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

这是一个美国-波兰合作研究项目，将集中在遍历理论和超越整函数和亚纯函数的几何。 主要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教授和华沙大学的安娜Zdunik教授。这类映射的Julia集的拓扑结构最近得到了深入的研究。 在Hausdorff维数的水平上的分形性质也进行了处理。本项目的研究将深入到非双曲指数映射的Hausdorff和packing测度。 研究双曲正切族的Hausdorff维数的真实的解析性。 研究的主要工具将包括一个共形措施的概念，基本上属于阿森纳的热力学形式主义超越一致双曲系统的紧凑空间。各种传输操作符也将经常使用。 非双曲指数映射类的工作将主要关注的Julia集的子集，其中携带的所有经常性和混沌的动力学部分。证明了该子集的适当Hausdorff测度是正的和有限的，而填充测度是无限的。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。 当参数lambda等于1/e时的情况也将被广泛研究，以及当lambda增加到1/e时Hausdorff维数的行为。其他要处理的问题是多重分形分析和最大化轨道问题在这方面。超越亚纯函数的研究主要集中在非回归椭圆函数和非双曲正切函数族上。在非递归椭圆函数的背景下，主要目标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。后一种情况将打开大门，广泛的研究遍历性的一个西格玛有限不变的措施相当于这个保形措施。对于双曲正切族，我们的目标是证明Julia集的Hausdorff维数以实解析的方式依赖于参数λ，该项目的完成将为非紧相空间混沌系统的演化提供新的思路。它将增强对此类系统长期行为的了解。Julia集的性质，分形出现在许多流行的出版物，将进一步理解。该数学研究项目旨在实现美国和中欧/东欧的主要专家在平等、互惠和互利的基础上，在具有强烈共同利益和能力的领域，将联合收割机互补的努力和能力结合在一起的计划目标。

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.