Smooth Shifts

Let M be a smooth manifold and Φ be a smooth flow on M . We study the maps M → M generated by smooth shifts along trajectories of Φ, i.e the maps of a form h(x) = Φ(x, σ(x)) where σ is some smooth function on M . We will classify smooth shifts which induce diffeomorphisms ofM and prove contractibility of the set of such diffeomorphisms for large class of flows. These results will be applied to the study of homotopy groups of orbits of Morse maps of surfaces under the action of diffeomorphisms group.