Attractors, Smooth Dynamics and Combinatorics in Low-Dimension

低维吸引子、平滑动力学和组合学

• 批准号: 9970363
• 负责人: Alexander Blokh
• 金额: $ 7.13万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970363&HistoricalAwards=false
• 关键词: Attractors; Smooth; Dynamics; Combinatorics; Low

Proposal: DMS-9970363Principal Investigator: Alexander M. BlokhAbstract: Blokh plans to describe Milnor attractors for certain rational functions, to determine their conformal measures, and to use the new notion of a growing tree as a tool for studying dynamics on the Julia set (jointly with Misiurewicz and Levin). In one-dimensional dynamics, Blokh and Misiurewicz introduced the notion of super-persistent recurrence. The lack of such recurrence at a point is a mild expanding property. This behavior implies recent results by Blokh and Misiurewicz according to which wild attractors with negative Schwarzian derivatives are super-persistently recurrent and limit sets of critical points in a dense family of smooth interval maps are minimal. Blokh intends to use this device to study further properties of attractors and stability for smooth interval maps. He also hopes to extend the rotation theory to graph maps and then study forcing among periodic orbit portraits.Many processes in biology, the physical sciences, and economics can be modeled by dynamical systems. Studying these models allows one to draw valid conclusions about the processes themselves and their long-term behavior. This project is aimed at studying a variety of such dynamical systems. Blokh intends to verify that either the behavior of the system is on average chaotic (but almost the same for the majority of initial conditions) or it coincides with the behavior of systems starting at a special finite set of so-called critical initial conditions. Also, a system is said to behave periodically if the same conditions recur with the same frequency all the time (e.g., the motion of the planets about the sun). Periodic dynamics are easier to study than more general motions, which is why it is important to understand periodic systems as fully as possible before probing the mysteries of more complicated systems. Blokh wishes to characterize all periodic behaviors of dynamical systems using only finitely many parameters. The results obtained will help to describe the possible outcomes for a variety of processes and to explain the prevalence of certain types of processes in both theoretical settings and the natural world.

提案：DMS-9970363主要研究者：亚历山大M. BlokhAbstract：Blokh计划描述Milnor吸引某些合理的功能，以确定其共形措施，并使用新的概念，不断增长的树作为一种工具，研究动态的朱莉娅集（与Misiurewicz和莱文）。在一维动力学中，Blokh和Misiurewicz引入了超持续递归的概念。在某一点上没有这种重现性是一种温和的膨胀性质。这种行为暗示了Blokh和Misiurewicz最近的结果，根据这些结果，具有负Schwarzian导数的野生吸引子是超持久常返的，并且在光滑区间映射的稠密族中临界点的极限集是最小的。Blokh打算利用这一工具进一步研究光滑区间映射的吸引子和稳定性的性质。他还希望将旋转理论扩展到图形映射，然后研究周期轨道肖像之间的强迫。生物学，物理科学和经济学中的许多过程都可以用动力系统建模。研究这些模型可以让人们对过程本身及其长期行为得出有效的结论。该项目旨在研究各种这样的动力系统。Blokh打算验证系统的行为平均而言是混沌的（但对于大多数初始条件几乎相同），或者它与系统在所谓的临界初始条件的特殊有限集合上开始的行为一致。此外，如果相同的条件总是以相同的频率重复出现，则系统被称为周期性地行为（例如，行星绕太阳的运动）。周期动力学比一般运动更容易研究，这就是为什么在探索更复杂系统的奥秘之前，尽可能全面地了解周期系统是很重要的。Blokh希望仅用200多个参数就能刻画动力系统的所有周期行为。所获得的结果将有助于描述各种过程的可能结果，并解释某些类型的过程在理论环境和自然世界中的普遍性。