systems, fixed point theorem.]{}

Introduction {#secintro}
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*Topological data analysis* (TDA), especially in the form of persistent homology, is rapidly developing into a widely used tool for the analysis of high dimensional data associated with nonlinear structures. That topological tools can play a role in this subject should not be unexpected given the central role of nonlinear functional analysis in the study of geometry, analysis, and differential equations. What is perhaps surprising is that, to the best of our knowledge, there has been no systematic attempt to make use of nonlinear functional analysis as a tool to process information obtained via persistent homology.

Persistent homology is often used as a means of data reduction. A typical example takes the form of a complicated scalar function defined over a fixed domain, where the geometry of the sub-(super)-level sets is encoded via homology. Of particular interest to us are settings in which the scalar function arises as a solution to a partial differential equation, we are interested in tracking the evolution of the function, but experimental data only provides information on the level of digital images of the process. Furthermore, for many problems capturing the dynamics requires a long time series of 