The self-similar evolution of stationary point processes via persistent homology

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point processes. We provide variants of notions of ergodicity and investigate measures on the space of persistence diagrams. In particular we introduce the notion of self-similar scaling of persistence diagram expectation measures and prove a packing relation for the occurring dynamical scaling exponents. As a byproduct we generalize the strong law of large numbers for persistent Betti numbers proven in [Hiraoka et al., Ann. Appl. Probab. 28(5), 2018] for averages over cubes to arbitrary convex averaging sequences.