Lifting of almost periodicity of a point through morphisms of flows

Abstract

Let $f:\mathcal{X}\rightarrow \mathcal{Y}$ be a morphism of flows, $y$ an almost periodic point of $\mathcal{Y}$, and $x\in f^{-1}(y)$. In general $x$ is not ne\-cessa\-rily almost periodic, but several conditions are known under which that happens. They fall into either ``compact" or ``noncompact" conditions, depending on whether $\mathcal{X}$ and $\mathcal{Y}$ are assumed to be compact or not. In ``noncompact" conditions other assumptions are restrictive. We find a criterion for almost periodicity of $x$, which generalizes both ``compact" and ``noncompact" statements at the same time. We deduce theorems of Ellis, Markley, Kutaibi-Rhodes and Pestov as corollaries.