Serge Tabachnikov
We consider the dual billiard map with respect to a smooth strictly convex closed hypersurface in linear 2m-dimensional symplectic space and prove that it has at least 2m distinct 3-periodic orbits.
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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 rigorously investigate closed Minkowski/Finsler billiard trajectories on $n$-dimensional convex bodies. We outline the central properties in comparison and differentiation from the Euclidean special case and establish two main results for length-minimizing closed Minkowski/Finsler billiard trajectories: one is a regularity result, the other is of geometric nature. Building on these results, we develop an algorithm for computing length-minimizing closed Minkowski/Finsler billiard trajectories in the plane.
We study length-minimizing closed generalized Euclidean billiard trajectories in convex bodies in $\mathbb{R}^n$ and investigate their relation to the inclusion minimal affine sections that contain these trajectories. We show that when passing to these sections, the length-minimizing closed billiard trajectories are still billiard trajectories, but their length-minimality as well as their regularity can be destroyed. In light of this, we prove what weaker regularity is actually preserved under passing to these sections. Based on the results, we develop an algorithm in order to calculate length…
We prove that a bounded domain $Ω$ in $\R^n$ with smooth boundary has a periodic billiard trajectory with at most $n+1$ bounce times and of length less than $C_n r(Ω)$, where $C_n$ is a positive constant which depends only on $n$, and $r(Ω)$ is the supremum of radius of balls in $Ω$. This result improves the result by C.Viterbo, which asserts that $Ω$ has a periodic billiard trajectory of length less than $C'_n \vol(Ω)^{1/n}$. To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.
In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan-Ostrover in 2012.
We give the asymptotic growth of the number of primitive periodic trajectories of a two dimensional dispersive billiard, when we prescribe their number of bounces on one of the obstacles.
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.
Non-periodic not everywhere dense trajectories in triangular billiards
利用诱导映射与区间交换变换，构造等腰三角形单参数族，其非周期轨迹非处处稠密，从而否定性解决三角台球中长期未决的稠密性问题。
Closed billiard trajectories in a polygon in the hyperbolic plane can be coded by the order in which they hit the sides of the polygon. In this paper, we consider the average length of cyclically related closed billiard trajectories in ideal hyperbolic polygons and prove the conjecture that this average length is minimized for regular hyperbolic polygons. The proof uses a strict convexity property of the geodesic length function in Teichmüller space with respect to the Weil-Petersson metric, a fundamental result established by Wolpert.
In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Then collisions are rare. We study trajectories of degenerate billiards which have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajector…
Symplectic billiards for pairs of polygons
研究多边形对的辛台球，给出使其映射完全周期的准则以构造新例，还展示了有任意大周期轨道、有孤立周期轨道及无周期轨道的例子，最后证明光滑严格凸曲线代替多边形时有无限多周期轨道。
We introduce the iteration theory for periodic billiard trajectories in a compact and convex domain of the Euclidean space, and we apply it to establish a multiplicity result for non-iterated trajectories.
We consider the billiard map inside a polyhedron. We give a condition for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a periodic orbit of length four (generalization of Fagnano's orbit for triangles), moreover we can study completly the orbit of points along this coding.
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。
Periodic billiard orbits are dense in the phase space of an irrational right triangle. A stronger pointwise density result is also proven.
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 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.
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。
In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincaré.
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.
We study periodic billiard trajectories on a compact Riemannian manifold with boundary, by applying Morse theory to Lagrangian action functionals on the loop space of the manifold. Based on the approximation method due to Benci-Giannoni, we prove that nonvanishing of relative homology of a certain pair of loop spaces implies the existence of a periodic billiard trajectory. We also prove a parallel result for path spaces. We apply those results to show the existence of short billiard trajectories and short geodesic loops. We also recover two known results on the length of a shortest periodic bi…
Morse Theory and Stability of Periodic Billiard Trajectories
本文通过莫尔斯指标定理计算台球轨迹的莫尔斯指标，并利用该指标判断周期轨迹的线性稳定性，给出了稳定2周期轨迹存在的充分条件。
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 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 paper, extending the works of Milena Radnović and Serge Tabachnikov, we establish conditions for two different non-symmetric norms to define the same billiard reflection law.
We study resonant billiard trajectories within quadrics in the $d$-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of $d$ intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to $s$-weak billiard trajectories, we com…
Symplectic billiards as Minkowski billiards
研究建立闵可夫斯基台球与辛台球的联系，通过从典范辛结构约化描述闵可夫斯基台球映射，将辛台球视为其辛版本“平方根”，恢复并推广相关结果，如证明\(2n\)维中\(2r\)周期辛台球轨道的存在性。
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…
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.
We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we obtain the lower bound $(d-2)(p-1)+2$, which is much better than the previous estimates.
Billiard trajectories inside Cones
本文研究凸锥内台球轨迹的反射次数问题，证明存在C²凸锥允许有限时间内无限多次反射，并利用两个首次积分估计了R³中椭圆锥内轨迹的反射次数。
We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results.
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…
This paper shows that the method of Galperin which had been widely accepted in the literature since 1983 for constructing a non-dense and non-periodic trajectory in a triangle can never work. This is done by showing that all possible examples that can be produced by Galperin's method of which there are infinitely many all produce a periodic trajectory.
Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from the boundary making the angle of incidence equal the angle of reflection. The billiard motion can happen to be a closed (or periodic) one when the billiard ball rebounds k times and then gets to the initial position with the same speed vector as in the beginning. The study of closed billiard trajectories is due to George Birkhoff who in 1927 proved a lower e…
Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to -1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N+2. We also give a precise description of the exceptional family of trajectories (of zero measure) : …
We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Finsler billiard trajectories correspond to $r$-gons inscribed in $M$ and having extremal Finsler length. The cyclic group ${\mathbb Z}_r$ acts on these extremal polygons, and one counts the ${\mathbb Z}_r$-orbits. Using Morse and Lusternik-Schnirelmann theories, we prove that if $r…
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。
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.
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。
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…
Star-shaped trajectories of certain billiards around a triangle
本研究在Beltrami-Klein模型框架下，基于已有研究考察双曲平面中三角形外台球映射旋转数为2/5的情况，给出充分条件并证明其对大等腰三角形的必要性，最后提出相关猜想。
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。
We provide a method to straighten each billiard trajectory in a flat disk. As an application, we show that for each point on the disk and for almost all directions, the closure of the corresponding billiard trajectory contains a vertex. We generalize Katok-Zemlyakov construction to the flat disks with rational angle data: for each rational flat disk we obtain a translation surface and a projection to the doubling of the disk. We calculate the Euler characteristics of this translation surface.
Some remarks on periodic billiard orbits in rational polygons
一个多边形被称为有理的，如果每对边之间的角度是$π.$的有理倍数。我们要证明的主要定理是定理1：对于有理多边形，台球流的周期点在台球流的相空间中是稠密的。这是对Masur定理的加强，Masur定理证明了任何有理多边形都有“许多”周期性台球轨迹;更确切地说，周期性轨迹的方向集在速度方向集$§^1.$中是稠密的。我们还将证明定理1的一些改进：周期轨道在多边形中的"良好分布“和点$q \in Q$具有稠密周期方向集的剩余性。
Self-intersection Points of Billiard Trajectories in a Square with Small Pockets
该研究针对四角带小口袋的正方形台球轨迹，建立了自交点数量各阶矩的渐近公式及其分布，揭示了小口袋对轨迹自交行为的影响。
A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line. By leveraging deep, but yet not widely known results of the Krein-Levin-Nudelman theory of generalized Chebyshev polynomials, fundamental properties of billiard dynamics are proven for any $d$, viz., that the sequences of winding numbers are monotonic. By employing the potential theory we pr…
We study the rate of mixing of observables of Z^d-extensions of probability preserving dynamical systems. We explain how this question is directly linked to the local limit theorem and establish a rate of mixing for general classes of observables of the Z^2-periodic Sinai billiard. We compare our approach with the induction method.
We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is als…
Any periodic trajectory on an isosceles triangle gives rise to a periodic trajectory on a right triangle obtained by identifying the halves of the original triangle. We examine the relationship between periodic trajectories on isosceles triangles and the trajectories on right triangles obtained in this manner, and the consequences of this relationship for the existence of stable trajectories on isosceles triangles and the properties of their orbit tiles.
Existence of a Periodic Orbit for Billiards in Polygons
通过反证法，结合Galperin等人关于台球流动力学的基本结果、单位切丛上黎曼度量的单参数缩放几何以及缩放度量的骨架或割迹拓扑，证明任意有限多边形中台球流至少存在一条周期轨道。
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.
Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards--planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain. In this paper we greatly extend previous results on boundedness of orbits (Broomhead and Gutk…
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.
For billiards with $N$ obstacles on a torus, we study the behavior of specific kind of its trajectories, \emph{the so called admissible trajectories}. Using the methods developed in \cite{1}, we prove that the \emph{admissible rotation set} is convex, and the periodic trajectories of admissible type are dense in the admissible rotation set. In addition, we show that the admissible rotation set is a proper subset of the general rotation set.
We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a $Z^2$-periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding billiard flow is not erg…
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 $4$.
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