  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.
