No Attracting Points or Cycles

Although we will spend most of our effort looking at the effects of attracting fixed points and cycles, we first consider a few functions which have neither. All of the G functions have two things in common: G(0) = 0 and G'(0) = 1. So if we look closely enough at the origin, we will see a pinwheel of color with the red wedge to the right and rotating through green and blue and back to red as we go counterclockwise. In graphs with no attracting points or cycles as we look away from the origin we frequently see what appears to be a hectic mishmash of color. (Other times we find very interesting patterns.) Even when things seem most hectic we should remember that G is meromorphic and looking closely enough at any point in the complex plane will show a smooth function with just a few poles if any.