Intersection of triadic Cantor sets with their translates. II. Hausdorff measure spectrum function and its introduction for the classification of Cantor sets

Initiated by the purpose of classification of sets having the same fractal dimension, we continue, in this second paper of a series of two, our investigation of intersection of triadic Cantor sets and their use in the classification of fractal sets. We exploit the infinite tree structure of translation elements to give the exact expressions of these elements. We generalize this result to a family of uniform Cantor sets for which we also give the Hausdorff measure spectrum function (HMSF). We develop three algorithms for the construction of HMSF of triadic Cantor sets. Then, we introduce a new method based on HMSF as a way for tracing the geometrical organization of a fractal set. The HMSF does carry a huge amount of information about the set to likely be explored in a chosen way. To extract this information, we develop a one by one step method and apply it to typical fractal sets. This results in a complete identification of fractals.