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Fatou and Julia


Iteration; background to the work of Fatou and Julia
Benoît Mandelbrot Mathematician
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Much of my life was devoted to the problem of iteration, of Julia sets and what later became known as the Mandelbrot set. It's a very long story. It started way back when I was a student at École Polytechnique. My teacher in geometry, differential geometry, was Gaston Julia, a man whom my uncle knew very well. And my uncle kept telling me about the marvellous works that Julia had done before and after 1920. That was background, otherwise my uncle and Julia were very, very hostile to each other. Then after many misadventures and when I was thinking that perhaps I should go back and do mathematics, I went back to my uncle and again he spoke to me about Julia. He said that this marvellous work that he and Fatou, Julia and Fatou had done in 1917 had lain fallow for thirty years and if I could bring it back to life I would get myself a marvellous Ph.D. and a marvellous mathematical career. Now what was the problem that Julia and Fatou were working on? It's one of the simplest and the oldest, the most elementary and the most complicated questions in mathematics. You take a function, f(z). z is a complex variable, (x+iy). f(z) can be z2, z2+c and can be a rational function, z2+c divided by (z-1). Rational functions are functions that are ratios of polynomials. The simplest. It can be also a transcendental function, exponential of z, logarithm of z, sine of z. But the rational functions were the most challenging. The question is the following: you take a point z0; you define z1 as being f(z0); z2 as being f(z1); then z3 as being f(z2), and so on. Therefore you get a sequence of points, z0, z1, z2 and z3, on to infinity. Those points create an orbit. Question: what happens to this orbit? Does it get lost in infinity? Does it converge to a point? Does it at the end oscillate between two points? What happens to it? Again, the question was very old, early nineteenth century. Arthur Caley made very considerable contributions to it under rather strange auspices. He was trying to show what happened if you solved the equation z2 -1 = 1 by Newton's method. Now, any sane person in mathematics knows that 'z2 -1 = 0' has two solutions: one, and minus one. That's fine, but Caley said, "What would happen if I did not know the solutions and applied a method due to Newton and Rafson to find them?" He showed that if you started on the right-half plane as an approximation, you go to z = 1; left half plane you go to z -1. Very simple. Then he asked himself the question what happens if you taken the question of z3 - 1 = 0, and found that that somehow instead of being a little bit more complicated it was gigantically more complicated. First he totally miscalculated the complication; he said, "Well, in my next paper I'm going to examine the case of z3 - 1 = 0," then he came back to it and found that it was very difficult. If you start with a reasonably simple value, close to the limits you want to have, you converge quite quickly. But then there was what he called a 'grey zone' where he couldn't say anything, and he gave up. He couldn't do anything because there was enormous difference between these iterations as they're called if you do them locally, if they are close to your limit; or globally. Local theory was easy; global theory in general was much more complicated. In 1905 a French mathematician named Fatou did much new work about it, and then in 1912 an event occurred, an event of a purely mathematical nature. A person named Paul Montel, who was then a very young and brilliant mathematician, produced a mathematical technique that he called 'the theory of normal families of functions', and that technique was hailed as a great advance in analysis. War came. During the war two mathematicians in France, Fatou again and Julia, became interested in the problem.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Bernard Sapoval Daniel Zajdenweber

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Duration: 5 minutes, 15 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008