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

这是一个美国-波兰的合作研究项目，将专注于超越整函数和亚纯函数的遍历理论和几何。主要研究人员是北德克萨斯大学的Mariusz Urbanski博士、华沙工业大学的Janina Kotus教授和华沙大学的Anna Zdunik教授，本项目的主要研究对象是超越整函数和亚纯函数。这类映射的Julia集的拓扑结构最近得到了广泛的研究。文中还讨论了Hausdorff维上的分形性质。本课题的研究将深入到非双曲指数映射的Hausdorff和Packing测度的更深层次。我们将研究双曲正切族的Hausdorff维度的真实解析性。研究的主要工具将包括本质上属于热力学形式主义武器库的共形度量的概念，该概念超越了紧致空间上的一致双曲系统。各类转接操作员也将频繁使用。关于这类非双曲指数映射的工作将主要涉及Julia集的子集，该子集承载了动力学的所有递归和混沌部分。我们猜想这个子集的适当Hausdorff测度是正的和有限的，而填充测度是无穷的。下一步将是证明与共形测度等价的不变测度的存在性，并探索其遍历性质。还将广泛研究当参数lambda等于1/e时的情况，以及当lambda增加到1/e时Hausdorff维度的行为。其他需要处理的问题是多重分形分析和在此背景下的最大化轨道问题。超越亚纯函数的研究将集中在非递归椭圆函数和非双曲正切族上。在非递归椭圆函数的背景下，主要目的是解释共形测度的性质，特别是确定这些测度是纯原子的还是无原子的。后一种情况将为广泛研究与这种共形测度等价的sigma-有限不变测度的遍历性质打开大门。对于双曲正切族，我们的目标是证明Julia集的Hausdorff维依赖于参数lambda，该项目的完成将为研究非紧相空间混沌系统的演化提供新的线索。它将增强对此类系统长期行为的了解。朱莉娅集的性质，即出现在许多流行出版物上的分形图，将被进一步理解。这个数学研究项目实现了计划的目标，即将美国和中东欧的顶尖专家聚集在一起，在平等、互惠和互利的基础上，在共同利益和能力强的领域结合互补的努力和能力。

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.