UC Riverside

Space-filling curves have been colloquially referred to as "fractals" since the term was coined and defined by Benoit Mandelbrot in the late 1970s. However, space-filling curves themselves do not satisfy Mandelbrot's definition, and other definitions of fractality also neglect to properly classify space-filling curves as fractals. This is due to the fact that, as sets, they are topologically simple because they fill N-dimensional space and, thus, coincide with N-dimensional space. In the 1990s through the present, Distinguished Professor Michel L. Lapidus and various collaborators have developed a more accurate definition of fractality based on a larger theory of complex dimensions of fractal sets. These complex dimensions correspond to the existence of singularities of certain complex-valued zeta functions associated with fractal sets. These singularities are termed complex dimensions due to their connection with "fractal dimensions" such as the Minkowski dimension, and the complex-valued zeta functions used in analytic number theory. The theory has been incredibly fruitful over the years, even allowing for the Riemann hypothesis to be recast in terms of the complex dimensions of the geometric zeta function of an appropriate fractal string. Today, it has allowed the present author to prove that a class of space-filling curves are, indeed, fractals, and to provide insight into the oscillatory properties of these curves via their complex dimensions.

In this dissertation, Chapter 1 provides the motivating background for the study of complex dimensions of fractal sets. Chapter 2 introduces the theory of fractal strings in R and the complex dimensions of fractal strings. Chapter 3 generalizes the theory to dimensions greater than 1. Chapter 4 provides all the necessary background in the theory space-filling curves required to make sense of the results of this dissertation. Chapter 5 details the construction of a class of relative fractal drums (RFDs) associated to a class of plane-filling curves. This construction allows us to detect the complex dimensions of such curves for the first time, and thereby properly classify them as fractals in a rigorous way. Chapter 6 details how this class of RFDs also allows us to investigate the oscillatory properties of these curves, in particular in the points and the volume of the tubular epsilon-neighborhood of the curves. Chapter 7 concludes the dissertation with some conjectures about future results involving different classes of space-filling curves.