Theorem \[thmmaindyn\] shows that given a flow with a global compact attractor ${{\mathcal{A}}}\subset{\mathbb{R}}^N$ and a neighborhood of a particular form in the space of persistence diagrams, if under the dynamics the neighborhood is mapped into itself, then there exists a fixed point for the flow. Returning to the partial differential equation analogy, one could consider a system defined over an interval where the dynamics on the global attractor is obtained via a finite difference approximation.

The challenge, even in the simplified finite dimensional setting, is that to obtain results one must understand the topology of pre-images of sets under persistent homology, a topic for which there are only limited results. That the structure of pre-images is complicated follows directly from the fact that persistent homology can provide tremendous data reduction, but in a highly nonlinear fashion. We emphasize that simple examples show that the preimage set of a straight line path in a persistence diagram is nonconvex, see Figure \[figpre\].

With this in mind the primary goal of this paper is to show that for a reasonable class of bounded piecewise monotone continuous functions on a bounded interval the pre-image of a persistence diagram is composed of contractible, simplicial sets. 