Fiber of Persistent Homology on Morse functions

Let $f$ be a Morse function on a smooth compact manifold $M$ with boundary. The path component $\mathrm{PH}_f^{-1}(D)$ containing $f$ of the space of Morse functions giving rise to the same Persistent Homology $D=\mathrm{PH}(f))$ is shown to be the same as the orbit of $f$ under pre-composition $\phi \mapsto f\circ \phi$ by diffeomorphisms of $M$ which are isotopic to the identity. Consequently we derive topological properties of the fiber $\mathrm{PH}_f^{-1}(D)$: 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.