Fiber of persistent homology on morse functions

Let f be a Morse function on a smooth compact manifold M with boundary. The path component PH^-1_f(D) containing f of the space of Morse functions giving rise to the same Persistent Homology D=PH(f) is shown to be the same as the orbit of f under pre-composition ϕ↦ f∘ϕ by diffeomorphisms of M which are isotopic to the identity. Consequently we derive topological properties of the fiber PH^-1_f(D) : In particular we compute its homotopy type for many compact surfaces M . In the 1-dimensional settings where M is the unit interval or the circle we extend the analysis to continuous functions and show that the fibers are made of contractible and circular components respectively.