In this paper we define and study the Julia set and the Fatou set of an arbitrary polynomial f, which is defined on the closed complex plane and whose degree is at least two. We are especially interested in the structure of these sets and in approximating the size of the Julia set.
First, we define the Julia and Fatou sets by using the concepts of normal families and equicontinuity. Then we move on to proving many of the essential facts concerning these sets, laying foundations for the main theorems of this paper presented in the fifth chapter. By the end of this chapter we achieve quite a good understanding of the basic structure of the Julia set and the Fatou set of an arbitrary polynomial f.
In the fourth chapter we introduce the Hausdorff measure and dimension along with some theorems regarding them. In this chapter we also say more about fractals and self-similar sets, for example the Cantor set and the Koch curve. The main goal of this chapter is to prove a well-known result which allows to easily determine the Hausdorff dimension of any self-similar set that fulfils certain conditions. We end this chapter by calculating the Hausdorff dimension of the one-third Cantor set and the Koch-curve by using the result described earlier and notice, that their Hausdorff dimension is not integer-valued.
In the fifth chapter we study the structure of the Julia set further, concentrating on its connectedness, and introduce the Mandelbrot set. In this chapter we also prove the three main theorems of this paper. First we show a sufficient condition for the Julia set of a polynomial to be totally disconnected. This result, with some theorems proven in the third chapter, shows that in this case the Julia set is a Cantor-like set. The second result shows when the Julia set of a quadratic polynomial of the form f(z) = z^2 + c is a Jordan curve. The third and final result shows that given an arbitrary polynomial f, there exists a lower bound for the Hausdorff dimension of the Julia set of the polynomial f, which depends on the polynomial f. This is the most important result of this paper.
