动力系统中的涌现性
Emergence in dynamical systems
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中文总结 AI 辅助
本文综述了动力系统中涌现性的最新研究，涵盖其定义、与量化及熵的联系，以及高涌现性在各类动力学中的实例与开放问题。
中文摘要 AI 辅助
在文献[Be17]中，我们引入了涌现性的概念，用以量化动力系统的统计复杂性。粗略地说，涌现性衡量的是描述大多数轨道在给定精度ε下的统计行为所需的概率测度的数量。当这个数量随着ε趋近于0而超多项式增长时，该系统被称为具有高涌现性。我们综述了旨在理解可微动力系统中高涌现性普遍性的研究项目的最新进展。我们回顾了涌现性的几种概念及其与量化、遍历分解和熵的联系，并讨论了在保守、辛、解析和耗散动力学以及单峰、Henon和有理映射等受限族中展现高或最大涌现性的例子。我们还介绍了关于度量和拓扑涌现性的变体、凸性性质、局部涌现性和变分原理的新结果，以及关于高涌现性典型性的一系列开放问题。
英文摘要
In [Be17], we introduced the notion of emergence to quantify the statistical complexity of a dynamical system. Roughly speaking, emergence measures the number of probability measures required to describe, up to a given precision $ε$, the statistical behavior of most orbits. A system is said to have high emergence when this number grows super-polynomially as $ε\to0$. We survey recent developments in a program aimed at understanding the prevalence of high emergence in differentiable dynamics. We review several notions of emergence and their connections with quantization, ergodic decompositions, and entropy, and discuss examples exhibiting high or maximal emergence in conservative, symplectic, analytic, and dissipative dynamics, as well as in constrained families such as unimodal, Hénon, and rational maps. We also present new results concerning variants of metric and topological emergence, convexity properties, local emergence, and variational principles, together with a collection of open problems on the typicality of high emergence.
发表机构
- IMJ-PRG, CNRS, Sorbonne Université, Université Paris Cité（巴黎数学研究所，法国国家科学研究中心，索邦大学，巴黎西岱大学）
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