Dynamics of Surface Maps

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。该项目的目标是了解哪些通用机制解释了这种转变，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，其中复杂性最大。换句话说，我们的目标是了解系统从周期性行为向混沌行为演变的过程，因为一个或多个控制系统行为的参数发生了变化。这只有在低维动力学中才能理解（低维动力学就像涉及一个变量的系统一样简单）。本项目提出了一种新颖的方法，使我们能够摆脱这些限制。此外，该项目还为数学和科学的新进展带来了一些观点，不一定是带来技术或定理，而是思想和概念方法。该项目还将支持直接参与本研究的研究生的培训。目前的项目提出了一个尝试性的全球框架，以描述一个大类的面积收缩表面动力学的零和低熵的光盘，部分灵感来自于一维理论的发展区间映射。更确切地说，该项目提出了一类介于一维和更高维之间的中间光滑动力学（包括雅可比矩阵小于1 /4的Henon家族），其中可以开发类似的一维类型方法，有助于理解零或低拓扑熵的动力学。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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). The goal of this project 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, where the complexity is maximal. In other words, the goal is to understand the processes by which a system evolves from periodic toward chaotic behavior as one or more parameters governing the behavior of the system are varied. This has only been understood in low dimensional dynamics (which are as simple as a system involving one variable). The present project presents a novel approach that allows us to move away from those limitations. Moreover, the project brings some perspectives to new advances in mathematics and science in general, not necessarily in terms of bringing techniques or theorems, but in terms of ideas and conceptual approaches. The project will also support the training of graduate students, directly involved in this research.The current project presents a tentative global framework toward describing a large class of area contracting surface dynamics of the disc with zero and low entropy, inspired partially by the developments in the one-dimensional theory of interval maps. More precisely, the project proposes a class of intermediate smooth dynamics between one and higher dimensions (that includes the Henon family with Jacobian smaller than 1 /4 ) where it could be possible to develop a similar one-dimensional type approach that could help to understand dynamics with zero or low topological entropy. In particular, it presents a general framework to understand the transition from zero entropy to positive entropy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.