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

点击翻译按钮获取中文摘要

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").