Jayadev S. Athreya, Giovanni Forni
We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.
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We study polygonal billiards with one-sided vertical mirror scattered on a square billiard table. We associate trajectories of these kinds of billiards with double rotations and study orbit behavior and questions of complexity.
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 discuss a recent result by C. Culter: every polygonal outer billiard has a periodic trajectory.
We prove that a polygonal billiard with one-sided mirrors has zero topological entropy. In certain cases we show sub exponential and for other polynomial estimates on the complexity.
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…
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.
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.
We introduce the class of piecewise convex transformations, and study their complexity. We apply the results to the complexity of polygonal billiards on surfaces of constant curvature.
We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two sequences of footpoints of these orbits have the same combinatorial order. We study this equivalence relation with additional regularity conditions on the orbit.
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.
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.
We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.
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。
Two polygons $P,Q$ are code equivalent if there are billiard orbits $u,v$ which hit the same sequence of sides and such that the projections of the orbits are dense in the boundaries $\partial P, \partial Q$. Our main results show when code equivalent polygons have the same angles, resp. are similar, resp. affinely similar.
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.
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…
We introduce a new method for estimating the growth of various quantities arising in dynamical systems. We apply our method to polygonal billiards on surfaces of constant curvature. For instance, we obtain power bounds of degree two plus epsilon in length for the number of billiard orbits between almost all pairs of points in a planar polygon.
For a given rotation number we compute the Hausdorff dimension of the set of well approximable numbers. We use this result and an inhomogeneous version of Jarnik's theorem to show strong recurrence properties of the billiard flow in certain polygons
\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 …
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…
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.
Consider two $k$-gons $P$ and $Q$. We say that the billiard flows in $P$ and $Q$ are homotopically equivalent if the set of conjugacy classes in the fundamental group of $P$ which contain a periodic billiard orbit agrees with the analogous set for $Q$. We study this equivalence relationship and compare it to the equivalence relations, order equivalence and code equivalence, introduced in \cite{BT1,BT2}. In particular we show if $P$ is a rational polygon, and $Q$ is homotopically equivalent to $P$, then $P$ and $Q$ are similar, or affinely similar if all sides of $P$ are vertical and horizontal…
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…
We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.
We discuss the interplay between the piece-line regular and vertex-angle singular boundary effects, related to integrability and chaotic features in rational polygonal billiards. The approach to controversial issue of regular and irregular motion in polygons is taken within the alternative deterministic and stochastic frameworks. The analysis is developed in terms of the billiard-wall collision distribution and the particle survival probability, simulated in closed and weakly open polygons, respectively. In the multi-vertex polygons, the late-time wall-collision events result in the circular-l…
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 study the notion of Fagnano orbits for dual polygonal billiards. We used them to characterize regular polygons and we study the iteration of the developing map.
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.
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…
We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic case, we establish some general results. For a particular family of $\Z^2$-periodic polygonal surfaces, known in the physics literature as the wind-tree model, assuming certain restrictions of geometric nature, we obtain the ergodic decomposition of directional billiard dynamics for a dense, countable set of directions. This is a consequence of our results …
In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called cylindric scatterers) have been removed. We prove that every such system is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for the ergodicity is present.
In a previous paper (nlin.CD/0107041) the following class of billiards was studied: For $f: [0, +\infty) \longrightarrow (0, +\infty)$ convex, sufficiently smooth, and vanishing at infinity, let the billiard table be defined by $Q$, the planar domain delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. For a large class of $f$ we proved that the billiard map was hyperbolic. Furthermore we gave an example of a family of $f$ that makes this map ergodic. Here we extend the latter result to a much wider class of functions.
The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional bill…
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.
We investigate the dynamics of no-slip billiards, a model in which small rotating disks may exchange linear and angular momentum at collisions with the boundary. We give new results on periodicity and boundedness of orbits which suggest that a class of billiards (including all polygons) is not ergodic. Computer generated phase portraits demonstrate non-ergodic features, suggesting chaotic no-slip billiards cannot readily be constructed using the common techniques for generating chaos in standard billiards.
Consider the non-compact billiard in the first quandrant bounded by the positive $x$-semiaxis, the positive $y$-semiaxis and the graph of $f(x) = (x+1)^{-α}$, $α\in (1,2]$. Although the Schnirelman Theorem holds, the quantum average of the position $x$ is finite on any eigenstate, while classical ergodicity entails that the classical time average of $x$ is unbounded.
In a previous paper we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given decreasing sequence of non-negative numbers $\{p_{n}$, there corresponds a table $\Bi := \bigcup_{n\in\N} [n,n+1] \times [0,p_{n}]$. In this article, first we generalize the main result of the previous paper to a wider class of examples. That is, a.s. there is a unique escape orbit which belongs to the alpha and omega-limit of every other trajectory. Then, following a recent work of Troubetzkoy, we prove that generically these systems are ergodic for almost all initial velocities,…
Boltzmann's Billiard Systems: Computation of the Billiard Mapping and Some Numerical Results
本文针对 Boltzmann 提出的台球系统，通过几何与数值分析研究其动力学，发现多数系统呈现混沌，其中一个系统可能遍历，从而检验其遍历性断言。
Outer Billiards is a geometrically inspired dynamical system based on a convex shape in the plane. When the shape is a polygon, the system has a combinatorial flavor. In the polygonal case, there is a natural acceleration of the map, a first return map to a certain strip in the plane. The arithmetic graph is a geometric encoding of the symbolic dynamics of this first return map. In the case of the regular octagon, the case we study, the arithmetic graphs associated to periodic orbits are polygonal paths in R^8. We are interested in the asymptotic shapes of these polygonal paths, as the period …
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 planar polygonal billiard $¶$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $¶$ there exists a finite number of ``blocking'' points $B_1, ..., B_n$ such that every billiard trajectory from $O$ to $A$ meets one of the $B_i$'s. As a counter-example to a theorem of Hiemer and Snurnikov, we construct a family of rational billiards that lack the finite blocking property.
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…
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 consider a class of mechanical particle systems interacting with thermostats. Particles move freely between collisions with disk-shaped thermostats arranged periodically on the torus. Upon collision, an energy exchange occurs, in which a particle exchanges its tangential component of the velocity for a randomly drawn one from the Gaussian distribution with the variance proportional to the temperature of the thermostat. In the case when all temperatures are equal one can write an explicit formula for the stationary distribution. We consider the general case and show that there exists a uniqu…
Linear Flows on Translation Prisms
本文研究平移棱柱上的线性流，利用Furstenberg和Veech思想建立平移曲面弱混合性与棱柱流遍历性的联系，得到唯一遍历性结果，并构造Pisot扩张因子方向上的显式特征函数及非遍历棱柱流和台球流例子。
Semi-deterministic processes with applications in random billiards
本研究刻画了交替随机游走与随机台球系统的遍历性条件，并给出产生遍历动力学的粗糙微结构实例，填补了扩散与确定性系统之间的空白。
Random circular billiards on surfaces of constant curvature: Pseudo integrability and mixing
本文在常曲率曲面上定义了具有不变Liouville测度的随机台球映射，并证明了随机圆形台球的遍历性等动力学性质。
This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory" (from 4 to 16 July, 2011). In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the SL(2,R)-action on these moduli spaces and the Kontsevich-Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flo…
This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces, and in particular the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi. We also discuss the context and applications of this result, and connections to other areas of mathematics such as algebraic geometry, Teichmuller theory, and ergodic theory on homogeneous spaces.
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。
We prove Knudsen's law for a gas of particles bouncing freely in a two dimensional pipeline with serrated walls consisting of irrational triangles. Dynamics are randomly perturbed and the corresponding random map studied under a skew-type deterministic representation which is shown to be ergodic and exact.
Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about mixing properties for billiard tables with non-compact cusps. There is no consensual definition of mixing for systems with infinite invariant measure. In this paper we study geometric and ergodic properties of billiard tables with a non-compact cusp. The goal of this text is, using the definition of mixing p…
Is every triangle a trajectory of an elliptical billiard?
本文运用 Marden 定理证明每个三角形、平行四边形及“Darboux butterflies”自交折线均为某唯一椭圆内的台球轨迹，并有效计算出边界椭圆的焦点，揭示了 Marden 定理与 Poncelet Porism 间的联系。
We prove that flow of a generic geodesic on a flat surface with finite holonomy group is ergodic. We use this result to prove that flows of generic billiards on certain flat surfaces with boundary are also ergodic.
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…
In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent. In this paper we focus on the problem of recurrence for elements of smooth curves in the moduli space. We give an effective criterion for the recurrence of almost every element of a smooth curve. The criterion relies on results developed by Minsky-Weiss in \cite{Mi-We}. Next we apply the criterion to the billiard flow on planar tables confined by arcs of confocal conics. The phase space of such billiard flow splits into invariant subsets determined by caustics.…
We prove almost sure ergodic theorems for a class of systems called quasistatic dynamical systems. These results are needed, because the usual theorem due to Birkhoff does not apply in the absence of invariant measures. We also introduce the concept of a physical family of measures for a quasistatic dynamical system. These objects manifest themselves, for instance, in numerical experiments. We then verify the conditions of the theorems and identify physical families of measures for two concrete models, quasistatic expanding systems and quasistatic dispersing billiards.
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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