Number systems, iterated function systems and fractals

In this thesis we investigate number systems of the form {∑_j=-∞^mα_jb^j} . We begin our investigation by studying various bases and digit sets that generate the real number system. Next, we introduce the concept of an iterated function system which consists of a set of invariant functions that correspond to a ratio list. Then, with A as a natural number; we study the complex number system generated by complex bases and digit sets in the form of -A±i and {0,1,2,...,A²} respectively. Having established the idea of fractal images and showing how a fractal image together with its copies cover the complex plane, we proceed by introducing symmetric fractals generated by the number system with a complex base and a digit set consisting of 0 and the powers of a primitive nth root of unity. We then examine the covering of the complex plane by a symmetric fractal and its copies by finding an estimate of the base of the number system in terms of n. We find an estimate of such bases in the complex plane by studying the similarity dimension and induction dimension of a symmetric fractal. Then, by looking at the fundamental geometric properties of the symmetric fractals with n ≥ 5, we find an estimate of the real bases in terms of n that do not lead to the covering of the entire complex plane as mentioned above.