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

Abstract

Let $F:\; M×R→M$ be a continuous flow on a manifold $M$, let $V ⊂ M$ be an open subset, and let $ξ:\; V→R$ 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 $\mathcal{C}^1$-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.