Mathematical Sciences: The Dynamical Theory of Tilings and Multi-dimensional Symbolic Dynamics

罗宾逊将继续他对欧几里得空间的周期性平铺的研究。该方法是将平铺集视为紧化度量空间，并通过平移研究作用在这些空间上的R^n的动力学。在这种情况下，许多标准的动力学性质对应于瓷砖的重要性质。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。通过这种方式，遍历理论在寻求对均匀空间上的流进行分类时与几何接触。

Robinson will continue his study of a periodic tilings of Euclidean space. The approach is to view sets of tilings as compact metric spaces, and to study the dynamics of R^n acting on such spaces by translation. In this context, many standard dynamical properties correspond to important properties of tilings. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way, ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.