27. The filled-in Julia set of z → z2 − 0.9 with its exterior shaded according to the escape time
The zoo of Julia sets
So what happens if we iterate our function
z → z2 + c
for other complex numbers c = a + bi? (As noted before, this may be written in terms of coordinates as (x, – y) → (x2 – y2 + a, 2 xy + b).) Geometrically, this function squares the magnitude and doubles the angle of z and shifts by c. We have seen that when c = 0 the Julia set is a circle, and when c = 0.1 + 0.05i it is a fractal loop or circuit. Is the Julia set always a loop? For complex numbers c = a + bi of fairly small magnitude, including all c with magnitude less than 0.25, the effect of the shift by c is not enough for the picture to change much, and the Julia set is still a loop, though its irregularities become more prominent as the magnitude of c increases, reflecting a larger dimension, see Figure 28(a),(b). However, if the magnitude of c gets larger still and the shifting part of the function becomes more significant, strange things may happen. For example, with c = –1, the Julia set of the function z → z2 – 1 consists of a central fractal loop surrounded by many smaller touching loops which in turn are surrounded by even smaller loops, and so on, see Figure 28(c). Moreover, this configuration is stable: for complex numbers c close to –1 the Julia set has a similar structure of touching loops.
28. Julia sets for a range of complex numbers c, the product and power rules for. pegiven by (a) 0.2–0.2i, (b) –0.6–0.3i, (c) –1, (d) –0.1 + 0.75i, (e) 0.25 + 0.52i, (f) –0.5 + 0.55i, (g) 0.66i, (h) –i
Investigating Julia sets of the function z → z2 + c for other complex numbers c yields more surprises. With c close to –0.1 + 0.75i the Julia set is again formed by a hierarchy of loops, but now they meet three at a point, Figure 28(d)—this is known as a Douady rabbit after the French mathematician, Adrien Douady. With c near 0.25 + 0.52i the loops meet four at a point and around –0.5 + 0.55i they meet five at a point, Figure 28(e),(f). On the other hand, by choosing specific values of c such as −i the loops collapse to give a dendrite or ‘twig-like’ Julia set, Figure 28(h).
As c becomes larger still, there is a further dramatic change in the nature of the Julia set—it ceases to be a connected entity and breaks up completely. The Julia set becomes a fractal dust, and is totally disconnected, so one cannot travel between any two distinct points of the set without going outside the set, for instance when c = 0.66i, Figure 28(g). In this situation, the Julia set is identical to the filled-in Julia set—any point not in the Julia set itself iterates to infinity, with no other destinations possible.
How can we make sense of what has been called the ‘zoo’ of Julia sets, which have many different characters that result from iterating the function z → z2 + c for different complex numbers c? How can we tell which c will give a Julia set that is a single loop, with loops meeting 2, 3, or 4 at a time, or something else? The key to these questions is an even more complicated object, namely the Mandelbrot set.
The Mandelbrot set
To find the filled-in Julia sets, for each fixed complex number c we identified and coloured black the various initial points from which the itineraries under z → z2 c did not wander off to infinity. As an alternative we could always take the same initial point and examine how the itineraries under z → z2 c change as c varies.
29. The Mandelbrot set comprises those complex numbers a + bi in the blackened portion of the complex plane
So, for each complex number c = a + bi we now examine the itinerary under
z → z2 + c
starting at Arnold
Figure 29—the Mandelbrot set. A rough picture may be computed by examining the iterates of 0 under z → z2 + c for complex numbers c ranging across the plane—if the first 100 iterates remain close to the origin then c can be deemed to be in the Mandelbrot set.
Although defined by iterating functions of the simple form z → z2 + c, the Mandelbrot set is extraordinarily complicated, far more so than any individual Julia set. There is a prominent cardioid or ‘heart-shaped region’ with many nearly circular buds touching its perimeter. The largest bud is on the left of the cardioid and is exactly circular, the next largest buds are a pair located at the top and bottom of the cardioid, and there are many smaller buds around the cardioid. These in turn have much smaller buds surrounding them, and so on. But there is far more than just the buds. Very fine ‘hairs’ grow out from various points on the buds and within these hairs are minute copies of the entire Mandelbrot set. Close examination near the edge of the Mandelbrot set reveals a wealth of fine structure: high magnification shows shapes such as seahorses, multi-armed spirals, and windmills, Figure 30. In principle, one can continue to zoom into the boundary of the Mandelbrot set indefinitely and the variety of shapes to be found is unending.
