Mathematical Sciences: Ergodic Theory and Combinatorics

点击翻译按钮获取中文摘要

Professors Ornstein, Katznelson, and Adams will investigate a variety of matters in ergodic theory. Problems in combinatorics and combinatorial number theory will be attacked using ergodic methods. The stability under perturbation of basic ergodic- theoretical properties of some specific classical dynamical systems is another focus of effort. The investigators will also consider smooth conjugacy of differentiable dynamical systems and its relevance to twist maps. Other topics include Hausdorff dimensions of orbit closures, combinatorial structures associated with equivalence relations, and the ergodic theory of foliations. Ergodic theory is concerned with what happens in the long run when a transformation of a space is iterated many times. (The space in question need not be anything particularly elaborate; something like a line segment will do.) Via the transformation, one can think of the group Z of integers acting on the space, or of time passing in discrete steps. Continuous time, similarly, is modeled by an action of the group R of real numbers, called a flow. This situation is also of interest in ergodic theory. A good example is a geodesic flow on a surface, where points on the surface are imagined as moving at constant speed along paths of shortest distance. A principal goal of research in ergodic theory is to classify the various sorts of long-run behavior that are possible in a given situation. Typically, there will be not too many, and all of them will be realizable within an appropriate standard model. Fitting ergodic-theoretical phenomena into such models is part of the work of this project. Ergodic theory has recently come to play an important role in the study of complicated combinatorial structures. The principal investigators hope to continue to make contributions in this area as well.