Thermodynamic Formalism, Dynamics and Dimensions

这个项目将进一步发展一个重要的数学子领域，称为动力系统。这个领域起源于19世纪末和20世纪初，伟大的数学家亨利·庞加莱和雅克·哈达玛的工作，仅举两例。他们的研究与物理学、经典力学和微分几何相重叠；目前的这个项目也是多学科的。除动力系统外，它还涉及数论和概率论等数学的其他几个子领域。在这个项目中，PI将建立在这些领域的基础上，并通过制定适用于动态系统等领域的严格方法来促进它们的发展。该项目的目的是让来自三大洲的至少11名学者作为合作者。其中包括4名美国学者、1名加拿大学者、5名欧洲学者和1名日本学者。拟议研究的结果也将成为进一步研究的来源，并为高级研究生课程的教学提供材料。被资助的学者将访问得克萨斯州丹顿的PI机构进行研究，发表研讨会演讲，并与那里的教职员工和研究生互动。主要研究人员在七个独立但相互关联的子项目中提出，推进丢番图近似、转移(Perron-Frobenius)算子、离散群、可数字母表共形迭代函数系统、全纯动力学、随机动力系统和遍历平均的理论。丢番图逼近包括有限维欧氏空间中的几何极值测度和极差逼近向量。转移算符包括阿波罗圆填充和有洞的动力系统。离散群包括Klein群和Gromov双曲空间的几何刚性。共形迭代函数系统包括调和测度、精确维数、Hausdorff测度的连续性和非自治的共形迭代函数系统。全纯动力学包括加倍测度和有理半群。随机动力系统包括收缩目标、随机共形动力系统中的动力刚性、具有横截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。这些子项目将需要动力系统、遍历理论、泛函分析、几何测度论、数论、复数分析和概率论的方法。该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在整个研究生涯中所获得的方法和结果的基础上，主要集中在过去五年中写的大约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.