Classification of Fractal Sets and Concluding Comments

Abstract

In this last chapter, we first introduce a refinement of the classification of bounded sets in \( \mathbb{R}^{N} \) which had begun with the well-known distinction between Minkowski nondegenerate and Minkowski degenerate sets. Further distinction will be made by classifying fractals according to the properties of their tube functions and allowing, in particular, more general scaling laws than the standard power laws. We then provide a short historical survey concerning notions pertaining to Minkowski measurability and related topics which play an important role in this work. We conclude the book with a few remarks, a long list of open problems, and propose several directions for future research. The research problems and directions proposed here connect many different aspects of fractal geometry, number theory, complex analysis, functional analysis, harmonic analyis, complex dynamics and conformal dynamics, partial differential equations, mathematical physics, spectral theory and spectral geometry, as well as nonsmooth analysis and geometry.