Almost periodicity and B-equicontinuity in topological dynamics

In the previous paper(8), we considered a property of families of functions we termed. ‘B-equicontinuity’. It was shown that B-equicontinuity is stronger than the usual equicontinuity, and is weaker than the equicontinuity defined by Bartle (3). In this paper we consider the concept of B-equicontinuity on topological transformation groups. The net characterization of equicontinuity obtained in (8) is used in discussion. It is proved in (1) that if (X, T, π) is almost periodic, the transition group {πt|t ∈ T} is equicontinuous. One might wonder whether this conclusion can be strengthened to say that {πt|t ∈ T} is B-equicontinuous; we show here by an example that this is not true and a partial solution to this problem is given. Some relations between almost periodicity and B -equicontinuity are also discussed.