33. Progression between the loops of the Julia set for z → z2 – 1 under iteration
Julia sets are fractals, so it is natural to ask about their dimensions. If c = 0 the Julia set is a circle, so has dimension 1. Elsewhere in the main cardioid, the loop becomes more wiggly and the dimension increases as c approaches the boundary of the Mandelbrot set. There are points on the boundary of the Mandelbrot set where the Julia sets have dimensions arbitrarily close to 2, corresponding to extremely complicated behaviour of the itineraries. However, once c leaves the Mandelbrot set the Julia set is a dust, which becomes very sparse and has small dimension when c is far away from the origin.
A historical note
Julia sets and the Mandelbrot set were studied long before their fractal form was appreciated. In 1915, the Acadйmie des Sciences in Paris declared that the topic for its 1918 Grand Prix was to be the global properties of iteration of functions. The War was raging, and whilst serving with the French army, the mathematician Gaston Julia (1893–1978) received serious facial injuries. Nevertheless, he managed to continue his mathematical work in his hospital bed, and in December 1918 was awarded the Prize. At the same time, another French mathematician Pierre Fatou (1878–1929) was working along similar lines, although he did not enter the competition. Thus in 1919, Julia and Fatou independently published major papers on the iteration of functions on the plane, including those of the form z → z2 + c. These papers contain the definitions of what are now known as Julia sets and the Mandelbrot set as w0" aid="8ILIR"
Chapter 5 Random walks and Brownian motion
A ‘random walk’ or ‘drunkard’s walk’ is a simple idea that has many mathematical consequences and applications, and in particular leads to fractal graphs and other fractal objects. A walker sets out at time 0 from a point, which we take to be the origin, on a straight road. Each second he takes a step of length 1 unit (yard or metre, say) either forwards or backwards, the direction being chosen at random with a 50 per cent chance of going in each direction, perhaps by tossing a coin or just by a drunken whim. The graph in Figure 34 indicates the typical progress of the walker, showing his position each second in steps relative to the starting point.
34. Progress of a typical random walk
With an equal chance of moving forwards or backwards at each step,
the walker does not make very fast overall progress in either
direction, but it is quite likely that after a time he will have
travelled some way from the origin. Figure
35 shows the first 100 steps of several independent random
walks—as time progresses, the range of locations of the various
walkers spreads out on either side of the origin. The position after
20 steps of 130 independent random walkers is plotted in Figure
36. These have the ‘bell-shaped’ or ‘normal’
distribution which frequently occurs in statistical samples. Heads
and tails being equally likely, most walkers will have taken roughly
the same number of steps forwards as backwards and so will be
unlikely to be more than about 5 steps from the starting point.
However, one or two walkers will have thrown a run of heads or tails,
and so will be rather further from the starting point. Since steps
forward and backward are equally likely the average position
of all the walkers will be at 0, but most walkers will be some
distance away from the start, typically about 4–5 steps. To express
this more precisely, the standard deviation, which gives the ‘width’
of the bell curve and indicates the ‘typical’ distance of a
walker from the origin, is
Continuing the random walk for a longer period of time, the average
position after T steps will sti a conductor in the form of a
fat the o ll be 0, but the ‘typical’ distance from the origin
will increase and be about
.
