From zero to positive entropy
From zero to positive entropy
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从零到正熵
DOI：
10.1090/noti2478
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发表时间：
2022
影响因子：
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通讯作者：
Sylvain Crovisier
Enrique Pujals
中科院分区：
文献类型：
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作者：
Sylvain Crovisier
Enrique Pujals
在一般的科学中，“通往混沌之路”这个短语已经成为一个隐喻，当一些物理，生物，经济或社会系统从一个展示秩序过渡到一个展示随机性（或混沌）时。有时，我们的目标是了解哪些普遍机制可以解释这种转变，以及如何描述在有序和完全混乱之间的区域中运行的系统。换句话说，我们的目标是理解一个系统从一个递归集是有限的系统向另一个表现出混沌行为的系统演化的数学过程，因为控制系统行为的参数是变化的。这只在一维动力学中才被理解。本说明介绍了能够摆脱这些限制的新办法。一个尝试性的全球框架，对描述一个大类的二维动力学，启发部分的发展，在一维理论的区间映射进行了讨论。更准确地说，我们提出了一类中间光滑的动力学之间的一个和更高的维度。在这种情况下，有可能开发类似的一维类型的方法，特别是理解从零熵到正熵的转变。
In the sciences in general, the phrase “route to chaos” has come to refer to a metaphor when some physical, biological, economic, or social system transitions from one exhibiting order to one displaying randomness (or chaos). Sometimes the goal is to understand which universal mechanisms explain that transition, and how one can describe systems that operate in a region between order and complete chaos. In other words, the goal is to understand the mathematical processes by which a system evolves from one whose recurrent set is finite towards another one exhibiting chaotic behavior as parameters governing the behavior of the system are varied. This has only been understood for one-dimensional dynamics. The present note exposes new approaches that allow one to move away from those limitations. A tentative global framework toward describing a large class of two-dimensional dynamics, inspired partially by the developments in the one-dimensional theory of interval maps is discussed. More precisely, we present a class of intermediate smooth dynamics between one and higher dimensions. In this setting, it could be possible to develop a similar one-dimensional type approach and in particular to understand the transition from zero entropy to positive entropy.