Thermodynamic Formalism, Dynamics and Dimensions

热力学形式主义、动力学和尺寸

• 批准号: 1001874
• 负责人: Mariusz Urbanski
• 金额: $ 23.99万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001874&HistoricalAwards=false
• 关键词: Thermodynamic; Formalism; Dynamics; Dimensions

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.

在这个项目中，主要研究者建议进一步推进和发展希尔伯特双曲离散群，随机动力系统，一维格子气，几何和亚纯超越函数的动力学，紧致复流形的全纯自同态，保形迭代函数系和保形扩张排斥子的Hausdorff测度的连续性。借助于动力系统的概念和技术，遍历理论，统计物理，泛函分析，几何测度理论，复分析，概率论，代数和微分几何，适当形式的热力学形式主义，确定性和随机，为这些系统将被构建和研究。该项目将涉及转移算子的分析，吉布斯和平衡状态，亚纯函数的Julia集，希尔伯特离散群的极限集，以及图导向马尔可夫系统的吸引子的Hausdorff测度和维数。事实上，该项目的概念，技术和方法，虽然本质上是动态的，然而，通过上述数学和物理学分支的相互作用而产生的，将产生有趣的后果。该项目将阐明这些领域本身，可能会刺激这些领域的技术和方法的发展，特别是，可能会导致它们的增长，以响应来自动力系统理论的需求。沿着这些思路，该项目假定主要调查员与这些领域的若干专家合作。这种联合工作预计将扩大他们的共同专业知识，并应导致加强调查领域。研究生的积极参与是拟议工作的一个组成部分。希望学生逐步掌握拟研究的课题，了解更多几何测度理论、超越亚纯函数和整函数理论、代数几何等学科知识，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，并吸引丹顿学者谁通过提供座谈会和研讨会讲座将互动，并科学地刺激研究生和教师在丹顿。