Periodic points on the regular and double $n$-gon surfaces

Polygonal Bicycle Paths and the Darboux Transformation

A Bicycle $(n,k)$-gon is an equilateral $n$-gon whose $k$ diagonals are of equal length. In this paper we consider periodic bicycle $(n,k)$-paths, which are a natural variation in which the polygon is replaced with a periodic polygonal path.

关于由正n边形与中心质量形成的中心配置的退化性

On the Degeneracy of the Central Configuration Formed by a Regular n-Gon with a Central Mass

研究正n边形等质量与中心质量形成的中心配置退化性，利用对称性分解Hessian矩阵，揭示退化值的结构与多模态特性。

Digit sequences of rational billiards on tables which tile $\mathbb{R}^2$

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.

正多边形的边几何——第二部分

Edge Geometry of Regular Polygons -- Part 2

本文研究正N边形边几何中奇点集W的局部结构，通过外台球映射、数字滤波器映射和双中心映射的剪切与旋转等价性，分析S[1]和S[2]第一族瓦片及其演化，并依据边猜想提出正多边形的“八重方式”。

Ω\mathcal M_g(2g-2)^{\mathrm{hyp}} 中的代数本原 Teichmüller 曲线

Algebraically primitive Teichmüller curves in Ω\mathcal M_g(2g-2)^{\mathrm{hyp}}

本文完整分类了 g≥5 时 Ω\mathcal M_g(2g-2)^{\mathrm{hyp}} 中的代数本原 Teichmüller 曲线，指出它们恰由双正则 (2g+1) 边形（2g+1 为素数）或正则 4g 边形（g 为 2 的幂）生成。

Outer billiards

In this paper outer, or dual, billiards outside regular polygons are studied; in particular, periodic points for cases of strictly convex "tables" and for regular n-gons with n = 3,4,6,8,12 are discussed. The main results of the paper are: 1) proof of an existence of aperiodic points for outer billiards outside regular octagon and dodecagon; 2) proof of the fact that for regular octagon, set of periodic points is of full measure; 3) list of all possible periods for case of regular octagon.

正多边形的边缘几何 —— 第1部分

The Edge Geometry of Regular Polygons -- Part 1

本文研究正N边形的边缘几何，通过外弹球等三种映射证明其共享相同的局部网格几何，提出正多边形可分为8类的“八重道”，并扩展了局部不变区域的代数判据。

Unchained polygons and the N-body problem

The simplest solutions of the N-body problem --symmetric relative equilibria-- are shown to be organizing centers from which stem some recently studied classes of periodic solutions. We focus on the relative equilibrium of the equal-mass regular N-gon, assumed horizontal, and study the families of Lyapunov quasi-periodic solutions bifurcating from them in the vertical direction. The proof of the local existence of such solutions relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We then discuss the possibility of co…

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most exponentially fast (call it the A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in Diff^r(M). Our proof of this result is much simpler than the original proof of Artin-Mazur. The second main result is that the A-M property is not (Baire) generic. Moreover, in a Newhous…

Periodic trajectories in the regular pentagon

We study periodic linear trajectories in the double pentagon and periodic billiard trajectories in the regular pentagon.

Periodic trajectories in the regular pentagon, II

In our recent paper, we studied periodic billiard trajectories in the regular pentagon and closed geodesic on the double pentagon, a translation surface of genus two. In particular, we made a number of conjectures concerning symbolic periodic trajectories. In this paper, we prove two of these conjectures.

双正多边形上的连接点

Connection points on double regular polygons

研究双正多边形平移曲面上的连接点，对奇数 n≥7 构造非连接点族，并在 n 为素数时给出中心点非连接点的显式证明。

Outer billiards outside regular decagon: periodicity of almost all orbits and existence of an aperiodic point

The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.

Long and Short Periodic Billiard Trajectories in the Regular Pentagon

In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the periodic trajectory through a given point is long, short, or a saddle connection. In this paper we present an algorithm resolving this question for trajectories emanating from the midpoints of the pentagon.

哪些Meyer集是正则模型集？一种通过殆周期性的刻画

Which Meyer sets are regular model sets? A characterization via almost periodicity

该研究在σ-紧局部紧阿贝尔群的一般框架下，证明了Meyer集是正则模型集的充要条件为其是殆周期图案，完善了Meyer相关结论的逆命题刻画。