Off-center Reflections: Caustics and Chaos

On cusps of caustics by reflection: a billiard variation on Jacobi's Last Geometric Statement

A point source of light is placed inside an oval. The $n$-th caustic by reflection is the envelope of the light rays emanating from the light source after $n$ reflections off the curve. We show that each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard variation on Jacobi's Last Geometric Statement, concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

二维射影 Finsler 度量中反射焦散线的尖点

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

本文研究射影 Finsler 度量下凸平面区域中的 Finsler 台球，证明经历固定反射次数的反射焦散线至少具有四个尖点，推广了欧氏台球的相关结果。

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。

Chaos in the square billiard with a modified reflection law

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the stand…

A Random Billiard Map in the Circle

We consider a random billiard map, the one in which the standard specular reflection rule is replaced by a random reflection given by a Markov operator. We exhibit an invariant measure for random billiards on general tables. In the special case of a circular table we show that almost every (random) orbit is dense in the boundary as well as in the circular ring formed between the circle boundary and the random caustic. We additionally prove Strong Knudsen's Law for a particular case of families of absolutely continuous measures with respect to Liouville.

Physical Versus Mathematical Billiards: From Regular Dynamics to Chaos and Back

In standard (mathematical) billiards a point particle moves uniformly in a billiard table with elastic reflections off the boundary. We show that in transition from mathematical billiards to physical billiards, where a finite size hard sphere moves at the same billiard table, virtually anything may happen. Namely a non-chaotic billiard may become chaotic and vice versa. Moreover, both these transitions may occur softly, i.e. for any (arbitrarily small) positive value of the radius of a physical particle, as well, as by a "hard" transition when radius of the physical particle must exceed some c…