Scale Covariance of Fractal Sets and Measures, A Differential Approach to the Box-Counting Function of a Fractal, with Applications
- Author(s): Quinn, John Roosevelt
- Advisor(s): Lapidus, Michel L
- et al.
Abstract: The scale symmetry of self-similarity is a fundamental one in physics and in geometry. We develop a calculus of the scale space evolution of self-similar fractal sets via an analysis of box-counting functions on these structures utilizing the theories of distributions and hyperfunctions. A differential study of the Box-Counting function can account for the oscillations of the local geometry of some examples of such structures, paralleling the theory of the complex dimensions of fractal strings. The algebraic structure on the iterates of the unit interval under an Iterated Function System admits a tensor product representation we develop to define an intrinsic geometry of fractals and an integral calculus on these objects.