Serge Troubetzkoy
There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.
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We provide explicit lower estimates on the complexity growth in typical directions for a class of irrational triangle billiards
Periodic billiard orbits are dense in the phase space of an irrational right triangle. A stronger pointwise density result is also proven.
We give an explicit sub-exponential estimate on the growth rate of periodic orbits and generalized diagonals for typical triangle billiards.
We find an upper estimate for a splitting time of a thin parallel beam for irrational triangle billiards in terms of some number-theoretic function of angles. We provide an upper estimate on this function for some class of angles.
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.
Any periodic trajectory on an isosceles triangle gives rise to a periodic trajectory on a right triangle obtained by identifying the halves of the original triangle. We examine the relationship between periodic trajectories on isosceles triangles and the trajectories on right triangles obtained in this manner, and the consequences of this relationship for the existence of stable trajectories on isosceles triangles and the properties of their orbit tiles.
Star-shaped trajectories of certain billiards around a triangle
本研究在Beltrami-Klein模型框架下，基于已有研究考察双曲平面中三角形外台球映射旋转数为2/5的情况，给出充分条件并证明其对大等腰三角形的必要性，最后提出相关猜想。
We classify the periodic digit strings which arise from periodic billiard orbits on the four convex $n$-gons $Δ$ which tile $\mathbb{R}^2$ under reflection, answering problem a posed by Baxter and Umble. $Δ$ is either an equilateral triangle, a $45-45-90$ triangle a $30-60-90$, triangle or a rectangle.
Billiards in a circle with trajectories circumscribing a triangle
本文在双曲几何的Klein-Beltrami模型中，以欧几里得几何方式重新表述了三角形被归类为“大”的条件，提出并评估了其欧几里得大小度量，并为等腰三角形给出显式公式。
Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to -1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N+2. We also give a precise description of the exceptional family of trajectories (of zero measure) : …
The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials. Additionally, given a fixed set of angles, the orbit closure of the unfolding of all unit area rational parallelograms, isosceles trapezoids, and right trapezoids outside of a discrete set is determined.
This paper had a serious error. In fixing the error the emphasis of the paper has changed completely, thus meriting a new name: ``Periodic orbits in right triangles''. I have made a new submission to arXiv with this name.
Explicit Families and Distribution of Triangular Billiards of Weakly Exponential Growth
本文构造了第一个显式的无限族三角形，其组合复杂度函数具有弱指数增长，并找到了单参数族使得一般参数下复杂度弱指数增长。
We prove Knudsen's law for a gas of particles bouncing freely in a two dimensional pipeline with serrated walls consisting of irrational triangles. Dynamics are randomly perturbed and the corresponding random map studied under a skew-type deterministic representation which is shown to be ergodic and exact.
We provide a lower bound on the complexity function of a typical (in the Lebesgue measure sence) right triangular billiard.
We give a complete solution of the following problem: Find, classify and count the (classes of) periodic orbits on an equilateral triangle. We prove that Fagnano's period 3 orbit is the only periodic orbit with odd period. A periodic orbit is either prime or some d-fold iterate thereof. We count prime and iterate periodic orbits of period $2n$ via a bijection with a certain partition of $n$, then count only prime orbits using the prime factorization of $n$.
Snell meets Fagnano. Path optimization through an imperfect mirror
本文研究了在不完美镜面中通过 Snell 法则优化路径的问题，提出了一种基于加权周长的三角形优化方法。
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.
A periodic trajectory on a polygonal billiard table is stable if it persists under any sufficiently small perturbation of the table. It is a standard result that a periodic trajectory on an $n$-gon gives rise in a natural way to a closed path on an $n$-punctured sphere, and that the trajectory is stable iff this path is null-homologous. We present a novel proof of this result in the language of covering spaces, which generalizes to characterize the stable trajectories in neighborhoods of a polygon. Using this, we classify the stable periodic trajectories near the 30-60-90 triangle, giving a ne…
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 …
In this work we completely describe the dynamics of triangle tiling billiards. In the first part of this work, we propose a geometric approach of dynamics by introducing natural foliations associated to it. In the second part, we exploit the relationship between triangle tiling billiards and a family of fully flipped $3$-interval exchange transformations on the circle. We give a combinatorial approach of dynamics via renormalization. By uniting the two approaches, we prove several conjectures on the dynamics of triangle tiling billiards. First, we prove the Tree Conjecture and the 4n+2 Conject…
We consider the billiard map inside a polyhedron. We give a condition for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a periodic orbit of length four (generalization of Fagnano's orbit for triangles), moreover we can study completly the orbit of points along this coding.
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.
We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behavior is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experime…
Non-periodic not everywhere dense trajectories in triangular billiards
利用诱导映射与区间交换变换，构造等腰三角形单参数族，其非周期轨迹非处处稠密，从而否定性解决三角台球中长期未决的稠密性问题。
Is every triangle a trajectory of an elliptical billiard?
本文运用 Marden 定理证明每个三角形、平行四边形及“Darboux butterflies”自交折线均为某唯一椭圆内的台球轨迹，并有效计算出边界椭圆的焦点，揭示了 Marden 定理与 Poncelet Porism 间的联系。
This paper shows that the method of Galperin which had been widely accepted in the literature since 1983 for constructing a non-dense and non-periodic trajectory in a triangle can never work. This is done by showing that all possible examples that can be produced by Galperin's method of which there are infinitely many all produce a periodic trajectory.
By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, M. Möller and the author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families: $Z(s) = \det(1-\mc L_s^+)\det(1-\mc L_s^-)$. In this article we show that the operator families $\mc L_s^\pm$ arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for $s\in\C$, $\Rea s=\tfrac12$, the operator $\mc L_s^+$ (resp. $\mc L_s^-$) has a 1-eigenfunction if and on…
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…
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
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.
Given a planar compact convex billiard table $T$, we give an algorithm to find the shortest generalised closed billiard orbits on $T$. (Generalised billiard orbits are usual billiard orbits if $T$ has smooth boundary.) This algorithm is finite if $T$ is a polygon and provides an approximation scheme in general. As an illustration, we show that the shortest generalised closed billiard orbit in a regular $n$-gon $R_n$ is 2-bounce for $n \ge 4$, with length twice the width of $R_n$. As an application we obtain an algorithm computing the Ekeland-Hofer-Zehnder capacity of the four-dimensional domai…
A periodic orbit on a frictionless billiard table is a piecewise linear path of a billiard ball that begins and ends at the same point with the same angle of incidence. The period of a primitive periodic orbit is the number of times the ball strikes a side of the table as it traverses its trajectory exactly once. In this paper we find and classify the periodic orbits on a billiard table in the shape of a 120-isosceles triangle, a 60-rhombus, a 60-90-120-kite, and a 30-right triangle. In each case, we use the edge tessellation (also known as tiling) of the plane generated by the figure to unfol…
Aperiodic points for outer billiards
本文证明非三角形、正方形或六边形的正多边形上的欧几里得外弹球座存在非周期点，解决了R. Schwartz在ICM 2022提出的问题。
It has been known since Fagnano in 1775 that an acute triangle always has a periodic billiard path, namely the orthic triangle. It is currently unknown whether every obtuse triangle has a periodic path. In 2006, Schwartz showed that every obtuse triangle with obtuse angle at most 100 degrees has a periodic path. The aim of this paper is to show that every obtuse triangle with obtuse angle at most 112.3 degrees has a periodic path using a computer assisted proof.
We introduce a new notion of stability for periodic orbits in polygonal billiards. We say that a periodic orbit of a polygonal billiard is $λ$-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as $λ$ $\rightarrow$ 1. This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We give sufficient and necessary conditions for a periodic orbit to be $λ$-stable and prove that the set of d-gons having at most finite number of $λ$-stable periodic orbits is dense is the space of d-gons. Moreover, we also determine comp…
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…
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.
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 $1/2$.
A piecewise contractive map on triangles
研究 $\mathbb{R}^2$ 中三条直线上分段压缩映射的动力学，证明任意初始点轨道必收敛于不动点或周期轨道，且存在无穷多种依赖于空间参数的周期轨道。
We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. The proof is based on a formula for the Lyapunov exponents of the geodesic flow on the moduli space, which gives a recursive formula from which the volumes could be recovered. An appendix with an ergodic theorem proved by Jon Chaika …
We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.
It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincare disk model. Our tables have m>=3 ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group PSU^\pm_{1,1}\Zbb[i]. The resulting billiard map \tilde B acts on the de Sitter space x_1^2+x_2^2-x_3^2=1, and has a natural factor B on the unit circle, the pythagorean triples appearing as the B-preimages of fixed points. We…
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