Topological bar-codes of fractals: a new characterization of symmetric binary fractal trees

Abstract

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-hulls as 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-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-hulls is equal to the similarity dimension.