Veech曲面中的切割序列

Cutting sequence and Sturmian sequence in billiard

The winning rule of billiards is to drive the billiard ball on the table into the designated holes. We try to study the trajectory of the billiard ball, so that we can predict the direction of the ball. For rational slopes, we got cutting sequence by setting up the square torus. We simplified cutting sequence using shearing and flipping and we obtain the transformation between trajectory slope and cutting sequence. For irrational slopes, we look at some properties of Sturmian sequence, which help us distinguish between cutting sequence and Sturmian sequence. In conclusion, in the case of diffe…

具有坡度$θ=[0;\dot{d}]$的采伐序列的缺口序列

Gap Sequence of Cutting Sequence with Slope $θ=[0;\dot{d}]$

本文研究了一类特殊类型的具有坡度的切削序列$θ=[0;\dot{d}]$（记为$F_{d,\infty}$）的因子性质和间隙序列。设$ω$是$F_{d,\infty}$的因子，则它在序列中出现无限次。令$ω_p$为$ω$的第$p$次出现，$G_p(ω)$为$ω_p$和$ω_{p+1}$之间的间隙。定义了核词和包络词的$d$类型，给出了核分解性质唯一性的两个版本。利用它们，我们证明了空位序列$\{G_p(ω)\}_{p\geq1}$对于每个$ω$恰好有两个不同的元素，并完全确定了空位的表达式。此外，我们证明了缺口序列是$σ_i(F_{d,\infty})$，其中$σ_i$是仅取决于$Ker(ω)$类型的替换，即$ω$的核心字。我们还确定了所有$(ω,p)$的$ω_p$的位置。作为应用，我们研究了$ω_p$和$ω_{p+1}$的幂、重叠和分离等组合性质，并找到了$F_{d,\infty}$中的所有回文。

正八边形Teichmueller圆盘上的测地流、切割序列和八边形连分数映射

Geodesic flow on the Teichmueller disk of the regular octagon, cutting sequences and octagon continued fractions maps

在本文中，我们给出了一个几何解释的重整化算法和连分数映射，我们介绍了arxiv：0905.0871中给出的线性流的符号序列的特征，在正八边形。我们将此算法解释为八边形Teichmueller圆盘上的重整化，并解释了与Teichmueller测地流的关系。这种联系类似于Sturmian序列、连分数和模曲面上的测地流之间的经典关系。利用这种联系，我们构造了连分数映射的自然扩张和不变测度。定义了具有有限不变测度的连分数映射的加速度。

方形瓷砖表面上的切割顺序

Cutting sequences on square-tiled surfaces

我们通过考虑曲面上特定区间的区间交换刻画了正方形拼接曲面上无限测地线的切割序列。这些区间交换可以被认为是旋转的斜交产品，我们将切割序列转换为这些区间交换的符号轨迹，以表明特殊类型的组合升降机的Sturmian序列完全描述所有切割序列的方形瓷砖表面。我们的研究结果扩展了家庭的表面切割序列被理解为一个稠密的子集的所有平移表面的模空间。

New Kronecker-Weyl type equidistribution results and diophantine approximation

An interesting result of Veech more than 50 years ago is a parity, or mod $2$, version of the Kronecker--Weyl equidistribution theorem concerning the irrational rotation sequence $\{qα\}$, $q=0,1,2,3,\ldots.$ If $α$ is badly approximable and $b\in(0,1)$ satisfies $b\ne\{mα\}$ for any $m\in\mathbb{Z}$, then the parity of cardinalities of the sets $\{1\le q\le N:\{qα\}\in[0,b)\}$ as $N\to\infty$ is evenly distributed. We first answer a question of Veech and establish a stronger form of the mod $n$ analog of his result (Theorem 3.1). Furthermore, for irrational $α$ and $b=\{mα\}$ for some $m\in\m…

模曲面上测地流的截割序列与连分数

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

本文描述了模曲面H/PSL（2，Z）上测测地流关于PSL（2，Z）的标准基本区F = {z=x+iy：-1/2 <= x <= 1/2 and |z| >=1}的切割序列.垂直测地线{θ+it：t > 0}的切割序列与θ的一维连分数展开有关，称为一维Minkowski测地线连分数（MGCF）展开，它与二维格族的参数化约化基族有关。所有测地线的切割序列的集合在符号空间{L，R，J}中形成双侧移位，符号空间{L，R，J}具有与垂直测地线相同的禁止块集合。我们证明了这种位移不是sofic位移，它表征了基本区F直到双曲平面H的等距。给出了垂直测地线{θ+it：t > 0}的切割序列、θ的MGCF展开式和θ的加性普通连分数（ACF）展开式之间的转换方法。我们证明了切割序列和MGCF展开都可以通过有限自动机从另一个计算出来，并且θ的ACF展开可以通过有限自动机从垂直测地线θ+it的切割序列计算出来。然而，垂直测地线的切割序列不能通过任何有限自动机从ACF展开计算，但是当给定ACF展开的前O（l）个符号作为输入时，有一种算法可以计算其前l个符号，这需要时间O（l^2）和空间O（l）。

秩1保测度系统的构造性符号表示

Constructive symbolic presentations of rank one measure-preserving systems

给定一个秩为1的测度保持系统，通过间隔子的切割和叠加，构造一个秩为1的二元序列，使得它的轨道在推移变换下的闭合，以其唯一的{非原子}不变概率，同构于给定的系统.特别是，经典的二进里程表中的两个符号的字母表$\{0,1\}$的块的递归序列。该构造是使用在adic或Bratteli-Vershik系统的设置中的秩1的定义来完成的。

Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces

A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be relat…