U.S.-Polish Collaborative Research: Ergodic Theory and Geometry of Transcendental Entire and Meromorphic Functions

美波合作研究：遍历理论和超越整体和亚纯函数的几何

• 批准号: 0306004
• 负责人: Mariusz Urbanski
• 金额: --
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306004&HistoricalAwards=false
• 关键词: U.S.; Polish; Collaborative; Research; Ergodic

This is a U.S.-Polish cooperative research project that will focus on Ergodic theory and geometry of transcendental entire and meromorphic functions. The principal investigators are Dr. Mariusz Urbanski from the University of North Texas, Professor Janina Kotus from Warsaw University of Technology and Professor Anna Zdunik from Warsaw University.The main objects of investigation in this project are transcendental entire and meromorphic functions. The topological structure of the Julia sets of such maps has recently been intensively investigated. Fractal properties on the level of Hausdorff dimension have also been dealt with. The research to be done in this project goes to the deeper level of Hausdorff and packing measures for non-hyperbolic exponential maps. Real analyticity of the Hausdorff dimension for the hyperbolic tangent family will be investigated. The main tool of the research will consist of the concept of a conformal measure essentially belonging to the arsenal of thermodynamic formalism going beyond the uniformly hyperbolic systems on compact spaces. Various kinds of transfer operators will also be used frequently. The work on the class of non-hyperbolic exponential maps will primarily concern the subset of the Julia set which carriers all the recurrent and chaotic part of the dynamics. It is conjectured that the appropriate Hausdorff measure of this subset is positive and finite whereas the packing measure is infinite. The next step would be to prove the existence of an invariant measure equivalent with the conformal measure, and to explore its ergodic properties. The case when the parameter lambda is equal to 1/e will also be extensively studied as well as the behavior of the Hausdorff dimension when lambda increases to 1/e. Other problems to be dealt with are the multifractal analysis and the maximizing orbit problem in this context. The investigations of transcendental meromorphic functions will be focused on non-recurrent elliptic functions and non-hyperbolic tangent family. In the context of non-recurrent elliptic functions the main goal is to explain the nature of conformal measures, in particular, to determine whether these measures are purely atomic or atomless. The latter case would open the door to an extensive study of ergodic properties of a sigma-finite invariant measure equivalent with this conformal measure. For the hyperbolic tangent family the goal is to show that the Hausdorff dimension of the Julia set depends on the parameter lambda in a real-analytic manner.The completion of the project will shed a new light on the evolution of chaotic systems with non-compact phase space. It will enhance the knowledge of long-term behavior of such systems. The nature of the Julia sets, the fractals appearing in many popular publications, will be understood further. Its fundamental geometric properties will be investigated.This project in mathematics research fulfills the program objectives of bringing together leading experts in the U.S. and Central/Eastern Europe to combine complementary efforts and capabilities in areas of strong mutual interest and competence on the basis of equality, reciprocity, and mutuality of benefit.

这是一个美国 - 派别的合作研究项目，它将重点介绍整个整体和男性函数的千古理论和几何形状。 首席研究人员是北德克萨斯大学的Mariusz Urbanski博士，华沙理工大学的Janina Kotus和华沙大学的Anna Zdunik教授。该项目的主要调查对象是整个和Meromormormorphthic功能。最近对此类地图的朱莉娅集合的拓扑结构进行了深入研究。 还处理了Hausdorff维度水平上的分形特性。该项目要做的研究涉及到更深层次的Hausdorff和非纤维指数图的包装措施。 将研究双曲线切线家族的Hausdorff维度的实际分析。 该研究的主要工具将包括一个结构测量的概念，基本上属于热力学形式主义的武器库，而不是紧凑型空间上均匀双曲线系统。各种转移操作员也将经常使用。 非纤维指数图的类别的工作将主要涉及朱莉娅集合的子集，朱莉娅集合载体所有复发和混乱的动力学部分。据推测，该子集的适当Hausdorff度量为正且有限，而包装度量是无限的。下一步将是证明存在与共形度量等效的不变度度量的存在，并探索其厄尔贡特性。 当参数lambda等于1/e的情况下，当lambda增加到1/e时，也将进行广泛研究以及Hausdorff尺寸的行为。在这种情况下，要解决的其他问题是多重分析分析和最大化轨道问题。先验仿药功能的研究将集中在非椭圆形功能和非纤维细胞切线家族上。在非电流椭圆函数的背景下，主要目标是解释保形度量的性质，特别是确定这些措施是纯原子还是无原子。后一种情况将打开对Sigma-Finite不变性测量与这种保形度量相等的千古特性的广泛研究的大门。对于双曲线切线家族的目的是证明朱莉娅集合的豪斯多夫维度以实用方式取决于参数lambda。它将增强对此类系统的长期行为的了解。将进一步理解朱莉娅套装的性质，即在许多流行出版物中出现的分形。将研究其基本的几何特性。这项数学研究项目实现了将美国和中欧的主要专家汇集在一起​​的计划目标，以结合互补的努力和能力，以相互利益和能力在平等，互惠，互惠和相互利益的基础基础上。