Holomorphic Dynamics, Small Divisors and Related Topics

全纯动力学、小除数及相关主题

• 批准号: 0202494
• 负责人: Ricardo Perez-Marco
• 金额: $ 15万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202494&HistoricalAwards=false
• 关键词: Holomorphic; Dynamics; Small; Divisors; Related

Proposal Number: DMS-0202494PI: Ricardo Perez-MarcoABSTRACTResearch will be conducted on several projects on the theoryof Dynamical Systems, and more specifically, in HolomorphicDynamics and Small Divisor theory. Using heuristic ideas fromHolomorphic Dynamics it is proposed to investigate an effectiveselection of polynomials as iterators in Pollard rho method of factorization. Factorization algorithms are a central topicin Cryptography. Numerical explorations show that there isroom for improvement in this field. In Holomorphic Dynamicsand Small Divisors, it is proposed to extend the generaltheory of dynamics on Hedgehogs. The focus of the proposedresearch is on weak forms of stability when arithmeticconditions, typical of Small Divisors, fail. It is proposed to investigate Siegel disks and their boundary, and to attackthe hard problem of classifying invariant annular continua.It is anticipated that the techniques of tube-log Riemannsurfaces should play an important role. Another projectconsists in extending geometric techniques to higherdimensional Small Divisors problems. In particular, a newgeometric construction of invariant tori is envisioned. Inthe theory of Renormalization in Holomorphic Dynamics itis proposed to develop the new theory of renormalization.This involves a better understanding of DegenerateQuasi-Conformal theory (i.e. for non bounded Beltramicoefficients). A last topic in Complex Analysis on whichresearch is proposed is the theory of Borel Monogenic functions. We project to develop a theory that is flexibleenough to apply to problems in Small Divisors.Dynamical Systems is a branch of Mathematics with ancientroots linked to Celestial Mechanics. It was founded as adistinct branch of Mathematics more than a century ago byH. Poincare. It is a rich field with multiple interactionswith other parts of Mathematics and Science. The main goalis the study of the long time behavior of evolution processes.Some of the most important problems come from fields asBiology, Physical Sciences and Chemistry. One of the central questions is the Problem of Stability. For example, is theSolar System stable according to Newton laws ? K.A.M. theory(also called Small Divisor theory) is one of the majorbranches in Dynamical Systems. It was founded in the secondhalf of the XXth century and had an important impact inDynamical Systems and related fields. It is the very firsttheory that provides stability results in non-linearconservative problems. It continues its expansion to fieldsas diverse as Partial Differential Equations, the theory of foliations, Holomorphic Dynamics, etc. In the proposedprojects some of the most difficult open questions in KAMtheory will be explored. What happens when stability fails?Can we still recover some traces of stability that may beuseful in the applications?

研究将在动力系统理论的几个项目上进行，更具体地说，在全纯动力学和小因子理论方面。利用全纯动力学中的启发式思想，研究了波拉德分解法中多项式作为迭代器的有效选择。因数分解算法是密码学中的一个核心课题。数值探索表明，该领域仍有改进的余地。在全纯动力学和小除数中，对刺猬动力学的一般理论进行了推广。所提出的研究重点是弱形式的稳定性当算术条件，典型的小除数，失败。研究了西格尔盘及其边界，解决了不变环连续体的分类难题。预计管-对数黎曼曲面技术将发挥重要作用。另一个项目是将几何技术扩展到高维小除数问题。特别提出了一种新的不变环面几何结构。在全纯动力学重整化理论中，提出了一种新的重整化理论。这涉及到对简并拟共形理论（即无界贝特拉米系数）的更好理解。复分析中提出的最后一个研究课题是Borel单基因函数理论。我们计划发展一种足够灵活的理论，以适用于小除数问题。动力系统是数学的一个分支，其古老的根源与天体力学有关。它是一个多世纪前由h。庞加莱。这是一个丰富的领域，与数学和科学的其他部分有多种相互作用。主要目标是研究进化过程的长时间行为。一些最重要的问题来自生物学、物理科学和化学等领域。其中一个核心问题是稳定问题。例如，根据牛顿定律，太阳系是稳定的吗？K.A.M.理论（又称小因子理论）是动力系统的主要分支之一。它成立于20世纪下半叶，在动力系统及相关领域产生了重要影响。它是第一个在非线性保守问题中提供稳定性结果的理论。它继续扩展到不同的领域，如偏微分方程，叶理论，全纯动力学等。在拟议的项目中，将探讨KAMtheory中一些最困难的开放问题。当稳定性失效时会发生什么？我们还能恢复一些可能对应用程序有用的稳定性吗？