Smooth shifts along flows

Abstract

Let $\Phi$ be a flow on a smooth, compact, finite-dimensional manifold $M$. Consider the subsets $E(\Phi)$ and $D(\Phi)$ of $C^{\infty}(M,M)$ consisting of smoothh mappings and diffeomorphisms (respectively) of $M$ preserving the foliation of the flow $\Phi$. Let also $E_{0}(\Phi)$ and $D_{0}(\Phi)$ be the identity path components of $E(\Phi)$ and $D(\Phi)$ with compact-open topology. We prove that under mild conditions on fixed points of $\Phi$ the inclusion $D_{0}(\Phi) \subset E_{0}(\Phi)$ is a homotopy equivalence and these spaces are either contractible or homotopically equivalent to the circle.