Smooth shifts along trajectories of flows

Let Φ be a flow on a smooth, compact, finite-dimensional manifold M. Consider the subset D(Φ) of C∞(M,M) consisting of diffeomorphisms of M preserving the foliation of the flow Φ. Let also D0(Φ) be the identity path component of D(Φ) with compact-open topology. We prove that under mild conditions on fixed points of Φ the space D0(Φ) is either contractible or homotopically equivalent to S1.