Topological bar-codes of fractals: a new characterization of symmetric binary fractal trees
The goal of this paper is to provide foundations for a new way to classify and characterize fractals using methods of computational topology. The fractal dimension is a main characteristic of fractal-like objects, and has proved to be a very useful tool for applications. However, it does not fully characterize a fractal. We can obtain fractals with the same dimension that are quite different topologically. Motivated by techniques from shape theory and computational topology, we consider fractals along with their $\epsilon$-hulls as $\epsilon$ ranges over the non-negative real numbers. In particular, we develop theory for the class of non-overlapping symmetric binary fractal trees that can be generalized to broader classes of fractals. We investigate various features of the $\epsilon$-hulls of the trees, based on the holes in these hulls. We determine the hole sequence of these trees together with the persistence intervals of the holes as the `topological bar-codes' of these fractals. We provide quantitative results for a selection of specific trees to illustrate the theory. Finally, we prove that for non-overlapping symmetric binary fractal trees, the growth rate of holes in $\epsilon$-hulls is equal to the similarity dimension.