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

研究将在三个相关领域进行：自仿射集和扩展映射；离散群的概率；和热力学形式主义。最终目标是有助于理解非保形膨胀和双曲动力系统的统计行为，特别是当它涉及到驱蚊器的“分形”几何形状时。更直接的目的是阐明自仿射集的结构，例如：， Hausdorff维和Bouligand维之间的关系以及与这些集合相关的自然动力系统。为此，更好地理解随机矩阵乘积，特别是它们的大偏差，似乎是必要的。随机矩阵积的研究，反过来又引出了关于非abel群上的随机游走、吉布斯测度理论和热力学形式论的问题：特别是，研究工作的一个重要组成部分将直接用于产生离散群和半群上随机行走转移概率的精确近似值，并发展适合于此目的的类ruelle算子理论。将对某些混沌动力系统的行为进行研究，特别是那些相空间在不同方向上以不同速率拉伸的系统。在这些动力系统的“相空间”中出现的不寻常的几何物体，称为“驱虫剂”，以及它们与动力学性质的联系将被密切研究。概率论在这些研究中占主导地位：通过“随机”选择系统的初始状态来研究涉及“典型”轨道行为的问题。这种观点导致了概率论固有的相关问题，主要涉及具有高度非欧几里德几何（“矩阵群”）的状态空间上随机行走的某些特征。

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").