Strongly Reversible Flows on Connected Manifolds

Let $G = \{h_t \ | \ t \in \mathbb R\}$ be a flow of homeomorphisms of a connected $n$-manifold and let $L(G)$ be its limit set. The flow $G$ is said to be strongly reversed by a reflection $R$ if $h_{-t} = R h_t R$ for all $t \in \mathbb R$. In this paper, we study the dynamics of positively equicontinuous strongly reversible flows. If $L(G)$ is nonempty, we discuss the existence of symmetric periodic orbits, and for $n=3$ we prove that such flows must be periodic. If $L(G)$ is empty, we show that $G$ positively equicontinuous implies $G$ strongly reversible and $G$ strongly reversible implies $G$ parallelizable with global section the fixed point set $Fix(R)$.