AI 中文总结
研究有限伯努利卷积，通过数值探索其作为湍流模型的特性，基于β展开交叉点分布及N元二进制词序的概念研究其混沌行为，还给出序图及与二进制反射格雷码的联系。
AI 中文摘要
在本笔记中，我们对有限伯努利卷积进行了数值研究。我们表明，通过适当选择参数，它可作为充分发展湍流中间歇能量级联的一个简化模型。接着展示了β展开的交叉点在β中的分布情况，并表明这可能突出与奇异连续测度相关的具有增强重叠结构的参数。随后基于N元二进制词的字典序为β展开引入了一种序的概念，观察到当β = 2时，对于大多数采样的相邻对，当β从2减小到1时，它们序中的距离呈指数增加，这表明存在‘混沌’行为，‘李雅普诺夫指数’聚集成几个依赖于N的簇。最后我们给出了一些‘序图’以及有限β紧集（β = g，g为黄金比例）与二进制反射格雷码之间的有趣联系。
英文摘要
In this note we explore numerically the finite Bernoulli convolutions. We show that with a suitable choice of parameter, it might serve as a toy model for intermittent energy cascade in fully developed turbulence. We then show how the crossings of $β$-expansions distribute in $β$, and suggest that it might highlight the parameters with enhanced overlap structure that are related to measures that are singular continuous. We later introduce a notion of order to the $β$-expansions based on the lexicographical order of the $N$-binary words, and observe that for most sampled adjacent pairs when $β=2$, the distance in their order increases exponentially when $β$ decreases from 2 to 1. This suggests 'chaotic' behavior, with the 'Lyapunov exponents' bunched into several clusters that depend on $N$. We end the note with some 'order plots' and an interesting connection between the finite $β$-compactum with $β=g$ (g being the golden ratio) and binary reflected Gray code.
Comments13 pages, 9 figures
