Mathematical Sciences: Self-Affine Sets, Random Walks on Discrete Groups, and Thermodynamic Formalism

将在三个相关领域进行研究： 自仿射集与扩张映射;离散概率 群;和热力学形式主义。 最终目标是 有助于理解统计行为， 非共形扩张和双曲动力系统， 特别是当它涉及到排斥体的“分形”几何形状时。 更直接的目的是阐明 自仿射集，例如，豪斯多夫和 Bouligand维数与自然动力系统 这样的设置。 为了更好的理解随机矩阵 产品，特别是它们的大偏差，似乎是必不可少的。的 随机矩阵乘积的研究反过来又引出了一些问题 关于非交换群上的随机游动和 吉布斯措施和热力学形式主义：特别是， 研究工作的重要组成部分将针对 产生转移概率的精确近似值 离散群和半群上的随机游动， 一个理论的Ruelle类运营商适合于这一目的。 对某些混沌系统行为的研究 动力系统将进行，特别是那些在 其中相空间以不同的速率拉伸， 个方向 不寻常的几何物体出现在 这些动力系统的“相空间”，称为 “排斥者”，以及他们与自然界的联系。 将密切研究动态。 概率论 在这些调查中占主导地位： 通过选择一个初始轨道， 系统的状态”。“这一观点导致了相关的 概率论中的问题，主要涉及 状态空间上随机游动的某些特征， 非欧几里得几何（“矩阵群”）。

Research will be conducted in three related areas: Self-Affine Sets and Expanding Maps; Probability on Discrete Groups; and Thermodynamic Formalism. The ultimate goal is to contribute to the understanding of statistical behavior in nonconformal expansive and hyperbolic dynamical systems, especially as it relates to the "fractal" geometry of repellers. The more immediate aim is to elucidate the structure of self-affine sets,e.g., the relationship between Hausdorff and Bouligand dimensions and the natural dynamical systems associated with such sets. For this a better understanding of random matrix products, especially their large deviations, seems essential. The study of random matrix products, in turn, leads to questions concerning random walks on nonabelian groups and the theory of Gibbs measures and thermodynamic formalism: in particular, a significant component of the research effort will be directed to producing accurate approximations to the transition probabilities of random walks on discrete groups and semigroups, and developing a theory of Ruelle-like operators appropriate for this purpose. Investigations into the behavior of certain chaotic dynamical systems will be conducted, in particular, those in which the phase space is stretched at different rates in different directions. Unusual geometric objects that arise in the "phase spaces" of these dynamical systems, called "repellers," and their connections with the nature of the dynamics will be closely studied. The theory of probability predominates in these investigations: questions involving the behavior of "typical" orbits are studied by choosing an initial state of the system "at random." This viewpoint leads to related problems in probability theory proper, mostly concerned with certain features of random walks on state spaces with a highly noneuclidean geometry ("matrix groups").