Billiard Table Research Articles

Linear and nonlinear stability of periodic orbits in annular billiards.

An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary and also a circular scatterer in the interior of the disk. We investigate the stability properties of some periodic orbits in annular billiards in which the scatterer is touching or close to the boundary. We analytically show that there exist linearly stable periodic orbits of an arbitrary period for scatterers with decreasing radii that are located near the boundary of the disk. As the position of the scatterer moves away from a symmetry line of a periodic orbit, the stability of periodic orbits changes from elliptic to hyperbolic, corresponding to a saddle-center bifurcation. When the scatterer is tangent to the boundary, the periodic orbit is parabolic. We prove that slightly changing the reflection angle of the orbit in the tangential situation leads to the existence of Kolmogorov-Arnold-Moser islands. Thus, we show that there exists a decreasing to zero sequence of open intervals of scatterer radii, along which the billiard table is not ergodic.

Lagrangian relations and linear point billiards

Motivated by the high-energy limit of the N-body problem we construct non-deterministic billiard process. The billiard table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed polygonal curve with vertices on the subspaces and change of direction upon hitting a subspace governed by ‘conservation of momentum’ (mirror reflection). The itinerary of a trajectory is the list of subspaces it hits, in order. (A) Are itineraries finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In a beautiful series of papers Burago–Ferleger–Kononenko [BFK] answered (A) affirmatively by using non-smooth metric geometry ideas and the notion of a Hadamard space. We answer (B) by proving that this space of trajectories is diffeomorphic to a Lagrangian relation on the space of lines in the Euclidean space. Our methods combine those of BFK with the notion of a generating family for a Lagrangian relation.

Estimating the fractal dimension of sets determined by nonergodic parameters

Given fixed and irrational $0<α, θ<1$, consider the billiard table $B_{α}$ formed by a $\frac{1}{2}×1$ rectangle with a horizontal barrier of length $α$ emanating from the midpoint of a vertical side and a billiard flow with trajectory angle $θ$. In 1969, Veech introduced two subsets $K_{0}(θ)$ and $K_{1}(θ)$ of $\mathbb{R}/\mathbb{Z}$ that are defined in terms of the continued fraction representation of $θ∈\mathbb{R}/\mathbb{Z}$, and Veech showed that these sets have Hausdorff dimension $0$ when $θ$ is rational. Moreover, the set $K_{1}(θ)$ describes the set of all $α$ such that the billiard flow on $B_{α}$ in direction $θ$ is nonergodic. We show that the Hausdorff dimension of the sets $K_{0}(θ)$ and $K_{1}(θ)$ can attain any value in $[0, 1]$ by considering the continued fraction expansion of $θ$. This result resolves an analogue of work completed by Cheung, Hubert, and Pascal in which they consider, for fixed $α$, the set of $θ$ such that the flow on $B_{α}$ in direction $θ$ is nonergodic.

An integrable deformation of an ellipse of small eccentricity is an ellipse

The classical Birkho conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity.

Nontrivial paths and periodic orbits of the T-fractal billiard table

We introduce and prove numerous new results about the orbits of the T-fractal billiard. Specifically, in section , we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In section , we examine the limiting behavior of particular sequences of compatible periodic orbits. Additionally, sufficient conditions for the existence of particular nontrivial paths are given in section . The proofs of two results of Lapidus and Niemeyer (2013 The current state of fractal billiards Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics (Contemporary Mathematics vol 601) ed D Carfi et al (Providence, RI: American Mathematical Society) pp 251–88 (e-print: arXiv:math.DS.1210.0282v2, 2013) appear here for the first time, as well. In section , an orbit with an irrational initial direction reaches an elusive point in a way that yields a nontrivial path of finite length, yet, by our convention, constitutes a singular orbit of the fractal billiard table. The existence of such an orbit seems to indicate that the classification of orbits may not be so straightforward. A discussion of our results and directions for future research is then given in section .

The ANAIS billiard experiment

The problem on the motion of a homogeneous ball on a horizontal plane is considered. The existence of the vector first integral independent of the friction character and the motion of the support plane in the horizontal directions is shown. In particular, the velocities of the ball center are identical at the moments of rolling on a motionless plane, which corresponds to experiments with the ANAIS billiard table.

Anosov Geodesic Flows, Billiards and Linkages

Any smooth surface in R^3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R^2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then the geodesic flow of the corresponding surface is Anosov. We apply this result to the theory of mechanical linkages and their dynamics: we provide a new example of a simple linkage whose physical behavior is Anosov. For the first time, the edge lengths of the mechanism are given explicitly.

Generic absence of finite blocking for interior points of Birkhoff billiards

Let $x$ and $y$ be points in a billiard table $M=M(\sigma)$ in$\mathbb{\mathbb{R}}^{2}$ that is bounded by a curve $\sigma$. Weassume $\sigma\in\Sigma_{r}$ with $r\geq2$, where $\Sigma_{r}$is the set of simple closed $C^{r}$ curves in $\mathbb{R}^{2}$ withpositive curvature. A subset $B$ of $M\setminus\{x,y\}$ is calleda blocking set for the pair $(x,y)$ if every billiard pathin $M$ from $x$ to $y$ passes through a point in $B$. If a finiteblocking set exists, the pair $(x,y)$ is called secure in$M;$ if not, it is called insecure. We show that for $\sigma$in a dense $G_{\delta}$ subset of $\Sigma_{r}$ with the $C^{r}$topology, there exists a dense $G_{\delta}$ subset $\mathcal{\mathcal{R}=R}(\sigma)$of $M(\sigma)\times M(\sigma)$ such that $(x,y)$ is insecure in$M(\sigma)$ for each $(x,y)\in\mathcal{R}$. In this sense, for thegeneric Birkhoff billiard, the generic pair of interior points isinsecure. This is related to a theorem of S. Tabachnikov [24]that $(x,y)$ is insecure for all sufficiently close points$x$ and $y$ on a strictly convex arc on the boundary of a smoothtable.

Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables

ABSTRACTLet q ≥ 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by studying the asymptotic behavior of the differences of the lengths of their axisymmetric (1, q)-periodic trajectories as q → +∞. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor q−3e−rq times either a constant or an oscillating function, and the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1, q)-periodic trajectories. Our experiments are focused on some perturbed ellipses and circles, so we can compare the numerical results with some analytical predictions obtained by Melnikov methods. We also detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.

On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables--asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than forty years ago.

Stochastic perturbations of convex billiards

We consider a strictly convex billiard table with C2 boundary, with the dynamics subjected to random perturbations. Each time the billiard ball hits the boundary its reflection angle has a random perturbation. The perturbation distribution corresponds to a situation where either the scale of the surface irregularities is smaller than but comparable to the diameter of the reflected object, or the billiard ball is not perfectly rigid. We prove that for a large class of such perturbations the resulting Markov chain is uniformly ergodic, although this is not true in general.

Devil's staircase – rotation number of outer billiard with polygonal invariant curves

ABSTRACTWe discuss rotation number associated with the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon η, we consider the outer billiard table constructed by cutting out a fixed area from the interior of η. is piecewise hyperbolic and the polygon η is an invariant curve of under the billiard map φ. We show that, if β ∈ η is a periodic point under φ with rational rotation number , then φq is not the local identity at β. This proves that the rotation number τ as a function of the parameter is a devil's staircase function.

Stability and ergodicity of moon billiards.

We construct a two-parameter family of moon-shaped billiard tables with boundary made of two circular arcs. These tables fail the defocusing mechanism and other known mechanisms that guarantee ergodicity and hyperbolicity. We analytically study the stability of some periodic orbits and prove there is a class of billiards in this family with elliptic periodic orbits. These moon billiards can be viewed as generalization of annular billiards, which all have Kolmogorov-Arnold-Moser islands. However, the novelty of this paper is that by varying the parameters, we numerically observe a subclass of moon-shaped billiards with a single ergodic component.

Andreyev, Alexandre. The Myth of the Masters Revived: The Occult Lives of Nikolai and Elena Roerich (review)

SEER, 93, 3, JULY 2015 556 net, Shachar Pinsker concludes that ‘The Viennese coffeehouse served perhaps as a paradigm, but it was also, like the East European Jewish migrants and like modernism, a transnational phenomenon that moved from one city to another’ (p. 94). Pinsker usefully observes that in many of the region’s cafés (including also Vienna’s) literary modernism developed not just in German but in Yiddish, Hebrew and other languages. In Zagreb, Ines Sabotič shows, the imprint of the Viennese café could be felt in the decorations, furniture and patterns of sociability of local cafés, even as they fostered an increasingly assertive and national-minded Croatian literary culture. Importantly, Sabotič analysesthemanylawsandregulationsthatgovernedcafés,taverns,restaurants and hotels. In Zagreb, these contributed to a sharp distinction between cafés clearly designated for the bourgeoisie (the kavana, which was required by law to have a billiard table) and more humble coffee establishments aimed at the working classes (the kavotočje). Sabotič’s attention to the legal framework and to coffeehouse owners points to aspects of ‘the Viennese café’ that would reward further research. In this volume, we hear surprisingly little about those who sold and served the coffee and almost nothing about the coffee itself (nor about food and other drinks). Amusements like chess, music and billiards likewise receive scant mention. Like a well-made Mélange, this volume is rich and satisfying; in the end, however, it may leave you wanting even more. Colgate University Robert Nemes Andreyev, Alexandre. The Myth of the Masters Revived: The Occult Lives of Nikolai and Elena Roerich. Eurasian Studies Library, 4. Brill, Leiden and Boston, MA, 2014. xxix + 502 pp. Illustrations. Notes. Bibliography. Index. €138.00: $179.00. This fascinating book describes the lives and travels of the painter Nikolai Roerich, his wife Elena and their sons Iuri and Sviatoslav, who from a privileged position in imperial St Petersburg, moved first to Finland in 1917 and from there to Britain, the United States, finally reaching India via France in 1923. In Asia, Nikolai, Elena and Iuri made two noted journeys: between 1925 and 1928 they travelled from India to Xinjiang, the Soviet Union, Mongolia and Tibet; and in 1934–35 they visited Japan, Manchuria and Inner Mongolia. Elena and Nikolai Roerich were products of Russia’s fin-de-siècle. They were influenced by Nietzscheanism, spiritualism, Blavatskii’s theosophy and buddhism; they felt that the meaning of life had to be rediscovered in ancient cultures, and were convinced that spiritual revolution was imminent. According to Andreyev, Elena was the driving mystic of the family. She began acting as a medium while still in Russia, channelling spirit messages and REVIEWS 557 experiencing visions. In 1920, during a walk in London’s Hyde park, Elena saw two Indians in British military uniforms, whom she immediately recognized as Helena Blavatskii’s spiritual mahatmas, masters Morya and Koot Hoomi. From this day on, master Morya would become her spirit guide and through daily communications influence every decision the family took. From Britain, the Roerichs travelled to New York, where they became recognized artistic and spiritual leaders and gained an impressive following: their American devotees not only supported the family throughout their many travels, but also staffed, ran and generously funded Roerich institutions. These included the Roerich Museum, the Master Institute of United Arts, the Corona Mundi Art Center, and the Roerich Museum Press, all of them eventually housed in a purpose-built twenty-nine-storey skyscraper in Manhattan, as well as the Urusvati Himalayan Research Institute at the family’s Indian residence. The aim of these institutions was to achieve universal brotherhood through beauty, develop a new religion and analyse the magical qualities of the Himalayan mountains. The ultimate aim of the Roerichs, however, was to bring about spiritual revolution, prepare the return of the Buddha and the establishment of the mystical kingdom of Shambhala. Initially, Nikolai expected these events to take place in the Altai mountains, but when Soviet authorities obstructed his plans for ‘cooperative Buddhism’, the promised land moved to Tibet. There, the Dalai Lama turned out to be difficult, so the spirit guides relocated their future residence to Manchuria and eventually to...

A Franks' lemma for convex planar billiards

Let γ be an orbit of the billiard flow on a convex planar billiard table; then the perpendicular part of the derivative of the billiard flow along γ is a symplectic linear map DP. This paper contains a proof of the following Franks’ lemma for a residual set of convex planar billiard tables: for any closed orbit, the map DP can be perturbed freely within a neighbourhood in Sp(1) by a C2-small perturbation in the space of convex planar billiard tables.

On a conjugacy problem in billiard dynamics

We study symmetric billiard tables for which the billiard map is locally (near an elliptic periodic orbit of period 2) conjugate to a rigid rotation. In the previous paper (Physica D 255, 31–34 (2013)), we obtained an equation (called below the conjugacy equation) for such tables and proved that if α, the rotation angle, is rationally incommensurable with π, then the conjugacy equation has a solution in the category of formal series. In the same paper there is also numerical evidence that for “good” rotation angles the series have positive radii of convergence. In the present paper we carry out a further study (both analytic and numerical) of the conjugacy equation. We discuss its symmetries, dependence of the convergence radius on α, and other aspects.

Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards.

We study diffusion on a periodic billiard table with an infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a Lévy walk combining exponentially distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients, which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [J. Machta and R. Zwanzig, Phys. Rev. Lett. 50, 1959 (1983)PRLTAO0031-900710.1103/PhysRevLett.50.1959].

PAKISTAN'S DANGEROUS GAME WITH RELIGIOUS EXTREMISM

Snooker is a popular game in Pakistan. Played on a billiards table, competitors wager on who can knock the most colored balls off into the side pockets. The winner is the one who wipes the table cl...

Polygonal Billiards with One Sided Scattering

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.

Periodic billiard orbits of self-similar Sierpiński carpets

We identify a collection of periodic billiard orbits in a self-similar Sierpiński carpet billiard table Ω(Sa). Based on our refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in Sa, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpiński carpet billiard tables Ω(Sa,n). The trivial limit of this sequence then constitutes a periodic orbit of Ω(Sa). We also determine the corresponding translation surface S(Sa,n) for each prefractal table Ω(Sa,n), and show that the genera {gn}n=0∞ of a sequence of translation surfaces {S(Sa,n)}n=0∞ increase without bound. Various open questions and possible directions for future research are offered.