Period functions for $ {mathcal{C}^0} $- and $ {mathcal{C}^1} $-flows

Let F:M ×mathbbR ? M {mathbf{F}}:M times mathbb{R} to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? mathbbR xi :V to mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C¹ {mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.