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If w is an attracting fixed point of f we define the basin of convergence of w, or B(w), as the set of those complex numbers z which under repeated applications of f approach w. That is B(w) = { z | lim fn(z) = w}. Consider the set of points where G takes on values in B(w). This is an open set and is made up of connected disjoint open components. We call those components bays. It is helpful to note that each point in a fixed cycle of order n of f is a fixed point of the nth iterate of f. When we look at the graph of G we see the bays corresponding to attracting fixed points of f and of fn for all n. We will be looking at the properties of these bays: The bays corresponding to each of the points in an attracting cycle appear symmetrically. Are there bounded or unbounded bays? How do the bays corresponding to one cycle interact with those corresponding to another? I would love to know what these properties say about f.
Attracting cycle of order three. There is one unbounded bay in each of the three colors, and infinitely many bounded bays.
Attracting cycle of order three. There is one unbounded bay in each of the three colors, and infinitely many bounded bays.
Attracting cycle of order two. There are no unbounded bays in either color but infinitely many bounded bays.
Attracting cycle of order two. There are no unbounded bays in either color but infinitely many bounded bays. The behavior near the origin is very different from the previous graph.
This is the same function but we are looking at an area away from the origin.
Attracting cycle of order 19. This does not look at all similar to the preceding graphs. The smooth region toward the right has 19 bays radiating out. This piece of the complex plane is at a distance of 10 to the power 80 away from the origin.
This a close up of a feature near the left edge of the previous graph. There are no unbounded bays and infinitely many bounded.
Attracting cycle of order 28.
Looking away from the origin and in the detail between the unbounded bays often shows surprising results.
In fact I find a trail of interesting images of the graph of this function including this and the next three.
Attracting cycle of order eleven.
Perhaps a guitar with a hint of some pretty wild music?
Attracting cycle of order 48. There are twelve rays out of the center, each with four shades of color.
The four shades separate into two pairs. One can barely notice that the vertical blue spike ends in a point from which twelve rays emerge.
Attracting cycle of order four. There are unbounded bays of each color although it is not obvious in this view
Attracting cycle of order 17.
This is detail between unbounded bays in the previous graph..
Attracting cycle of order seven.
Attracting cycle of order 48. There are no unbounded bays.
The 48 point cycle breaks up into two sets of 24 in the graph.
Attracting cycle of order five. The bays are quite isolated from each other.
Attracting cycle of order 89.
Detail from the previous function.
Attracting cycle of order eight. Mushrooms or toadstools?
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