Prolongations in topological dynamics

Abstract

In 1953, Taro Ura [20] introduced the prolongations D + prolongational x' limits j~ were defined in [i~; and regions of (weak) attraction in [3]. Since then the study of these concepts has been vigorously developed, and applied with considerable effect, within dynamical system theory (references are too numerous to give here; a systematic exposition will appear in [5]) • However, topological dynamics has largely not profited from these developments; surprisingly so~ since there had always been considerable inter-communication between the two disciplines. The present paper aims at alleviating this situation, by presenting the fundamentals, and by showing that many concepts introduced and studied earlier are intimately related to, and succintly described by, the prolongations (our list is intended to be suggestive rather than exhaustive). The exposition proceeds by what we call the dynamical relations [5,VI]; these seem to be a unifying concept, to some extent. It is also appropriate to point out two shortcomings of our approach, possibly obvious even at the outset. First, even in dynamical system theory, not 811 important concepts allow a nice characterization in terms of prolongations; e.g., parallelizability, or the strong (-regionally) mixing property [i0, 902], [9]. In point of fact~ the well-known theorem on parallelizahility [7] (actually proved earlier in [17, 2.4]) exhibits further assumptions under