Period functions for $C^0$ and $C^1$ flows

Let $(F_t)$ be a continuous flow on a topological manifold M. For every open $V \subset M$ denote by P(V) the set of all continuous functions $\alpha:V \to R$ such that $F_{\alpha(z)}(z)=z$ for all $z\in V$. Such functions vanish at non-periodic points and their values at periodic points are equal to the corresponding periods (in general not minimal). They can be used for reparametrizations of flows to circle actions. In this paper P(V) is described for connected open subsets V, which extends previous results of the author. It is also shown that there is a difference between the sets P(V) for $C^{0}$ and $C^1$ flows.