construction of an equivalence relation that captures the dynamics described in physical toy model of Andreev reflection are given. The continuous flow and discrete flow on the respective phase spaces. It is then shown that the continuous flow preserves the absolute value of the volume element $dx\wedge dy\wedge dθ$ and the billiard (collision) map preserves the measure $\cos ϕdr dϕ$, respectively. One can then characterize the dynamics of a rational polygonal Andreev billiard table. Finally, a discussions of the effect o…
\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate …
Devil's Staircase -- Rotation Number of Outer Billiard with Polygonal Invariant Curves
本文讨论单参数外台球桌族不变曲线上的旋转数。给定一个凸多边形$η$，我们可以通过从$η$的内部切出一个固定区域来构造一个外部台球桌$T$。$T$是分段双曲的，多边形$η$是$T$在台球映射$ϕ$下的不变曲线。证明了：若$β$是有理旋转数为$τ= p / q$的外台球映射下的周期点，则该台球映射的第$n$次迭代不是$β$上的局部恒等式.这证明了作为面积参数的函数的旋转数$τ$是魔鬼阶梯函数。
The bounce spectrum of a polygonal billiard table is the collection of all bi-infinite sequences of edge labels corresponding to billiard trajectories on the table. We give methods for reconstructing from the bounce spectrum of a polygonal billiard table both the cyclic ordering of its edge labels and the sizes of its angles. We also show that it is impossible to reconstruct the exact shape of a polygonal billiard table from any finite collection of finite words from its bounce spectrum.
Homotopical rigidity of polygonal billiards
考虑两个$k$ -双$P$和$Q$。我们说$P$和$Q$中的台球流是同伦等价的，如果包含周期台球轨道的$P$的基本群中的共轭类的集合与$Q$的类似集合一致。我们研究了这种等价关系，并将其与\cite{BT1,BT2}中引入的等价关系，顺序等价和代码等价进行了比较。特别是，我们表明，如果$P$是一个合理的多边形，和$Q$是同伦等价的$P$，然后$P$和$Q$是相似的，或affirmative相似，如果所有的边$P$是垂直和水平的。
On the complexity of polygonal billiards
本文研究典型多边形台球的复杂性，证明复杂性三次增长、鞍连接二次增长，并给出收敛速率多项式下界及慢熵上界，还推广到高维和任意不变测度。
We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards has a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probability measures (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is always equal to zero. Our main result states that for a generic polygon, if the reflection law has a Lips…
Veech groups, irrational billiards and stable abelian differentials
我们描述Veech集团的平面所产生的非理性的角度多边形台球或不可约的稳定阿贝尔微分。对于无理多边形台球，我们证明了这些群是SO（2，R）的非离散子群，并计算了它们的秩。
In this article, we study polygonal symplectic billiards. We provide new results, some of which are inspired by numerical investigations. In particular, we present several polygons for which all orbits are periodic. We demonstrate their properties and derive various conjectures using two numerical implementations.
Deviation of ergodic averages for rational polygonal billiards
我们证明多项式上界的遍历平均值的偏差的直线流在几乎每个方向上的每个平移表面，特别是对那些表面所产生的合理多边形台球。
Growth rates for geometric complexities and counting functions in polygonal billiards
本文介绍一种新的方法来估计动力系统中各种量的增长。我们将我们的方法应用于多边形台球表面的常曲率。例如，我们得到的权力范围的程度2加长度的台球轨道之间的几乎所有对点的平面多边形的数量。
Inhomogeneous Diophantine Approximation and Angular Recurrence for Polygonal Billiards
对于一个给定的旋转数，我们计算的豪斯多夫维数的一组很好的逼近数。我们利用这个结果和Jarnik定理的非齐次形式证明了某些多边形中台球流的强递归性质
We introduce a new notion of stability for periodic orbits in polygonal billiards. We say that a periodic orbit of a polygonal billiard is $λ$-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as $λ$ $\rightarrow$ 1. This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We give sufficient and necessary conditions for a periodic orbit to be $λ$-stable and prove that the set of d-gons having at most finite number of $λ$-stable periodic orbits is dense is the space of d-gons. Moreover, we also determine comp…
Periodic billiard orbits in right triangles II
周期台球轨道在无理直角三角形的相空间中是稠密的。一个更强的逐点密度结果也被证明。
Dual billiards, Fagnano orbits and regular polygons
我们研究了对偶多边形台球的Fagnano轨道的概念。我们用它们刻画了正多边形，并研究了展开图的迭代问题。
Weakly Mixing Polygonal Billiards
本文证明了存在非有理多边形的剩余集，使得台球流关于Liouville测度弱混合，其核心方法是通过Baire范畴论证及平移曲面方向对流的遍历性与特征值性质。
We consider polygonal billiards and we show the uniqueness of coding of non-periodic billiard trajectories in polygons whose holes have non-zero minimal diameters, generalising a theorem of Galperin, Krüger and Troubetzkoy.
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
研究了非紧多边形曲面上具有自由余紧作用$\Z$或$\Z^2$的台球的常返性和遍历性。在$\Z$ -周期性的情况下，我们建立复发的标准。在更困难的$\Z^2$ -周期的情况下，我们建立了一些一般的结果。对于一个特殊的家庭的$\Z^2$ -周期多边形表面，在物理文献中称为风树模型，假设一定的几何性质的限制，我们得到遍历分解的方向台球动力学的一个密集的，可数的方向。这是一个后果，我们的结果遍历$\ZZ$值上循环的无理旋转。
Knudsen's law and random billiards in irrational triangles
本文证明了在由无理三角形组成的锯齿形管壁的二维管道中自由弹跳的气体粒子的克努森定律。动力学随机扰动和相应的随机映射下的斜型确定性表示，这是遍历和准确的研究。
On Chaotic Dynamics in Rational Polygonal Billiards
我们讨论了有理多边形台球中与可积性和混沌特征相关的分段线正则边界效应和顶角奇异边界效应之间的相互作用。多边形中规则和不规则运动的有争议的问题的处理方法是在确定性和随机性的框架内进行的。分析的台球壁碰撞分布和粒子的生存概率，分别在封闭和弱开多边形模拟。在多顶点多边形中，后期的壁碰撞事件导致圆形的规则的周期性轨迹（滑动轨道），这在开放台球的情况下可能被转化为幸存的集体激发（涡）。在弱开多边形中，滑动轨道和涡旋与相应的圆台球中的规则轨道没有拓扑相似性，通过普适和非普适弛豫动力学可以很好地区分滑动轨道和涡旋.
Ergodicity of polygonal slap maps
多边形拍打映射是通过将多边形的边沿其法线投影到多边形的周界上而获得的区间的分段仿射扩展映射。这些地图出现在多边形台球与非镜面反射法律的研究。研究了几种多边形（包括正多边形和三角形）的拍击映射的绝对连续不变概率。我们还提出了一个一般的方法来构造多边形的拍打地图有一个以上的遍历绝对连续不变的概率。
Outer Billiards, Arithmetic Graphs, and the Octagon
外台球是一个几何启发动力系统的基础上凸形的平面。当形状是多边形时，系统具有组合的味道。在多边形的情况下，有一个自然加速的地图，第一个返回地图的某一条在平面上。算术图是第一个返回映射的符号动力学的几何编码。在我们研究的正八边形的情况下，与周期轨道相关的算术图是R^8中的多边形路径。我们感兴趣的是这些多边形路径的渐近形状，因为周期趋于无穷大。我们表明，这些图的任何序列的基本上重新缩放的限制收敛到一个分形曲线，同时项目的一种方式到一个变种的科赫雪花和另一种方式到一个变种的谢尔宾斯基地毯。在某种意义上，这给出了第一个返回映射的符号动力学的渐近行为的完整描述。使我们所有的证明工作是一个有效的（基本上众所周知的）动力学重整化方案。
In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that $C^\infty$ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is that we use Herman's result on Diophantine invariant curves to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.
Illumination in Rational Billiards
该研究证明有理多边形台球中不相互照明的点对集合有限，扩展了有限阻挡点对数量的相关结果，其方法依赖平移曲面阻挡性质研究及模空间动力学成果。
A counter-example to the theorem of Hiemer and Snurnikov
如果对于$¶$中的每对点$(O,A)$，存在有限数量的"阻塞“点$B_1, ..., B_n$，使得从$O$到$A$的每个台球轨迹满足其中一个$B_i$，则称平面多边形台球$¶$具有有限阻塞性质。作为一个反例Hiemer和Snurnikov定理，我们构造了一个家庭的理性台球，缺乏有限阻塞属性。
Billiards in Nearly Isosceles Triangles
证明了等腰三角形的任意充分小的扰动都有周期的台球路径.我们的证明涉及到与三角形台球有关的傅里叶级数的某些无限族的分析，并揭示了无理三角形台球中的一些自相似现象.我们的分析说明了一个令人惊讶的事实，即台球的三角形附近的维奇三角形是非常复杂的，即使台球的维奇三角形是非常好的理解。
In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary of the region is an oval. Using these properties and defining good perturbations for billiards, we show, in this new version, that having only a finite number of nondegenerate periodic orbits for each fixed period is an open property for billiards on surfaces of constant curvature and a dense one on the Euclidean and the hyperbolic planes. Fo…
Reflective projective billiards in the plane: odd periods, period five, and metric rigidity
本文研究反射律逐边恒定的多边形投影台球，给出显式单参数族覆盖所有至少三的周期（含奇周期），并完整刻画周期五情形，证明度量刚性，指出仅球面支持周期轨道开族。
The dynamics of a beam held on a horizontal frame by springs and bouncing off a step is described by a separable two degrees of freedom Hamiltonian system with impacts that respect, point wise, the separability symmetry. The energy in each degree of freedom is preserved, and the motion along each level set is conjugated, via action angle coordinates, to a geodesic flow on a flat two-dimensional surface in the four dimensional phase space. Yet, for a range of energies, these surfaces are not the simple Liouville-Arnold tori - these are tori of genus two, thus the motion on them is not conjugate…
Escape orbits and Ergodicity in Infinite Step Billiards
在前一篇文章中，我们定义了一类非紧多边形台球，即无限步台球：对于给定的非负数递减序列$\{p_{n}$，有一个对应表$\Bi := \bigcup_{n\in\N} [n,n+1] \times [0,p_{n}]$ .在本文中，首先我们将上一篇论文的主要结果推广到更广泛的例子类别。也就是说，A. S。存在唯一的逃逸轨道，其属于每个其它轨道的α和ω极限。然后，Troubetzkoy最近的工作后，我们证明，一般这些系统是遍历几乎所有的初始速度，和熵相对于广泛的一类遍历措施是零。
On the finite blocking property
如果对于$¶$中的每对点$(O,A)$，存在有限数量的"阻塞“点$B_1, ..., B_n$，使得从$O$到$A$的每个台球轨迹满足其中一个$B_i$，则称平面多边形台球$¶$具有有限阻塞性质。推广我们的建设一个反例的定理Hiemer和Snurnikov（见\cite{Mo}），我们表明，只有正多边形，具有有限的阻塞性质是正方形，等边三角形和六边形。然后我们将这个结果推广到平移曲面。证明了具有有限块性质的Veech曲面是环面分支覆盖。我们还提供了一个局部的充分条件的平移面失败的有限块性质。这使我们能够给出一个完整的分类的L-型曲面以及获得一个密度结果的空间中的平移曲面在每一个亏格$g\geq 2$。
We show that every GL(2, R) orbit closure of translation surfaces is either a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense GL(2,R) orbit.
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