Georgia Institute of Technology College of Sciences

Let (X, T) be a flow, that is a continuous left action of the group T on the compact Hausdorff space X. The proximal P and regionally proximal RP relations are defined, respectively (assuming X is a metric space) by P = {(x; y) | if \epsilon > 0 there is a t \in T such that d(tx, ty) < \epsilon} and RP = {(x; y) | if \epsilon > 0 there are x', y' \in X and t \in T such that d(x; x') < \epsilon, d(y; y') < \epsilon and t \in T such that d(tx'; ty') < \epsilon}. We will discuss properties of P and RP, their similarities and differences, and their connections with the distal and equicontinuous structure relations. We will also consider a relation V defined by Veech, which is a subset of RP and in many cases coincides with RP for minimal flows.