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First reading of the work of Fatou and Julia

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First reading of the work of Fatou and Julia

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The theory of Fatou and Julia

Benoît Mandelbrot
Mathematician

Views | Duration | ||
---|---|---|---|

71. Self-affining variability | 193 | 03:16 | |

72. Pathological shapes | 235 | 01:59 | |

73. Iteration; background to the work of Fatou and Julia | 229 | 05:14 | |

74. Fatou and Julia | 236 | 01:32 | |

75. The theory of Fatou and Julia | 225 | 04:09 | |

76. First reading of the work of Fatou and Julia | 197 | 02:23 | |

77. Return to iteration in 1977: Hadamard, Poincaré and... | 199 | 02:48 | |

78. Solving the problem of limit sets | 236 | 05:11 | |

79. Solving the problem of limit sets by using computers | 172 | 03:22 | |

80. Imitation of nature and creation of shapes | 207 | 03:03 |

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Both Fatou and Julia realised that Montel's theory could bring the theory of iteration of functions that I have described before- those sequences of z0, z1, could bring a very gigantic step forward. The fact that Fatou and Julia had for each other the most negative feelings would have been totally insignificant otherwise. It's well known that academics do not necessarily love each other, it was well documented in every country of the world, but the level of antipathy or hatred these two people had for each other was quite unusual. It was relevant because they used every trick in the book in that theory. They did effectively the same theory in parallel, in a different spirit and different style. Julia was more geometric; Fatou was in some ways perhaps deeper. Whatever the case, the theory of Fatou and Julia was published in 1917, in '17, '18, '19, in those years, and became a major landmark in the history of analysis. Now why was it a landmark and what happened to it? It had almost no follow up; that is over the thirty years between, say, 1917-47, a few papers appeared that clarified specific questions that Julia had found to be- had answered incorrectly. He was corrected, very important work was done, but by and large it can be said that very little happened in terms of starting the theory again. And that was my uncle's point; that the theory of normal families of functions had given rise to this immense advance and then nothing else happened. Now why was the advance interesting? At that time the reason was not formulated very strongly but it is very easy to formulate it today. Shortly before Julia and Fatou, Poincaré and Hadamard, and many others, but these two primarily - Poincaré was a great man of his time, Hadamard was nearly his equal - had demonstrated that solutions of very simple equations can have extraordinarily bizarre trajectories, that when the equation was simple and natural it did not mean that the solution was also simple. Hadamard in fact discovered this sensitivity to initial conditions, which was later, a hundred years later, to become a very key factor in physics and mathematics. Poincaré made a list of possible horrors, which may happen if one studied equations and emphasised them and the effect they would have. And Julia in his book was very much writing in this spirit. What he wanted to show is that even though the sequence of operations I described for simple functions, f(z), looked perfectly childish, very strange things could happen to such solutions. The orbits could behave in a very complicated fashion. This was something which was very much ahead of its time; that is, hardly anybody thought of these matters until again thirty years or two passed, when the theory of chaos started and went very rapidly to a very highly refined state. Why was it so? Why was not more attention paid? I don't know. Some people say it's just a matter of the state of the world, that the world was believed to be a steady comfortable place in 1900, that World War I was not enough to make people change their minds. There were several articles, not on the Fatou-Julia theory, but on others, on the Hadamard theory, over the years, but by and large, very, very few.

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.

**Title: **The theory of Fatou and Julia

**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:**
4 minutes, 10 seconds

**Date story recorded:**
May 1998

**Date story went live:**
24 January 2008