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Solving the problem of limit sets


Return to iteration in 1977: Hadamard, Poincaré and Kleinian groups
Benoît Mandelbrot Mathematician
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Chapter two occurred much later. I wrote my first book on fractals in '75 in French. Then there was a translation, which appeared in English in '77, and my second book appeared in 1982. The '77 book was not much different from the translation; it was like additions. But one addition that occurred was to state what I was going to do next, in '77. In the meantime I had read an obit. of Poincaré, written by Hadamard. Now this I think is a masterpiece among obits. There may be such obits. by statesmen about another statesman, but here was a man of extraordinary breadth, depth, style and strength, namely Hadamard, describing the whole life and works of Poincaré which he understood from end to end. The most striking part to me about that obit. was how Hadamard was dwelling on Poincaré's work in dynamics, and on Fuchsian functions, on Kleinian groups. These were the discoveries Poincaré had done very early in his career, which consisted also in repeating operations, but the operations were not the ones I mentioned, but simply ratios of linear functions, called Möbius transformations. And Möbius transformations, if you start with a certain number of transformations you create a group which is Kleinian in general, in some special cases they are called Fuchsian. Hadamard was very aware of the fact that in some cases the solutions came very simply; in other cases in a very irregular fashion. And again, for Poincaré and Hadamard these are examples of what later came to be called 'chaos'. And by reading this obituary, and not because of the environment, what was happening in science, I suddenly realised that fractals had been seen by Poincaré in the study of these groups but never explored, and I remembered that Julia had presented his work as being in the spirit of Kleinian groups of Poincaré and again this wasn't explored. Computers were available and I became, as soon as the book of '77 came out, extraordinarily involved in the exploration of Kleinian groups and of iterations of rational functions.

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: 2 minutes, 49 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008