A complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ can be viewed as mapping from two dimensional real plane to itself. Consequently, a graph of such function lies in a real four dimensional space, which means that it is hard to visualize in our three dimensional world.
In my thesis we investigate the behaviour and visualization of complex maps $q_c(z) =z^2+c$, $c \in \mathbb{C}$ and $R(z) = z^n+P(z)/Q(z)$, $n \geq 2$, where the degree of polynomial $Q$ is greater or equal to the degree of the polynomial $P$. More precisely, we examine the convergence of the functional sequence of iterates of a given function; to iterate $f$ is to apply $f$ repeatedly to an input. Those complex numbers which are bounded under iteration with $f$ form the filled Julia set, denoted as $K(f)$. We define sets $U(f)$ and $V(f)$ which contain the complex numbers with the real and imaginary components that remain bounded under iterations with the map $f$, respectively.
We prove that for the $q_c(z)$ the filled Julia set $K(q_c)$ equals $U(q_c)$ and that inequality $K(q_c) \neq V(q_c)$ holds. For the family of maps $R(z)$ we show that $K(R) = U(R)$ if $n$ is even, $K(R) \neq U(R)$ if $n$ is odd; the inequality $K(R) \neq V(R)$ always holds.
In the last section we examine more in detail the Newton method applied on holomorphic functions.
|
