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.

25 pages, final version