Periodic Billiard Trajectories Research Articles

Billiard partitions, Fibonacci sequences, SIP classes, and quivers

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d d -dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths.

Combinatorics of periodic ellipsoidal billiards

We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions and provide closed forms for the generating functions of these partitions.

Dan Reznik’s identities and more

Dan Reznik found, by computer experimentation, a number of conserved quantities associated with periodic billiard trajectories in ellipses. We prove some of his observations using a non-standard generating function for the billiard ball map. In this way, we also obtain some identities valid for all smooth convex billiard tables.

Counting Periodic Trajectories of Finsler Billiards

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 $r\ge 3$ is prime, then the number of $r$-periodic Finsler billiard trajectories is not less than $(r-1)(d-2)+1$. We also give stronger lower bounds when $M$ is in general position. The problem of estimating the number of periodic billiard trajectories from below goes back to Birkhoff. Our work extends to the Finsler setting the results previously obtained for Euclidean billiards by Babenko, Farber, Tabachnikov, and Karasev.

Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials

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 a 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 polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. Similarities and differences with respect to the previously studied Euclidean case are highlighted.

Periodic Ellipsoidal Billiard Trajectories and Extremal Polynomials

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 prove the injectivity of the frequency map. As a byproduct, for $d=2$ a new proof of the monotonicity of the rotation number is obtained. The case study of trajectories of small periods $T$, $d\le T\le 2d$ is given. In particular, it is proven that all $d$-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates $d+1$-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for $d=3$.

Quantitative error term in the counting problem on Veech wind-tree models

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We exhibit effective asymptotic formulas for the number of (isotopy classes of) periodic billiard trajectories (up to $\mathbb{Z}^2$-translations) on Veech wind-tree billiards, that is, wind-tree billiards whose underlying compact translation surfaces are Veech surfaces. We show that the error term depends on spectral properties of the Veech group and give explicit estimates in the case when obstacles are squares of side length $1/2$.

Caustics of Poncelet Polygons and Classical Extremal Polynomials

A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line and ellipsoidal billiards in d-dimensional space. Even in the planar case systematically studied in the present paper, it leads to new results in characterizing n periodic trajectories vs. so-called n elliptic periodic trajectories, which are n-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell’s equation. This new approach also sheds light on some classical results. In particular, we connect the search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which, as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for n = 3, 4, 5, 6 along with an effective search for caustics. As a byproduct, an intriguing connection between Cayley-type conditions and discriminantly separable polynomials has been observed for all those small periods.

Singular and Calabi–Yau varieties linked with billiard trajectories and diffusion operators

The images under a certain map of periodic billiard trajectories inside the fundamental region of the affine Weyl group of the root system $$A_{2}$$ are closed curves and configurations of lines described by a one-parameter family of polynomials. The polynomials are related with eigenvectors of symmetric diffusion operators connected with the deltoid. Several associated singular varieties and Calabi–Yau threefolds defined over the rationals are constructed.

УСТОЙЧИВОСТЬ ПЕРИОДИЧЕСКИХ БИЛЬЯРДНЫХ ТРАЕКТОРИЙ В ТРЕУГОЛЬНИКЕ

УСТОЙЧИВОСТЬ ПЕРИОДИЧЕСКИХ БИЛЬЯРДНЫХ ТРАЕКТОРИЙ В ТРЕУГОЛЬНИКЕ

On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different (p, q)-periodic orbits and every (p, q)-periodic orbit has its (p, q)-periodic action for suitable couples (p, q). We establish an exponentially small upper bound for the differences of (p, q)-periodic actions when the map is analytic on a (m, n)-resonant rotational invariant curve (resonant RIC) and p/q is ‘sufficiently close’ to m/n. The exponent in this upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem, in which the n-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay–Meiss–Percival action principle.We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the (1, q)-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period q. This improves some classical results of Marvizi, Melrose, Colin de Verdière, Tabachnikov, and others about the smooth case.

Smooth Billiards with a Large Weyl Remainder

We prove an estimate from below for the remainder in Weyl's law for smooth star-shaped planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. Our results confirm a prediction of P. Sarnak who proved a similar statement for surfaces without boundary. The proof is based on the analysis of the singularity in the wave trace, occurring at time T>0T>0 equal to the length of the closed trajectories belonging to the family. Such families of periodic orbits produce large singularities in the wave trace, which in turn imply estimates from below for the absolute value of the Weyl remainder averaged over a spectral window. We also obtain lower bounds for the error term in higher dimensions. In this case, the main contribution to the Weyl remainder typically comes from the singularity at zero of the wave trace. However, for certain domains, such as the Euclidean ball, the dimension of the family of periodic trajectories is large enough to dominate the contribution of the singularity at zero.

Periodic billiard trajectories and Morse theory on loop spaces

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 proposed by 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. Further, we recover two known results on the length of a shortest periodic billiard trajectory in a convex body: Ghomi’s inequality, and Brunn–Minkowski type inequality proposed by Artstein-Avidan–Ostrover.

Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances

We present very recent results related to the Poncelet Theorem on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of geometry, recalling for general mathematical audiences the dramatic historic circumstances which led to its discovery, a glimpse of its intrinsic appeal, and the importance of its relationship to dynamics of billiards within confocal conics. We focus on the three main issues: A) The case of pseudo-Euclidean spaces, for which we present a recent notion of relativistic quadrics and apply it to the description of periodic trajectories of billiards within quadrics. B) The relationship between so-called billiard algebra and the foundations of modern discrete differential geometry which leads to double-reflection nets. C) We present a new class of dynamical systems—pseudo-integrable billiards generated by a boundary composed of several arcs of confocal conics having nonconvex angles. The dynamics of such billiards have several extraordinary properties, which are related to interval exchange transformations and which generate families of flows that are minimal but not uniquely ergodic. This type of dynamics provides a novel type of Poncelet porisms—the local ones.

The existence of a billiard orbit in the regular hyperbolic simplex

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.

Displacement energy of unit disk cotangent bundles

We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle \(D^*M\) in a cotangent bundle \(T^*M\), when the base manifold \(M\) is an open Riemannian manifold. Our main result is that the displacement energy is not greater than \(C r(M)\), where \(r(M)\) is the inner radius of \(M\), and \(C\) is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.

Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra

We consider the relations between the lengths of periodic billiard trajectories in regular triangles, squares, and regular pentagons and those of closed geodesics on the surfaces of regular polyhedra. The cases of regular tetrahedra and octahedra are fully resolved in [3]. The cases of cubes and regular icosahedra are treated below. In the case of regular dodecahedra we can present only preliminary partial results.

Periodic Trajectories in the Regular Pentagon, II

In our recent paper [1], we studied periodic billiard trajectories in a regular pentagon and in the isosceles triangle with with the angles (π/5, π/5, 3π/5). We provided a full computation of the lengths of these trajectories, both geometric and combinatoric, and formulated some conjectures concerning symbolic periodic trajectories. The main goal of this article is to prove two of these conjectures. Technically, the study of billiard trajectories in a regular pentagon is essentially equivalent to the study of geodesics in the “double pentagon,” a translation surface obtained from two centrally symmetric copies of a regular pentagon by pairwise pasting the parallel sides. The result is a surface of genus 2 that has a flat structure inherited from the plane and a conical singularity. See [2, 3, 4, 5, 6, 7] for surveys of flat surfaces and rational polygonal billiards. Let us describe the relevant results from [1]. First of all, a periodic linear trajectory is always included into a parallel family of such trajectories, and when we talk about the period, length, symbolic orbit, etc., we always mean these parallel families. See Figure 1. Second, the double pentagon has an involution, the central symmetry that exchanges the two copies of the regular pentagon. This involution interchanges the linear trajectories that have the opposite directions. For this reason, we identify the opposite directions, so the set of directions is the real projective line RP. We identify this projective line with the circle at infinity of the hyperbolic plane in the Poincare disc model. It is clear from Figure 1 that the directions of the sides of the pentagons are periodic: every linear trajectory in this direction is closed. In fact, the socalled Veech dichotomy applies to the double pentagon: if there exists a periodic trajectory in some direction then all parallel trajectories are also periodic (and they form two strips, longer – shaded in Figure 1, and shorter – left unshaded,

Bounds for Minkowski Billiard Trajectories in Convex Bodies

In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiards.

Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

We study the geometry of confocal quadrics in pseudo-Euclidean spaces of arbitrary dimension d and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of our approach is the introduction of a new discrete combinatorial geometric structure associated with a confocal pencil of quadrics, a colouring in d colours. This is used to decompose quadrics of d+1 geometric types of a pencil into new relativistic quadrics of d relativistic types. A study of what we term discriminant sets of tropical lines Σ+ and Σ− and their singularities provides insight into the related geometry and combinatorics. This yields an analytic criterion describing all periodic billiard trajectories, including light-like trajectories as a case of special interest.