M. J. Dias Carneiro, S. Oliffson Kamphorst, S. Pinto-de-Carvalho
In this paper we show that, under certain generic conditions, billiards on ovals have only a finite number of periodic orbits, for each period, all non-degenerate and at least one of them is hyperbolic. Moreover, the invariant curves of two hyperbolic points are transversal. We explore these properties to give some dynamical consequences specially about the dynamics in the instability regions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Sufficiently differentiable oval billiards always have invariant rotational curves, but there are only two types of ovals with an invariant horizontal circle in its phase-space: the constant width ovals and some very special symmetric curves. In this work we study the dynamics near the horizontal circle for the billiard map of those two types of ovals, and show that the horizontal circle is approached, from both sides, by other invariant rotational curves. We will also describe some dynamical consequences.
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…
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.
Symmetric periodic orbits in symmetric billiards
本文研究具有旋转对称边界的卵形台球，证明具有相同对称性的周期轨道中一半为椭圆型且Moser稳定，另一半为双曲型且具有同（异）宿相交。
In this note we establish the existence of a (n+1)-periodic billiard trajectory inside an n-dimensional regular simplex in the hyperbolic space, which hits the interior of every facet exactly once.
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…
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.
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.
We study polygonal billiards with reflection laws contracting the reflected angle towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic SRB measures whose basins cover a set of full Lebesgue measure.
The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Simányi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. The goal of this work is to survey the key results of their theory and highlight some central problems which deserve particular attention and efforts.
We study billiards on polytopes in $\Rr^d$ with contracting reflection laws, i.e. non-standard reflection laws that contract the reflection angle towards the normal. We prove that billiards on generic polytopes are uniformly hyperbolic provided there exists a positive integer $k$ such that for any $k$ consecutive collisions, the corresponding normals of the faces of the polytope where the collisions took place generate $\Rr^d$. As an application of our main result we prove that billiards on generic polytopes are uniformly hyperbolic if either the contracting reflection law is sufficiently clos…
We prove that polygonal billiards with contracting reflection laws exhibit hyperbolic attractors with countably many ergodic SRB measures. These measures are robust under small perturbations of the reflection law, and the tables for which they exist form a generic set in the space of all polygons. Specific polygonal tables are studied in detail.
On another edge of defocusing: hyperbolicity of asymmetric lemon billiards
本文证明边界由两条圆弧组成的非对称柠檬形凸台球桌能产生双曲台球，揭示了散焦机制在聚焦组件极为接近时仍可成立的新边界。
This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.
\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 …
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.
We survey applications of the theory of hyperbolic (and to a lesser extent non hyperbolic) billiards to some fundamental problems of statistical physics and their mathematically rigorous derivations in the framework of classical Hamiltonian systems.
Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry
本文结合辛台球与平铺台球提出辛平铺台球，证明其特殊情形下的周期轨道结果，并结合Thurston构造赋予平面等边多边形模空间双曲结构，得出六边形平面连杆位形空间为10尖点双曲3-流形的推论。
Hyperbolicity and Ergodicity in Randomly Perturbed Billiard Systems
本文探究小随机扰动下随机台球系统的李雅普诺夫指数与双曲性，发现非圆形椭圆、柠檬台球经扰动后兼具遍历性与双曲性，圆形台球则仅遍历且保持零李雅普诺夫指数，相关结果为受噪动力学系统研究提供新视角。
Gouezel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.
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…
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。
We consider a convex curve $γ$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $γ$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve $Γ$. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.
A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.
Coding of billiards in hyperbolic 3-space
本文将符号动力学扩展至3维双曲空间中的理想多面体，构建了台球动力学的新型编码系统，建立了带基点轨迹空间与编码移位空间的共轭关系，并发现其闭包为有限型子移位。
Markov partitions designed by Sinai(1968) and Bowen(1970) proved to be an efficient tool for descibing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular, for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai(1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich-Chernov-Sinai, 1990), Markov towers (Young, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai's original idea is surveyed i…
Generic properties of planar symplectic billiards
研究平面辛台球的一般性质，证明对于\(C^\infty\)强凸域剩余集，周期轨迹相关性质，还证弗兰克斯引理等，得出\(C^\infty\)强凸辛台球一般有正拓扑熵的结论。
The connected configuration space of a so called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point particle in this configuration space with specular reflections at the boundaries of the removed cylinders. It is proven here that under a certain geometric condition --- slightly stronger than the necessary condition presented in [S-Sz(1998)] --- a cylindric billiard flow is completely hyperbolic. As a consequence, every hard ball system is completely hyperbolic --- a result strengthening the theorem of …
The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the so-called defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nons…
We investigate the decay rates of correlations for nonuniformly hyperbolic systems with or without singularities, on piecewise Hölder observables. By constructing a new scheme of coupling methods using the probability renewal theory, we obtain the optimal bounds for decay rates of correlations for a large class of such observables. We also establish the alpha-mixing property for time series generated by these systems, which leads to a vast ranges of limiting theorems. Our results apply to rather general hyperbolic systems with singularities, including Bunimovich flower billiards, semidispersin…
We prove convergence to a Levy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung & Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod J_1 topology. Our main results yield elementary geometric conditions for convergence (i) in M_1, (ii) in M_2 but not M_1. In general, we show for a large class of nonuniformly hyperbolic systems how to deduce functional limit laws once convergence to the corresponding stable law is known.
Complexity of Billiards in Polygons Associated to Hyperbolic $(p,q)$-Tilings
本文研究双曲平面中正多边形弹道语言的复杂性，计算当q为偶数时的指数增长速率，并在q为奇数时给出界。还为q偶数情况提供完整的语法规则，确定由p个字母组成的词是否由弹道路径实现。
We consider billiard ball motion in a convex domain on a constant curvature surface influenced by the constant magnetic field. We examine the existence of integral of motion which is polynomial in velocities. We prove that if such an integral exists then the boundary curve of the domain determines an algebraic curve in $\mathbf{C}^3$ which must be nonsingular. Using this fact we deduce that for any domain different from round disc for all but finitely many values of the magnitude of the magnetic field billiard motion does not have Polynomial in velocities integral of motion.
An approach due to Wojtkovski [9], based on the Jacobi fields, is applied to study sets of 3-period orbits in billiards on hyperbolic plane and on two-dimensional sphere. It is found that the set of 3-period orbits in billiards on hyperbolic plane, as in the planar case, has zero measure. For the sphere, a new proof of Baryshnikov's theorem is obtained which states that 3-period orbits can form a set of positive measure provided a natural condition on the orbit length is satisfied.
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。
Billiards in generic convex bodies have positive topological entropy
本文证明在一般凸体中，台球映射存在非平凡双曲基本集，从而推出其拓扑熵为正且周期轨道数量指数增长。
Dynamics of planar integrable Kepler billiards with a focused hyperbolic branch
本文研究以聚焦双曲分支为边界的平面开普勒台球的可积动力学，利用焦点-焦散圆识别椭圆曲线并线性化动力学，分析周期轨道的凯莱条件，并推广至抛物弧和直线边界情况。
Billiards in a circle with trajectories circumscribing a triangle
本文在双曲几何的Klein-Beltrami模型中，以欧几里得几何方式重新表述了三角形被归类为“大”的条件，提出并评估了其欧几里得大小度量，并为等腰三角形给出显式公式。
In this note we study caustic-free regions for convex billiard tables in the hyperbolic plane or the hemisphere. In particular, following a result by Gutkin and Katok in the Euclidean case, we estimate the size of such regions in terms of the geometry of the billiard table. Moreover, we extend to this setting a theorem due to Hubacher which shows that no caustics exist near the boundary of a convex billiard table whose curvature is discontinuous.
Every finite horizon Sinai billiard map has a unique measure of maximal entropy
本文研究了有限时间辛恩尼亚德映射的最大熵测度存在性和唯一性，通过新的证明方法不依赖稀疏复发条件，证明了其唯一性。
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。
We describe a one-parameter family of dispersing (hence hyperbolic, ergodic and mixing) billiards where the correlation function of the collision map decays as $1/n^a$ (here $n$ denotes the discrete time), in which the degree $a \in (1, \infty)$ changes continuously with the parameter of the family, $β$. We also derive an explicit relation between the degree $a$ and the family parameter $β$.
We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard fl…
Improved estimates of statistical properties in some non-uniformly hyperbolic dynamical systems
提出一种通用方案改进非一致双曲动力系统统计性质的界限，通过消除对数因子获得多项式可和混合速率下本质上最优的估计，并应用于多个物理系统。
We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young tower, such as Sinai billiard with finite horizon, suspensions over Pomeau-Manneville maps, and geometric Lorenz attractors.
We show that a compound Poisson distribution holds for scaled exceedances of observables $ϕ$ uniquely maximized at a periodic point $ζ$ in a variety of two-dimensional hyperbolic dynamical systems with singularities $(M,T,μ)$, including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form $ϕ(z)=-\ln d(z,ζ)$ where $d$ is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a Pólya-Aeppli distibution of index $θ$. We calculate $θ$ in terms of the derivative of the m…
We investigate superdiffusion for stochastic processes generated by nonuniformly hyperbolic system models, in terms of the convergence of rescaled distributions to the normal distribution following the abnormal central limit theorem, which differs from the usual requirement that the mean square displacement grow asymptotically linearly in time. We construct a martingale approximation that follows the idea of Doob's decomposition theorem. We obtain an explicity formula for the superdiffusion constant in terms of the fine structure that originates in the phase transitions as well as the geometry…
We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the condition of the so called ``Ansatz'' can be dropped. That condition assumed that almost every singular phase point had a hyperbolic trajectory after the singularity. Having this condition dropped, the cited theorem becomes much stronger and easier to apply. At the end of the paper two immediate corollaries of this improvement are discussed: One of them is the (fully hyperbolic) Bernoulli mixing property of every hard disk system (D=2), the other one claims…
Adapted Measures for Markov Interval Maps
为研究熵与适配性的关联，本文考察含单个奇点的马尔可夫区间映射，证明映射熵与奇点“强度”的关联条件决定最大熵测度是否适配该奇点，并指出 Hölder 条件下奇点回复性是非适配不变测度存在的必要条件。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。