Connectedness related to almost periodicity of compositions of flow homomorphisms (Q1103910)

Consider the homomorphisms \(\phi\) :X\(\to Y\) and \(\psi\) :Y\(\to Z\), where \(\phi\) is open and N-to-one, \(\psi\) is almost periodic. In the paper by \textit{R. J. Sacker} and \textit{G. R. Sell} [Trans. Am. Math. Soc. 190, 325- 334 (1974; Zbl 0288.34042)] it was shown that, under a certain condition on the phase group, the composition \(\psi\circ \phi:X\to Z\) is almost periodic (provided that Z is trivial and X is minimal). Almost periodicity of \(\psi\circ \phi\) is studied under connectedness conditions on the fiber of \(\psi\). For instance it is shown that if \(\psi\) is almost periodic with connected fibers then \(\psi\circ \phi\) is almost periodic. If \(\psi\) is almost periodic with locally connected fibers then \(\psi\circ \phi\) is locally almost periodic.