Holomorphic Dynamics, Small Divisors and Related Topics

建议编号：DMS-0202494PI：Ricardo Perez-Marco ABSTRACT将在动力系统理论的几个项目上进行研究，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，研究了在Pollard Rho因式分解方法中选择多项式作为迭代算子的有效性。因式分解算法是密码学中的一个中心话题。数值研究表明，这一领域仍有改进的空间。在全纯动力学和小约数中，推广了刺猬动力学的一般理论。所提出的研究的重点是当算术条件(典型的小因子)失效时的弱形式的稳定性。建议研究Siegel圆盘及其边界，并解决不变环状连续的分类难题，预计管状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推广到高维小因子问题。特别地，给出了不变环面的一种新的几何构造。在全纯动力学中的重整化理论中，提出了发展新的重整化理论，这涉及到更好地理解简并的拟共形理论(即对于无界的Beltram效率)。复变分析的最后一个主题是Borel正则函数理论。我们计划开发一种足够灵活的理论来应用于小除数的问题。动力系统是数学的一个分支，与天体力学有着古老的渊源。它是一个多世纪前由H创立的一个独特的数学分支。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互动。主要目标是研究进化过程的长期行为。一些最重要的问题来自生物、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，太阳系根据牛顿定律是稳定的吗？K.A.M.理论(又称小因子理论)是动力系统的主要分支之一。它成立于二十世纪下半叶，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守恒问题中提供稳定性结果的理论。它继续扩展到各种领域，如偏微分方程组、叶层理论、全纯动力学等。在提出的项目中，将探索KAM理论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?