发表机构
Sorbonne Université; Université Paris Cité; Ariel University; Ben-Gurion University of the Negev（索邦大学; 巴黎西岱大学; 阿里埃勒大学; 内盖夫本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究最长边二分过程的退化性，证明三维存在退化例、四维有退化开集，且高维随机单纯形渐近几乎必然退化，并揭示其双曲与非周期椭圆行为。
AI 中文摘要
我们将最长边二分（LEB）过程作为单纯形射影形状空间上的动力系统进行研究。一个长期存在的猜想可追溯到Adler和Rivara-Levin，其动机源于有限元网格细化，且常被作为基本假设，即该过程是非退化的，并且实际上在某种意义上是周期的。我们证明：\begin{itemize} \item 存在三维单纯形使得最长边二分算法退化。\item 存在一个四维单纯形的开集，在其上最长边二分算法退化。\item 若用独立的标准高斯向量参数化$d$维单纯形空间，则随着$d$增大，随机单纯形渐近几乎必然退化。\end{itemize} 这是通过展示LEB过程的双曲行为来实现的。我们还展示了非周期的椭圆行为。
英文摘要
We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.
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