arXiv subjects

Serge Tabachnikov

Publications and source records attributed to Serge Tabachnikov.

At least 19 recordsLinked to original sources

arXiv2026

Symplectic billiards as Minkowski billiards

We establish a connection between Minkowski billiards and symplectic billiards, two classes of dynamical systems that have been studied largely independently. We show that the Minkowski billiard map can be described in symplectic terms via reduction from the canonical symplectic structure on $V \times V^*$, and that symplectic billiards can be viewed as a ``square root'' of a symplectic version of Minkowski billiards. As an application, we recover several known results on symplectic billiards from the more general Minkowski setting, and extend some of them to higher dimensions and to periodic orbits of even period. In particular, we prove the existence of at least $(r-1)(n-1)$ $2r$-periodic symplectic billiard orbits in dimension $2n$.

math.DS↗

arXiv2026

On continuous 2-frieze patterns

We define and study a continuous version of 2-frieze patterns, a combinatorial structure closely related with frieze patterns of Coxeter and Conway. We describe the relation of continuous 2-friezes with the moduli space of projective curves and relate the (pre)symplectic structure on the space of closed 2-friezes, considered as a cluster variety, with the Adler-Gelfand-Dikii bracket on the space of 3rd order differential operators.

math.CO↗

arXiv2026

Remarks on the outer length billiards

We study outer length billiards; our main results are as follows. We prove 3- and 4-periodic versions of the Ivrii conjecture. We show that, for every period $n\ge 3$, there exists a functional space of billiard tables that possess invariant curves consisting of $n$-periodic points. For $n=4$, we explicitly parameterize such centrally symmetric billiard tables by functions of one variable and describe how to construct these tables geometrically, similarly to the known construction of Radon curves.

math.DS↗

arXiv2026

Open problems in billiards and quantitative symplectic geometry

This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.

math.SG↗

arXiv2025

Outer length billiards on a large scale

We present some foundational results about the outer length billiard system, including its generating function and the invariant area form. We describe the limiting behavior of the orbits far away from the billiard table: the orbits of the map lie on the origin-centered circles, and the second iteration of the map is approximated by the flow of a Hamiltonian function whose level curves are these circles. Furthermore, the orbits "at infinity" of the centers of the auxiliary circles involved in the definition of the map lie on the curves that are polar dual to the symmetrization of the billiard table, the curves which are traced by the second iteration of the usual outer billiard map "at infinity".

math.DS↗

arXiv2025

Outer symplectic billiard map at infinity

We show that the second iteration $T^2$ of the outer symplectic billiard map with respect to a convex domain $M$ in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from $M$. More precisely, denote by $N$ the symplectic polar dual of the symmetrization $M\ominus M$ of $M$. If we write $N$ as the unit level set of a 1-homogeneous function $H$, then the difference between $T^2$ and the time-2-Hamiltonian flow of $H$ applied to a point $x$ is smaller than $c/|x|$ for some constant $c$ depending only on $M$. Moreover, we show that if an orbit escapes to infinity, then its distance to the origin grows not faster than $\sqrt{k}$ in the number of iterations. Finally, we prove that a $k$-periodic orbit needs to be close, in terms of $k$, to $M$.

math.SG↗

arXiv2025

On cusps of caustics by reflection in two dimensional projective Finsler metrics

A Finsler, not necessarily symmetric, metric in the plane or its convex subset is called projective if its geodesics are straight segments. We consider Finsler billiards in a convex planar domain endowed with a projective Finsler metric. A caustic by reflection is the envelope of the oriented lines, the billiard trajectories, that start at a point inside the billiard and undergo a fixed number of reflections. We show that such a caustic has at least four cusps. This problem is motivated by the "Last Geometric Statement of Jacobi" that the conjugate locus of a non-umbilic point of a triaxial ellipsoid has exactly four cusps. The present note extends the recent results in this direction concerning Euclidean billiards.

math.DG↗

arXiv2024

A 4-point theorem: still another variation on an old theme

An old theorem, due to Graustein, asserts that the average curvature of a plane oval is attained at least at four points. We present a proof by way of wave propagation and extend this result to the spherical and hyperbolic geometries - in the latter case, to horocyclically convex curves only.

math.DG↗

arXiv2024

Outer symplectic billiards

A submanifold of the standard symplectic space determines a partially defined, multi-valued symplectic map, the outer symplectic billiard correspondence. Two points are in this correspondence if the midpoint of the segment connecting them is on the submanifold, and this segment is symplectically orthogonal to the tangent space of the submanifold at its midpoint. This is a far-reaching generalization of the outer billiard map in the plane; the particular cases, when the submanifold is a closed convex hypersurface or a Lagrangian submanifold, were considered earlier. Using a variational approach, we establish the existence of odd-periodic orbits of the outer symplectic billiard correspondence. On the other hand, we give examples of curves in 4-space which do not admit 4-periodic orbits at all. If the submanifold satisfies certain conditions (which are always satisfied if its dimension is at least half of the ambient dimension) we prove the existence of two $n$-reflection orbits connecting two transverse affine Lagrangian subspaces for every $n\geq1$. In addition, for every immersed closed submanifold, the number of single outer symplectic billiard ``shots" from one affine Lagrangian subspace to another is no less than the number of critical points of a smooth function on this submanifold. We study, in detail, the behavior of this correspondence when the submanifold is a curve or a Lagrangian submanifold. For Lagrangian submanifolds in 4-dimensional space we present a criterion for the outer symplectic billiard correspondence to be an actual map. We show, in every dimension, that if a Lagrangian submanifold has a cubic generating function, then the outer symplectic billiard correspondence is completely integrable in the Liouville sense.

math.SG↗

arXiv2024

When Gr\"unbaum meets Poncelet -- Infinite Classes of Movable $n_4$ Configurations

We study relations between $(n_4)$ incidence configurations and the classical Poncelet Porism. Poncelet's result studies two conics and a sequence of points and lines that inscribes one conic and circumscribes the other. Poncelet's Porism states that whether this sequence closes up after $m$ steps only depends on the conics and not on the initial point of the sequence. In other words: Poncelet polygons are movable. We transfer this motion into a flexibility statement about a large class of $(n_4)$ configurations, which are configurations where 4 (straight) lines pass through each point and four points lie on each line. A first instance of such configurations in real geometry had been given by Gr\"unbaum and Rigby in their classical 1990 paper where they constructed the first known real geometric realisation of a well known combinatorial $(21_4)$ configuration (which had been studied by Felix Klein), now called the Gr\"unbaum-Rigby configuration. Since then, there has been an intensive search for movable $(n_4)$ configurations, but it is very surprising that the Gr\"unbaum-Rigby $(21_4)$ configuration admits nontrivial motions. It is well-known that the Gr\"unbaum-Rigby configuration is the smallest example of an infinite class of $(n_4)$ configurations, the trivial celestial configurations. A major result of this paper is that we show that all trivial celestial configurations are movable via Poncelet's Porism and results about properties of Poncelet grids. Alternative approaches via geometry of billiards, in-circle nets, and pentagram maps that relate the subject to discrete integrable systems are given as well.

math.CO↗

arXiv2024

Explicit Constructions for Poncelet Polygons

We study the geometric structure of Poncelet $n$-gons from a projective point of view. In particular we present explicit constructions of Poncelet $n$-gons for certain $n$ and derive algebraic characterisations in terms of bracket polynomials. Via the connections of Poncelet polygons and $(N_4)$-configurations, the results of this article can be used to construct a large class of specific movable $(N_4)$-configurations, the trivial celestial 4-configurations, which up to this point were all thought to be rigid and to require regular polygons for their construction.

math.CO↗

arXiv2024

Cusps of caustics by reflection in ellipses

This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point $O$ inside an elliptic billiard table, one considers the family of rays emanating from $O$ and the caustic $\Gamma_n$ of the reflected family after $n$ reflections off the ellipse, for each positive integer $n$. It is known that $\Gamma_n$ has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of $\Gamma_n$, though we do not prove that these are the only cusps.

math.DG↗

arXiv2024

A family of maps and a vector field on plane polygons

We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of completely integrable transformation on closed plane curves, known as the bicycle correspondence, that is a geometric realization of the B\"acklund transformation of the planar filament equation. For odd-gons, we construct a symplectic form on the quotient space by parallel translations and show that the transformations are symplectic, and the vector field is Hamiltonian. In the case of triangles, we prove complete integrability of the respective vector field and provide evidence for the conjecture that the transformations are integrable as well.

math.DS↗

arXiv2023

Monotone twist maps and Dowker-type theorems

Given a planar oval, consider the maximal area of inscribed $n$-gons resp. the minimal area of circumscribed $n$-gons. One obtains two sequences indexed by $n$, and one of Dowker's theorems states that the first sequence is concave and the second is convex. In total, there are four such classic results, concerning areas resp. perimeters of inscribed resp. circumscribed polygons, due to Dowker, Moln\'ar, and Eggleston. We show that these four results are all incarnations of the convexity property of Mather's $\beta$-function (the minimal average action function) of the respective billiard-type systems. We then derive new geometric inequalities of similar type for various other billiard system. Some of these billiards have been thoroughly studied, and some are novel. Moreover, we derive new inequalities (even for conventional billiards) for higher rotation numbers.

math.DS↗

arXiv2023

Bicycling geodesics are Kirchhoff rods

A bicycle path is a pair of trajectories in ${\mathbb R}^n$, the `front' and `back' tracks, traced out by the endpoints of a moving line segment of fixed length (the `bicycle frame') and tangent to the back track. Bicycle geodesics are bicycle paths whose front track's length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for $n\geq 3$ each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily of Kirchhoff rods, a class of curves introduced in 1859 by G. Kirchhoff, generalizing the planar elastic curves of J. Bernoulli and L. Euler.

math.DG↗

arXiv2023

Iterating skew evolutes and skew involutes: a linear analog of the bicycle kinematics

The evolute of a plane curve is the envelope of its normals. Replacing the normals by the lines that make a fixed angle with the curve yields a new curve, called the evolutoid. We prefer the term ``skew evolute", and we study the geometry and dynamics of the skew evolute map and of its inverse, the skew involute map. The relation between a curve and its skew evolute is analogous to the relation between the rear and front bicycle tracks, and this connections with the bicycle kinematics (a considerably more complicated subject) is our motivation for this study.

math.DG↗

arXiv2022

Differential geometry of space curves: Forgotten chapters

We study evolutes and involutes of space curves. Although much of the material presented is not new and can be found in classic treatises, we believe that a modern and unified treatment, complemented with several novel observations, may be useful. The results are illustrated with the help of computer graphics, a tool that was not not available to the classical geometers.

math.DG↗

arXiv2022

Self-Bäcklund curves in centroaffine geometry and Lamé's equation

Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centoraffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centoraffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam's problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics. We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.

math.DG↗