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Julia sets


Beginning to work on the problems of Julia and Fatou
Benoît Mandelbrot Mathematician
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But again, this effort paled next to the effort of my coming back to the works of Julia and Fatou. Originally what I did was simply to play with them. An extraordinary situation- where Julia spoke of the complication of these shapes without being able to describe their simplicity, their regularity. He was very conscious that they represented something else, something beyond circles and other shapes, but he couldn't see them. The power of the computer to simply construct them became something that I enjoyed enormously, and we went through a long period of just changing a parameter - and first we had a rather complicated expression- of seeing what was happening. Now, being a physicist, or more precisely, a realist at heart, I did not ask myself abstract questions about iteration, but the very concrete questions, which were very close to what Julia had been motivated by originally. So you have a sequence again of an orbit, zO, z1, z2, etc., to infinity; that orbit under certain conditions converges to a limit; and under other conditions converges to a cycle of two, three, four, a finite cycle; under other conditions it does neither but just goes on wandering for ever. Now, I tried to do it experimentally by taking a nice slightly complicated function, f(z), because I felt that to get something interesting I needed something complicated, and we just played. It turns out that the problem I was asking is simply too ridiculous for words because you needed a very complicated calculation, computation, to get those points, those values of c, a parameter for which there was a limit; and then those which are a limit cycle of two, then a limit cycle of three, of four, and the convergence of these sets to the limit, I mean, was very, very slow. It was irritation of the the cumbersome of the argument that led me to think of something else, and this also led me to replace those very complicated functions that I was trying first with the simplest one, which is z2+c. In a way, to come back to what I was saying about inversion: inversions more or less 1/z, and iteration of squares, quadratic iteration, is z2+c. If you go to functions which are non-linear, these are the simplest, and that has very often been my policy: not to try to be general, not to try to be too over ambitious, to think about the strategy or to think about ultimate goals, but to try to understand simple cases absolutely thoroughly. In the simple case of 1/z again gave rise to these works on iteration and limit sets of Klenian groups. The simple case of z2+c turned out to be of a wealth, a richness, beyond anything I could have imagined, because I thought it was going to at most a tactical exercise and it became very much more than that.

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: Daniel Zajdenweber Bernard Sapoval

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

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.

Duration: 3 minutes, 43 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008