A description of surface dynamics

该项目旨在了解动力系统从有序状态到随机（混沌）状态转变的机制。换句话说，目标是理解当一个或多个控制参数变化时，系统行为从周期性向混沌演变的过程。一个相关的目标是确定导致动力系统所表现的质变的主要分岔。虽然这种理解以前已经获得了低维动力系统，但该项目引入了一种超越低维限制的新方法。该项目将提供新的概念思想和方法，为数学和科学的进步提供新的视角。此外，该项目将促进直接从事研究的研究生的培训，并将通过组织暑期学校介绍数学主题，包括与动力系统相关的主题，为本科生提供教育机会。一维动力系统理论成功地解释了混沌现象的深度和复杂性，并对典型地图的典型轨道动力学进行了描述。它显著的普适性用强大的几何工具补充了这种理解。在二维环境中，可能出现的动力情景的范围目前只被部分理解，而一维动力学中不会出现的新现象的一般框架仍有待开发。在之前由国家科学基金会支持的工作中，首席研究员引入了一个大的二维动力系统开放类，包括经典的Henon族，没有大面积收缩的限制，可以得到一维情况下的结果。此外，在利用重整化方案理解从零熵到正熵的转变方面取得了重大进展。目前的项目有几个组成部分。首先，现有的重整化方案将适应于正熵领域。接下来，将讨论在更一般的情况下对耗散微分同态进行表征的初步步骤。最后，首席研究员将寻求发展理论的可微重整化没有先验的假设接近一维设置。这些结果将为耗散微分同态及其在扰动下的行为的全局描述打开大门，为光滑动力系统理论带来新的工具和新的视角。该奖项反映了美国国家科学基金会的法定使命，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。

This project seeks to understand the mechanisms that underlie the transition of a dynamical system from an ordered state to a random (chaotic) state. In other words, the aim is to understand the processes through which a system's behavior evolves from periodicity toward chaos, as one or more governing parameters are varied. A related goal is to identify the primary bifurcation responsible for qualitative changes exhibited by a dynamical system. While such comprehension has previously been attained for low-dimensional dynamical systems, this project introduces a novel approach to transcend the low-dimensional limitation. The project will offer new conceptual ideas and approaches to provide fresh perspectives on advances in mathematics and science. Additionally, the project will facilitate the training of graduate students directly engaged in the research, and will afford educational opportunities to undergraduate students through the organization of a summer school presenting topics in mathematics, including topics related to dynamical systems.The theory of one-dimensional dynamical systems successfully explains the depth and complexity of chaotic phenomena in concert with a description of the dynamics of typical orbits for typical maps. Its remarkable universality properties supplement this understanding with powerful geometric tools. In the two-dimensional setting, the range of possible dynamical scenarios that can emerge is at present only partially understood, and a general framework for those new phenomena that do not occur for one-dimensional dynamics remains to be developed. In prior work supported by the NSF, the principal investigator introduced a large open class of two-dimensional dynamical systems, including the classical Henon family without the restriction of large area contraction, that is amenable to obtaining results as in the one-dimensional case. Moreover, major progress was reached to understand the transition from zero entropy to positive entropy using renormalization schemes. The present project has several components. First, existing renormalization schemes will be adapted to the positive entropy realm. Next, initial steps towards a characterization of dissipative diffeomorphisms in more general contexts will be addressed. Finally, the principal investigator will seek to develop the theory of differentiable renormalization without an a priori assumption of proximity to the one-dimensional setting. These results will open the door to a global description of dissipative diffeomorphisms and their behavior under perturbation, bringing both new tools and new perspectives to smooth dynamical systems theory.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.