Image of a shift map along the orbits of a flow Sergiy Maksymenko 37C10orbit preserving diffeomorphismshift mapreparametrization of time Let $(\mathbf{F}_t)$ be a smooth flow on a smooth manifold $M$ and $h: M \to M$ be a smooth orbit preserving map. The following problem is studied: suppose that for every $z \in M$ there exists a germ $\alpha_z$ of a smooth function at $z$ such that $h(x) = F_{\alpha(x)}(x)$ near $z$; can the germs $(\alpha_z)_{z \in M}$ be glued together to give a smooth function on all of $M$? This question is closely related to reparametrizations of flows. We describe a large class of flows $(\mathbf{F}_t)$ for which the above problem can be resolved, and show that they have the following property: any smooth flow $(\mathbf{G}_t)$ whose orbits coincide with the ones of $(\textbf{F}_t)$ is obtained from $(\mathbf{F}_t)$ by smooth reparametrization of time. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4213 10.1512/iumj.2010.59.4213 en Indiana Univ. Math. J. 59 (2010) 1587 - 1628 state-of-the-art mathematics http://iumj.org/access/