Thermodynamic Formalism, Dynamics and Dimensions

这个项目将进一步发展数学的一个重要分支——动力系统。这个领域起源于19世纪末和20世纪初的伟大数学家，如亨利·庞加莱和雅克·阿达玛尔，仅举两位。他们的研究与物理学、经典力学和微分几何有重叠；目前这个项目也是多学科的。除了动力系统外，它还涉及数学的其他几个子领域，如数论和概率论。在这个项目中，PI将建立在这些领域的基础上，并通过制定适用于这些领域和动力系统的严格方法来促进它们的发展。该项目旨在让来自三大洲的至少11位学者作为合作者。其中美国学者4人，加拿大学者1人，欧洲学者5人，日本学者1人。建议的研究结果也将成为进一步研究的来源，并为高级毕业课程的教学提供材料。获得资助的学者将访问PI在德克萨斯州丹顿的机构进行研究，发表研讨会讲座，并与那里的教师和研究生互动。主要研究者提出，在七个独立但相互关联的子项目中，推进丢番图近似理论，转移（Perron-Frobenius）算子，离散群，可数字母共形迭代函数系统，全纯动力学，随机动力系统和遍历平均。丢芬图近似包括几何极值度量和有限维欧几里得空间中难以近似的向量。传递算子包括阿波罗圈填料和带孔的动力系统。离散群体包括几何刚度克莱因理论的组织和格罗莫夫双曲空间。共形迭代函数系统包括谐波测度、精确维数、豪斯多夫测度的连续性和非自治共形迭代函数系统。全纯动力学包括双测度和有理半群。随机动力系统包括收缩目标、随机共形动力系统的动力刚性、具有横向性的随机共形迭代函数系统、随机有理函数和随机超越亚纯函数。这些子项目将需要来自动力系统、遍历理论、泛函分析、几何测量理论、数论、复分析和概率论的方法。本提案的每个子项目的目标是解决与这些领域相关的主要动力学、几何和分析问题。它建立在PI在其整个研究生涯中获得的方法和结果的基础上，主要集中在过去五年中撰写的约40篇论文。

This project will further develop an important subfield of mathematics called dynamical systems. This field originated at the end of nineteenth century and at the beginning of twentieth century with the work of great mathematicians such as Henri Poincare and Jacques Hadamard, to mention only two. Their research overlapped with physics, classical mechanics, and differential geometry; this current project is also multidisciplinary. It involves, in addition to dynamical systems, several other subfields of mathematics such as number theory and probability theory. In this project the PI will build on these fields and contribute to their development by laying down rigorous methods as applicable to those fields as to dynamical systems. The project is intended to involve at least eleven scholars from three continents as collaborators. Among them included are four scholars from the United States, one from Canada, five from Europe, and one from Japan. The results of the proposed research will also form a source for further research and provide material for teaching advanced graduated courses. The supported scholars will visit the PI's institution in Denton, Texas to conduct research, deliver seminar lectures, and interact with faculty and graduate students there.The principal investigator proposes, in seven separate but interrelated subprojects, to advance the theory of Diophantine approximations, transfer (Perron-Frobenius) operators, discrete groups, countable alphabet conformal iterated function systems, holomorphic dynamics, random dynamical systems, and ergodic averages. Diophantine approximations include geometrically extremal measures and badly approximable vectors in finite-dimensional Euclidean spaces. Transfer operators include Apollonian circle packings and dynamical systems with holes. Discrete groups include geometric rigidity for Kleinian groups and Gromov hyperbolic spaces. Conformal iterated function systems include harmonic measure, exact dimensionality, continuity of Hausdorff measure, and non--autonomous conformal iterated function systems. Holomorphic dynamics includes doubling measures and rational semigroups. Random dynamical systems include shrinking targets, dynamical rigidity in random conformal dynamical systems, random conformal iterated function systems with transversality, random rational functions, and random transcendental meromorphic functions. These subprojects will require methods from dynamical systems, ergodic theory, functional analysis, geometric measure theory, number theory, complex analysis, and probability theory. The goal of each subprojects of this proposal is to solve the major dynamical, geometric, and analytical problems related to these fields. It builds upon the methods and results the PI has obtained throughout his entire research career, focusing mainly on about 40 papers written in the past five years.