arXiv 2303.011282023-03-02math.NTmath.DG

关于有限指数和II的对称性

On the symmetry of finite sums of exponentials II

Florian Pausinger, Dimtris Vartziotis

备注：: 10 pages, 8 Figures

摘要：

在本文中，我们扩展了我们对定义为两个指数的加权和的曲线的图的丰富几何的研究。令$γ_{a,b}^{s}: [0,1] \rightarrow \mathbb{C}$定义为$$γ_{a,b}^s(t) = (1-s) \exp(2 πi a t) + (1+s) \exp(2πi b t) $$，其中$1\leq a < b$是两个正整数，而$s \in [-1,1]$。在第一部分中，我们确定了$γ_{a,b}:=γ_{a,b}^0$图的对称群。本文的主要目的是研究当$s$从$-1$变为$1$时曲线图形的连续过渡。作为主要结果，我们确定了$s \in [-1,1] \setminus \{ 0 \}$的缠绕数$\mathrm{wind}(γ_{a,b}^s,0)$以及每个这样的曲线的尖点的集合。这进一步阐明了我们最初的对称性结果，并提供了更多的非平凡的，但容易状态的几何和拓扑的高级概念的例子。

英文摘要：

In this note we extend our study of the rich geometry of the graph of a curve defined as the weighted sum of two exponentials. Let $γ_{a,b}^{s}: [0,1] \rightarrow \mathbb{C}$ be defined as $$γ_{a,b}^s(t) = (1-s) \exp(2 πi a t) + (1+s) \exp(2πi b t) $$ in which $1\leq a < b$ are two positive integers and $s \in [-1,1]$. In the first part we determined the symmetry groups of the graphs of $γ_{a,b}:=γ_{a,b}^0$. The main aim of this note is to study the continuous transition of the graph of the curve when $s$ changes from $-1$ to $1$. As a main result we determine the winding numbers $\mathrm{wind}(γ_{a,b}^s,0)$ for $s \in [-1,1] \setminus \{ 0 \}$ as well as the set of cusp points of each such curve. This sheds further light on our initial symmetry result and provides more non-trivial albeit easy-to-state examples of advanced concepts of geometry and topology.

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