Alexander Tumanov, Vadim Zharnitsky
It is shown that the set of 4-period orbits in outer billiard with piecewise smooth convex boundary has an empty interior, provided that no four corners of the boundary form a parallelogram.
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We present the first example of a hyperbolic outer billiard. More precisely we construct a one parameter family of examples which in some sense correspond to the Bunimovich billiards.
Integrable Outer billiards and rigidity
本文提出了一种新的平面外台球生成函数，并利用它证明了刚性结果：若正曲率光滑凸曲线邻域由外台球映射不变的连续曲线叶状化，则该曲线必为椭圆，证明中通过寻找合适权重克服相空间非紧性并归约为Blaschke-Santalo不等式。
In this paper we prove two asymptotic estimates for pairs of closed trajectories for open billiards similar to those established by Pollicott and Sharp for closed geodesics on negatively curved compact surfaces. The first of these estimates holds for general open billiards in any dimension. The more intricate second estimate is established for open billiards satisfying the so called Dolgopyat type estimates. This class of billiards includes all open billiards in the plane and open billiards in $\R^N, N \geq 3$ satisfying some additional conditions.
We discuss a recent result by C. Culter: every polygonal outer billiard has a periodic trajectory.
Bouncing Outer Billiards
本文提出三维反弹外台球系统，证明光滑凸体上至少存在四个单参数族固定点，完整刻画线段情形，并通过数值实验揭示正方形和椭圆等形状的复杂非遍历行为。
The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.
In the class of projective billiards, which contains the usual billiards, we exhibit counter-examples to Ivrii's conjecture, which states that in any planar billiard with smooth boundary the set of periodic orbits has zero measure. The counter-examples are polygons admitting a $2$-parameters family of $n$-periodic orbits, with $n$ being either $3$ or any even number grower than $4$.
We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behavior is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experime…
This paper had a serious error. In fixing the error the emphasis of the paper has changed completely, thus meriting a new name: ``Periodic orbits in right triangles''. I have made a new submission to arXiv with this name.
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。
We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse.
We show that every polynomially integrable planar outer convex billiard is elliptic.
In this paper we establish a kind of bijection between the orbits of a polygonal outer billiards system and the orbits of a related (and simpler to analyze) system called the pinwheel map. One consequence of the result is that the outer billiards system has unbounded orbits if and only if the pinwheel map has unbounded orbits. As the pinwheel map is much easier to analyze directly, we think that this bijection will be helpful in attacking some of the main questions about polyonal outer billiards.
Outer Contact Billiards
该研究提出外接触台球作为外辛台球的奇维对应物，证明其可生成接触同胚，实射影三维空间中二次曲面对应完全可积且无3周期轨道，仅一个二次表有4周期轨道。
In this paper, we discuss rotation number on the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon $η$, we can construct an outer billiard table $T$ by cutting out a fixed area from the interior of $η$. $T$ is piece-wise hyperbolic and the polygon $η$ is an invariant curve of $T$ under the billiard map $ϕ$. We will show that, if $β$ is a periodic point under the outer billiard map with rational rotation number $τ= p / q$, then the $n$th iteration of the billiard map is not the local identity at $β$. This proves that the rotation number $τ$ as a function…
In this paper, outer billiards outside regular octagon are studied in details. We described all periodic points and their periods; also, we proved that the periodic points form a set of full measure outside the octagon and found an aperiodic point.
We show that for almost every $(P,λ)$ where $P$ is a convex polygon and $λ\in(0,1)$, the corresponding outer billiard about $P$ with contraction $λ$ is asymptotically periodic, i.e., has a finite number of periodic orbits and every orbit is attracted to one of them.
Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has zero measure.
We study outer billiards with contraction outside regular polygons. For regular $n$-gons with $n = 3, 4, 5, 6, 8$, and $12$, we show that as the contraction rate approaches $1$, dynamics of the system converges, in a certain sense, to that of the usual outer billiards map. These are precisely the values of $n \geq 3$ with $[\mathbb{Q}(e^{2πi/n}):\mathbb{Q}] \leq 2$. Then we discuss how such convergence may fail in the case of $n=7$.
We give a simple proof of our previous result with V. Zharnitsky that the set of period 4 orbits in planar outer billiard with piecewise smooth convex boundary has empty interior, provided that no four corners of the boundary form a parallelogram. We also obtain results on period 5 and 6 orbits.
In this paper outer, or dual, billiards outside regular polygons are studied; in particular, periodic points for cases of strictly convex "tables" and for regular n-gons with n = 3,4,6,8,12 are discussed. The main results of the paper are: 1) proof of an existence of aperiodic points for outer billiards outside regular octagon and dodecagon; 2) proof of the fact that for regular octagon, set of periodic points is of full measure; 3) list of all possible periods for case of regular octagon.
Ivrii's conjecture for some cases in outer and symplectic billiards
本文证明平面外台球两类周期 Ivrii 猜想，并以新几何论证统一处理外台球与辛台球的 3、4 周期情形，且推广辛台球至高维。
Aperiodic points for outer billiards
本文证明非三角形、正方形或六边形的正多边形上的欧几里得外弹球座存在非周期点，解决了R. Schwartz在ICM 2022提出的问题。
In this paper we present new results regarding the periodicity of outer billiards in the hyperbolic plane around polygonal tables which are tiles in regular two-piece tilings of the hyperbolic plane.
Elliptic islands and zero measure escaping orbits in a class of outer billiards
本文研究一类圆形扇区外台球系统，证明半圆盘存在正比例椭圆岛（稳定性），并构造可数族扇区使逃逸轨道测度为零，展示稳定性与扩散共存。
Outer billiards is a simple dynamical system based on a convex planar shape. The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if there exists a planar shape for which outer billiards has an unbounded orbit. The first half of this monograph proves that outer billiards has an unbounded orbit defined relative to any irrational kite. The second half of the monograph gives a very sharp description of the set of unbounded orbits, both in terms of the dynamics and the Hausdorff dimension. The analysis in both halves reveals a close connection between outer billiards on kites …
Remarks on the outer length billiards
本文研究外长度弹奏，证明了Ivrii猜想的3和4周期版本，并给出了构造具有n周期点的弹奏桌的方法。
An existence of an aperiodic point for outer billiard outside regular dodecagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic, and all possible periods are listed explicitly. The proof is based on facts established with computer program implemented by author. All calculations of a program has been made with absolute accuracy; these calculations can be done "by hands" but it this case, amount of calculations would be too large.
In this paper, we give detailed analysis and description of periodic trajectories of the billiard system within an ellipsoid in the 3-dimensional Minkowski space, taking into account all possibilities for the caustics. The conditions for periodicity are derived in algebro-geometric, analytic, and polynomial form.
We consider outer billiard outside regular convex polygons. We deal with the case of regular polygons with $\{3,4,5,6,10\}$ sides, and we describe the symbolic dynamics of the map and compute the complexity of the language.
Outer billiards is a basic dynamical system, defined relative to a planar convex shape. This system was introduced in the 1950's by B.H. Neumann and later popularized in the 1970's by J. Moser. All along, one of the central questions has been: is there an outer billiards system with an unbounded orbit. We answer this question by proving that outer billiards defined relative to the Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that appears in the famous Penrose tiling. We also analyze some of the finer orbit structure of outer billiards on the penrose kite. This ana…
We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<λ<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $λ/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.
Outer length billiards on a large scale
本文研究外长度台球系统，给出生成函数与不变面积形式，揭示远离台球桌时轨道趋近于以原点为中心的圆，并阐明辅助圆圆心在无穷远处的轨道与台球桌对称化极对偶曲线的关系。
We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbit…
Starting the study of outer length billiards
本文开创外长度台球研究，证明椭圆情形完全可积，并给出Mather β-函数一阶项及4阶Lazutkin坐标的显式表示。
Outer billiards in the complex hyperbolic plane
本文研究复双曲平面中二次凸超曲面诱导的外台球映射，证明其外部具有双重测地叶状结构，且该映射为辛同胚，推广了双曲平面和复欧氏空间中的已知几何性质。
Outer billiards of symplectically self-polar convex bodies
本文研究了辛自极凸体的外Billiards映射，证明其存在由4周期轨道组成的不变超曲面，为更高维Birkhoff Billiards提供了非平凡不变结构。
Higher order terms of Mather's $β$-function for symplectic and outer billiards
本文利用仿射微分几何工具，显式计算了严格凸平面区域中辛台球与外部台球的Mather β-函数泰勒展开的高阶项，并确定了最佳逼近多边形距离渐近展开的第三项。
We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions. We provide closed forms for the generating functions of these…
Outer length billiards on polygons
本文研究多边形台面上的外长度台球系统，描述远离台面的轨道行为，并对正方形台面的逃逸轨道提出猜想，补充了处理光滑正曲率台面的相关研究。
We consider the outer billiards map with contraction outside polygons. We construct a 1-parameter family of systems such that each system has an open set in which the dynamics is reduced to that of a piecewise contraction on the interval. Using the theory of rotation numbers, we deduce that every point inside the open set is asymptotic to either a single periodic orbit (rational case) or a Cantor set (irrational case). In particular, we deduce existence of an attracting Cantor set for certain parameter values. Moreover, for a different choice of a 1-parameter family, we prove that the system i…
Let $v$ be a unit vector field on a complete, umbilic (but not totally geodesic) hypersurface $N$ in a space form; for example on the unit sphere $S^{2k-1} \subset \mathbb{R}^{2k}$, or on a horosphere in hyperbolic space. We give necessary and sufficient conditions on $v$ for the rays with initial velocities $v$ (and $-v$) to foliate the exterior $U$ of $N$. We find and explore relationships among these vector fields, geodesic vector fields, and contact structures on $N$. When the rays corresponding to each of $\pm v$ foliate $U$, $v$ induces an outer billiard map whose billiard table is $U$. …
Existence of stable periodic orbits in billiards close to lemon and moon billiards
本文研究了在接近柠檬和月球弹球的弹球台中，通过替换圆弧为一般小曲率边界，产生非线性稳定的周期轨道的条件。
This paper is the third in a series which explores a combinatorial method for generating lattice polygons in the plane. I call this method the plaid model. In this paper I prove the main result I had been aiming for since the beginning, which is to show that there is a coarse isomorphism between the plaid model and the so-called arithmetic graph for outer billiards on kites. The content of the theorem is that the plaid model predicts the symbolic dynamics of the outer billiards orbits, up to an error of one unit. This result combines with the work in the other papers to give a second proof tha…
Periodic billiard orbits are dense in the phase space of an irrational right triangle. A stronger pointwise density result is also proven.
There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.
Persistence of periodic billiard orbits under domain deformation
研究证明，若多边形存在满足特定组合准则的周期弹道轨道，则参数空间中存在路径，使路径上每个多边形都存在相同类型的周期弹道轨道。
We study the stability of periodic trajectories of planar inverse magnetic billiards, a dynamical system whose trajectories are straight lines inside a connected planar domain $Ω$ and circular arcs outside $Ω$. Explicit examples are calculated in circles, ellipses, and the one parameter family of curves $x^{2k}+y^{2k}=1$. Comparisons are made to the linear stability of periodic billiard and magnetic billiard trajectories.
We give an explicit sub-exponential estimate on the growth rate of periodic orbits and generalized diagonals for typical triangle billiards.
Coexistence of periodic-2 and periodic-3 caustics for nearly circular analytic billiard maps
本文研究了近圆解析台球映射中周期2与周期3焦散线的共存问题，证明了圆的对称解析形变需满足等距变换这一必要条件。
We prove that the set of periodic points of a generic C^1-billiard table is dense in the phase space.
In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that this compactification has a renormalization scheme. These two features allow us to make some sharp statements concerning the distribution, large-scale geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits. For instance, one of our results is that the union of the unbounded orbits has Hausdorff dimension 1. We give a computer-aided proo…
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…
In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the periodic trajectory through a given point is long, short, or a saddle connection. In this paper we present an algorithm resolving this question for trajectories emanating from the midpoints of the pentagon.
We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table. Based on a refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in a self-similar Sierpinski carpet, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpinski carpet billiard tables. The trivial limit of this sequence then constitutes a periodic orbit of a self-similar Sierpinski carpet billiard table. We also determine the corresponding translation surface for each prefractal billiard table, and s…
Twist Coefficients of Periodic Orbits of Minkowski Billiards
该研究引入相空间新坐标系探究闵可夫斯基台球基本性质，推导椭圆周期轨道扭转系数公式并分析其稳定性，为该领域周期轨道研究提供新方法与理论支撑。
Birkhoff Normal Form and Twist Coefficients of Periodic Orbits of Billiards
本文针对多种动力台球，显式构造椭圆周期点附近的Birkhoff变换，并用台球桌几何参数给出前两个扭转系数的显式公式，进而刻画非线性稳定性与局部解析可积性。
Non-Birkhoff periodic orbits in symmetric billiards
本文研究了对称平面弹奏器中非Birkhoff周期轨道的存在性，并给出了其存在条件及计算方法。
A polygon is called rational if the angle between each pair of sides is a rational multiple of $π.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has ``many'' periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $§^1.$ We will also prove some refinements of Theorem 1: the ``well distribution'' of periodic orbits in…
I announce a solution of the conjecture about the measure of periodic points for planar billiard tables. The theorem says that if $\Om\subset\R^2$ is a compact domain with piecewise $C^3$ boundary, then the set of periodic orbits for the billiard in $\Om$ has measure zero. Here I outline a proof. A complete version will appear elsewhere.
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