Holomorphic Dynamics, Small Divisors and Related Topics

提案编号：DMS-0202494 PI：Ricardo Perez-Marco摘要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。本文利用全纯动力学的启发式思想，研究了Pollard rho因式分解方法中迭代多项式的有效选取问题。因式分解算法是密码学中的一个核心问题。数值研究表明，这一领域还有改进的空间。在《全纯动力学与小因子》中，提出了将动力学的一般理论推广到Hedgehogs上。建议的研究重点是弱形式的稳定性时，arithmeticconditions，典型的小因子，失败。本文提出了研究Siegel圆盘及其边界，解决不变环形连续统分类的难题，并期望管-对数Riemann曲面技术在其中发挥重要作用。另一个项目是将几何技术扩展到高维小因子问题。特别是，一个新的几何结构的不变环面的设想。在《全纯动力学重整化理论》中，提出了发展新的重整化理论，这涉及到对退化拟共形理论（即对无界Beltramiccoefficients）的理解。最后一个议题在复杂的分析上提出的研究是理论的Borel单演函数。我们的目标是发展一个足够灵活的理论来应用于小因子问题。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。它作为数学的一个独特的分支而创立于世纪以前。庞加莱它是一个丰富的领域，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期行为，其中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性问题。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）是动力系统的一个重要分支。它创立于世纪后半期，对动力系统及其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。它继续扩大到不同的领域，如偏微分方程，叶理理论，全纯动力学等，在提出的项目中，一些最困难的KAM理论的开放问题将被探讨。当稳定性失败时会发生什么？我们还能恢复一些可能在应用程序中有用的稳定性痕迹吗？

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?