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.
