rather large digital images. Thus, rather than storing the full images, one can hope to work with a time series of persistence diagrams.

The simplest mathematical analogy is that of attempting to describe the evolution of a partial differential equation via observations of the evolution of finite dimensional vectors obtained through a Galerkin projection. Obviously, since this is a central problem in numerical analysis this problem has been addressed in a variety of ways, including functional analysis. However, we are interested in going a step further. One can imagine that a piecewise linear function is associated with each finite dimensional vector, and to continue the analogy, the sublevel set persistence diagram for this function is computed. We would like to draw conclusions about the dynamics of the original partial differential equation from the time series of the persistence diagrams. This is an extremely ambitious goal and far beyond our capabilities at the moment. A much simpler question is the following: if there is an attracting region in the space of persistence diagrams, under what conditions can we conclude that there is a fixed point for the partial differential equation.

This paper represents a first step towards answering the simpler question. 