Vol. 21, No. 3, 1967

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Stability in topological dynamics

James Walton England

Vol. 21 (1967), No. 3, 479–485

This paper is concerned with two types of stability in transformation groups. The first is a generalization of Lyapunow stability. In the past this notion has been discussed a setting where the phase group was either the integers or the one-parameter group of reals. In this paper it is defined for replete subsets of a more general phase group in a transformation group. Some connections between this type of stability and almost periodicity are given. In particular, it is shown that a type of uniform Lyapunov stability will imply Bohr almost periodicity. The second type of stability this paper is a limit stability. This gives a condition which is necessary and sufficient for the limit set to be a minimal set. Finally, these two types of stability are combined to provide a sufficient condition for a limit set to be the closure of a Bohr almost periodic orbit.

Mathematical Subject Classification
Primary: 54.82
Received: 21 February 1966
Published: 1 June 1967
James Walton England