Local inverses of shift maps along orbits of flows

Abstract

Let $\mathbf{F}$ be a smooth flow on a smooth manifold $M$ and $\mathcal{D}(\mathbf{F})$ be the group of diffeomorphisms of $M$ preserving orbits of $\mathbf{F}$. We study the homotopy type of the identity components $\mathcal{D}_{\id}(\mathbf{F})^{r}$ of $\mathcal{D}(\mathbf{F})$ with respect to distinct Whitney topologies $\mathsf{W}^{r}$, ($0 \leq r \leq \infty$). The main result presents a class of flows $\mathbf{F}$ for which $\mathcal{D}_{\id}(\mathbf{F})^{r}$ coincide for all $r$ and are either contractible or homotopy equivalent to the circle. The group $\mathcal{D}_{\id}(\mathbf{F})^{0}$ was studied in the author's paper [13]. Unfortunately that article contains a gap in estimations of continuity of local inverses of the so-called shift map. The present paper also repairs these estimations and shows that they hold under additional assumptions on the behavior of regular points of $\mathbf{F}$.