Thermodynamic Formalism, Dynamics and Dimensions

在这个项目中，首席研究者建议进一步推进和发展研究Hilbertian双曲离散群的动力学、统计和几何方面的方法，随机动力系统，一维晶格气体，亚纯超越函数的几何和动力学，紧复流形的全纯自同态，以及共形迭代函数系统和共形扩展的Hausdorff测度的连续性。在动力系统、遍历理论、统计物理、泛函分析、几何测量理论、复分析、概率论、代数和微分几何的概念和技术的帮助下，将为这些系统构建和研究确定和随机的适当形式的热力学形式。该项目将涉及转移算子、Gibbs态和平衡态、亚纯函数的Julia集和Hilbertian离散群的极限集的分析，以及图向马尔可夫系统的Hausdorff测度和吸引子的维度。项目的概念、技术和方法虽然本质上是动态的，但却是通过上述数学和物理分支的相互作用而产生的，这一事实将产生有趣的结果。该项目将阐明这些领域本身，可能刺激这些领域的技术和方法的发展，特别是可能导致它们的增长，以响应来自动力系统理论的需求。在这些方面，该项目假定首席研究员与这些领域的几位专家合作。预期这种联合工作将扩大它们的相互专业知识，并应加强所研究的领域。研究生的积极参与是拟议工作的一个组成部分。期望学生逐渐掌握所拟研究的课题，学习更多的几何测度理论、超越亚纯与全函数理论、代数几何等学科知识，并最终为项目贡献自己的创造性工作。拟议的研究预计将导致高级研究生课程，并通过举办座谈会和研讨会讲座吸引丹顿的学者，他们将与丹顿的研究生和教师进行互动，并科学地激励他们。

In this project the principal investigator proposes to further advance and develop the methods for investigations of dynamical, statistical and geometrical aspects of Hilbertian hyperbolic discrete groups, random dynamical systems, one-dimensional lattice gasses, geometry, and dynamics of meromorphic transcendental functions, holomorphic endomorphisms of compact complex manifolds, and continuity of Hausdorff measures for conformal iterated function systems and conformal expanding repellers. Aided by concepts and techniques of dynamical systems, ergodic theory, statistical physics, functional analysis, geometric measure theory, complex analysis, probability theory, and algebraic and differential geometry, appropriate forms of thermodynamic formalism, both deterministic and random, for those systems will be constructed and investigated. The project will involve the analysis of transfer operators, Gibbs and equilibrium states, Julia sets of meromorphic functions, and limit sets of Hilbertian discrete groups, as well as Hausdorff measures and dimensions of attractors of graph-directed Markov systems.The fact that the concepts, techniques and methods of the project, while dynamical in essence, are nevertheless created through the interplay of the branches of mathematics and physics indicated above, will have interesting consequences. The project will shed light on these fields themselves, may stimulate the development of techniques and methods in these areas, and in particular, may cause their growth in response to demands coming from the theory of dynamical systems. Along these lines, the project assumes cooperation of the principal investigator with several specialists in those fields. Such joint work is expected to broaden their mutual professional expertise and should give rise to enhancement of the investigated domains. The active involvement of graduate students is an integral part of the proposed work. The students are expected to gradually master the topics of the proposed research, to learn more about geometric measure theory, the theory of transcendental meromorphic and entire functions, algebraic geometry and other subjects, and finally to contribute to the project their own creative work. The proposed research is expected to result in advanced graduate courses and to attract to Denton scholars who by delivering colloquium and seminar lectures will interact with and scientifically stimulate graduate students and faculty in Denton.