New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were introduced by the authors in "Can the Elliptic Billiard Still Surprise Us?" (2020), Math. Intelligencer, 42(1): 6--17, some of which were generalized to $N>3$. Invariants mentioned there included ratios of radii and/or areas, sum of angle cosines, and a special stationary circle. Here we present some of the proofs omitted there as well as a few new related facts.
We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.
We consider 3-periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law or rfelection.
The locus of centers of inscribed circles in triangles, the 3-periodic orbits of an elliptic billiard, is also an ellipse. In this work we obtain the canonical equation of this ellipse, complementing the previous results obtained by O. Romaskevich.
Data Science Consulting（数据科学咨询公司）；Federal University of Goiás（戈亚斯联邦大学）；Federal University of Rio de Janeiro（里约热内卢联邦大学）
We introduce several-dozen experimentally-found invariants of Poncelet N-periodics in the confocal ellipse pair (Elliptic Billiard). Recall this family is fully defined by two integrals of motion (linear and angular momentum), so any "new" invariants are dependent upon them. Nevertheless, proving them may require sophisticated methods. We reference some two-dozen proofs already contributed. We hope this article will motivate contributions for those still lacking proof.
Data Science Consulting（数据科学咨询公司）；Inst. de Matemática e Estatística, Univ. Federal de Goiás（戈亚斯联邦大学数学与统计学院）
A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing invariances such as aspect ratio and pairwise ratio of focal lengths.
Ivrii's Conjecture states that in every billiard in Euclidean space the set of periodic orbits has measure zero. It implies that for every $k\geq2$ there are no k-reflective billiards, i.e., billiards having an open set of k-periodic orbits. This conjecture is open in Euclidean spaces, with just few partial results. It is known that in the two-dimensional sphere there exist 3-reflective billiards (Yu.M.Baryshnikov). All the 3-reflective spherical billiards were classified in a paper by V.Blumen, K.Kim, J.Nance, V.Zharnitsky: the boundary of each of them lies in three orthogonal big circles. In…
The dynamic geometry of the family of 3-periodics in the Elliptic Billiard is mystifying. Besides conserving perimeter and the ratio of inradius-to-circumradius, it has a stationary point. Furthermore, its triangle centers sweep out mesmerizing loci including ellipses, quartics, circles, and a slew of other more complex curves. Here we explore a bevy of new phenomena relating to (i) the shape of 3-periodics and (ii) the kinematics of certain Triangle Centers constrained to the Billiard boundary, specifically the non-monotonic motion some can display with respect to 3-periodics. Hypnotizing is …
Univ. Federal de Goiás（戈亚斯联邦大学）；Univ. Federal de Juiz de Fora（茹伊斯迪福拉联邦大学）；Data Science Consulting（数据科学咨询公司）
A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods to prove if the locus of a given center is an ellipse: one based on computer algebra, and another based on an algebro-geometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic
Inst. de Matemática e Estatística, Univ. Federal de Goiás（戈亚斯联邦大学数学与统计学院）；Data Science Consulting（数据科学咨询公司）
Discovered by William Chapple in 1746, the Poristic family is a set of variable-perimeter triangles with common Incircle and Circumcircle. By definition, the family has constant Inradius-to-Circumradius ratio. Interestingly, this invariance also holds for the family of 3-periodics in the Elliptic Billiard, though here Inradius and Circumradius are variable and perimeters are constant. Indeed, we show one family is mapped onto the other via a varying similarity transform. This implies that any scale-free quantities and invariants observed in one family must hold on the other.
Data Science Consulting（数据科学咨询公司）；Universidade Federal de Goiás（戈亚斯联邦大学）；Universidade Federal de Juiz de Fora（茹伊斯迪福拉联邦大学）
Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery of curves: ellipses, quartics, sextics, circles, and even a stationary point. Secondly, one notices this family conserves an intriguing ratio: Inradius-to-Circumradius. In turn this implies three conservation corollaries: (i) the sum of bounce angle cosines, (ii) the product of excentral cosines, and (iii) the ratio of excentral-to-orbit areas. Monge's Orth…
This paper addresses the question of genericity of existence of elliptic islands for the billiard map associated to strictly convex closed curves. More precisely, we study 2-periodic orbits of billiards associated to C5 closed and strictly convex curves and show that the existence of elliptic islands is a dense property on the subset of those billiards having an elliptic 2-periodic point. Our main tools are normal perturbations, the Birkhoff Normal Form for elliptic fixed points and Moser's Twist Theorem.
We study pinball billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls are non-elastic: the outgoing angle with the normal vector to the boundary is a uniform factor $λ< 1$ smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as t…
We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero system
We prove that any sufficiently small perturbation of an isosceles triangle has a periodic billiard path. Our proof involves the analysis of certain infinite families of Fourier series that arise in connection with triangular billiards, and reveals some self-similarity phenomena in irrational triangular billiards. Our analysis illustrates the surprising fact that billiards on a triangle near a Veech triangle is extremely complicated even though Billiards on a Veech triangle is very well understood.
Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed polygons--- are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This…
We introduce a class of billiards with chaotic unidirectional flows (or non-chaotic unidirectional flows with "vortices") which go around a chaotic or non-chaotic "core", where orbits can change their orientation. Moreover, the corresponding billiard tables are simply connected in difference with many attempts to build billiards with interesting and/or exotic dynamics by putting inside billiard tables various "scatterers" with funny shapes. Therefore the billiards in this new class are amenable to experimental studies in physics labs as well as to the rigorous mathematical ones, which may shed…
We compare invariants of N-periodic trajectories in the elliptic billiard, classic and new, to their aperiodic counterparts via a spatial integrals evaluated over the boundary of the elliptic billiard. The integrand is weighed by a universal measure equal to the density of rays hitting a given boundary point. We find that aperiodic averages are smooth and monotonic on caustic eccentricity, and perfectly match N-periodic average invariants at the discrete caustic parameters which admit a given N-periodic family.
The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exist exactly two confocal ellipses such that the triangular orbits of E are circumscribed about one of them, and each tangent line to one of those ellipses is a side of a triangular orbit. We also give a bound on the number of caustics for orbits with a fixed number of sides.
In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that $C^\infty$ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is that we use Herman's result on Diophantine invariant curves to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.
In this work we address the question of proving the stability of elliptic 2-periodic orbits for strictly convex billiards. Eventhough it is part of a widely accepted belief that ellipticity implies stability, classical theorems show that the certainty of stability relies upon more fine conditions. We present a review of the main results and general theorems and describe the procedure to fullfill the supplementary conditions for strictly convex billiards.
We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal …
We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of those billiard systems in terms of Fomenko invariants. We provide then periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions and classical Chebyshev and Zolotarev polynomials, as extremal polynomials over one or two intervals. These results are contrasted with the cases …
The arithmetic triangular billiards are classically chaotic but have Poissonian energy level statistics, in ostensible violation of the BGS conjecture. We show that the length spectra of their periodic orbits divides into subspectra differing by the parity of the number of reflections from the triangle sides; in the quantum treatment that parity defines the reflection phase of the orbit contribution to the Gutzwiller formula for the energy level density. We apply these results to all 85 arithmetic triangles and establish the boundary conditions under which the quantum billiard is \textquotedbl…
The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the re…
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…
The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii)…
We solve the longstanding problem of smoothing a stadium billiard. Besides our results demonstrate why there were no clear conjectures how much the stadium's boundary must be smoothened to destroy chaotic dynamics. To do that we needed to extend standard KAM theory to analyze stability of periodic orbits, because of the low smoothness of the system. In fact, the stadium has a $C^1$ boundary, and we show that $C^2$ smoothing results in appearance of elliptic periodic orbits.
On an elliptic billard, we study a complex reflexion law introduced by A. Glutsyuk in order to state that the locus of the circumcenters of triangular orbits is an ellipse.
In the class of strictly convex smooth boundaries, each of which not having strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the "normalized" Mather's $β$-function are invariants under $C^\infty$-conjugacies. In contrast, we prove that any two elliptic billiard maps are $C^0$-conjugated near their respective boundaries, and $C^\infty$-conjugated in the open cylinder, near the boundary and away from a plain passing through the center of the underlying ellipse. We also prove that if the billiard maps corresponding to two ellipses are topologically …
This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for families of Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary conditions, associated with C^1 families of billiard tables. We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on the i…
In this paper we are interested in the motion of a ball inside a billiard table bounded by a particular smooth curve. This table belongs to a family of billiards which can all be drawn by a common process: the so-called gardener's string construction. The classical elliptical billiard is, of course, the foremost member of this family. So it should come as no surprise that our hexagonal string billiard shares many basic properties with the latter, but, on the other hand, also exhibits some essential differences with it. We have gathered numerical evidence against the Birkhoff-Poritsky conjectur…
Non-Birkhoff periodic orbits in symmetric billiards
Heriot-Watt University（赫瑞瓦特大学）；Loughborough University（拉夫堡大学）；Vrije Universiteit Amsterdam - Faculty of Science（阿姆斯特丹自由大学理学院）；Università degli Studi di Trento（特伦托大学）；Politecnico di Torino（都灵理工大学）；Centre Inria d’Université Côte d’Azur（法国国家信息与自动化研究所蔚蓝海岸大学中心）
The main goal of the present paper is to evaluate the perturbed locations and investigate the linear stability of the triangular points. We studied the problem in the elliptic restricted three body problem frame of work. The problem is generalized in the sense that the two primaries are considered as triaxial bodies. It is found that the locations of these points are affected by the triaxiality coefficients of the primaries and the eccentricity of orbits. Also it is observed that the stability regions depend on the involved perturbations. In addition to this we studied the periodic orbits in t…
This article is concerned with the study of Mather's β-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum. After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both th…
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 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 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 …
We investigate the existence of elliptic islands for a special family of periodic orbits of a two-parameter family of maps corresponding to the billiard problem on the elliptical stadium. The hyperbolic or elliptical character of these orbits is also investigated. Depending on the parameters, we obtain upper bounds of ellipticity for this special family as a lower bound for chaos. On a different region of the parameter space, we can prove that there is no upper bound for the existence of elliptic islands. The main results we use are Birkhoff Normal Form and Moser's Twist Theorem.
