Almost periodic structures and the semiconjugacy problem (Q413453)

This paper studies a certain class of flows that preserve a one-dimensional foliation. The authors show, for this class, that a semiconiugacy to a minimal translation flow exists if and only if a boundedness condition holds on the distances between the orbits of the flow and those of the translation. This is used to address the question, concerning some dynamical systems originating from the study of quasi-periodic structures, whether the considered system is semi-conjugated to a rigid translation. In order to do so, the authors investigate a general setting that includes those cases but also allows the treatment of scalar differential equations that are almost-periodic both in space and time.