Pseudo-Euclidean Billiards within Confocal Curves on the Hyperboloid of One Sheet

Pseudo-integrable billiards and arithmetic dynamics

We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties of such billiards are studied. The novelty, caused by reflex angles on boundary, induces invariant leaves of higher genera and dynamical behaviour different from Liouville-Arnold's theorem. Its analogue is derived from the Maier theorem on measured foliations. A local version of Poncelet theorem is formulated and necessary algebro-geometric conditions for per…

On commuting billiards in higher-dimensional spaces of constant curvature

We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov. The main result is deduced from the classical theorem due to Marcel Berger saying that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the…

只有二次曲面才具有伪焦散线——关于高维黎曼、伪欧几里得与射影弹子球的焦散线研究

Only quadrics have pseudo-caustics -- on caustics of Riemannian, pseudo-Euclidean and projective billiards in higher dimensions

该文研究高维射影弹子球的焦散线问题，统一了多种度量下的反射定律，刻画了焦散线仅为二次曲面的射影弹子球类，并将结果推广到黎曼与伪欧几里得弹子球，证明后者的焦散线必为伪共焦二次曲面。

Closed geodesics and billiards on quadrics related to elliptic KdV solutions

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

Conformal Transformations and Integrable Mechanical Billiards

In this article we explain that several integrable mechanical billiards in the plane are connected via conformal transformations. We first remark that the free billiard in the plane are conformal equivalent to infinitely many billiard systems defined in central force problems on a particular fixed energy level. We then explain that the classical Hooke-Kepler correspondence can be carried over to a correspondence between integrable Hooke-Kepler billiards. As part of the conclusion we show that any focused conic section gives rise to integrable Kepler billiards, which brings generalizations to a…

Complex caustics of the elliptic billiard

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.

Recurrence for smooth curves in the moduli space and application to the billiard flow on nibbled ellipses

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.…

Wire billiards, the first steps

Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature $k$ in $\mathbb{R}^n$ (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In this paper we are investigating whether the basic features of conventional planar billiards extend to this more general situation. In particular, we extend Lazutkin's KAM result, as well as Mather's converse KAM result, to wire billiards. We address the notion of caustics: for wire billiards, it corresponds to striction curve of the ruled surface spanned by the…

Four equivalent properties of integrable billiards

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. Let us say that a geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed pl…

Poncelet 拨片：余弦空间中的和谐曲线

Poncelet Plectra: Harmonious Curves in Cosine Space

本文研究共焦椭圆台球中 Poncelet 三角形及其仿射像的余弦不变量，证明二者的余弦三元组扫过相同的拨片形三次曲线或球面曲线。