# Jean-Christophe Yoccoz

Jean -Christophe Yoccoz (* May 29, 1957 in Paris ) is a French mathematician. He was awarded in 1994 for his work in the field of dynamical systems with the Fields Medal.

## Life

Yocccoz won in 1974 the gold medal at the International Mathematical Olympiad, studied from 1975 at the École normale supérieure ( where he at the input entrance exams like those of the Ecole Polytechnique Best was the same), with a degree in 1977. 1981-1983 he did his military service in Brazil during a research stay at IMPA. He received his doctorate in 1985 at Michael Herman at the École Polytechnique in Paris, with a thesis on dynamic systems. In 1988 he became a professor at the Université Paris -Sud and also a member of the Institut Universitaire de France. Since 1996 he is professor at the Collège de France.

Yoccoz is a member of the Academie des Sciences ( 1994). He is a member of Bourbaki.

His doctoral Ricardo Pérez- Marco belongs.

## Services

Has His field of dynamical systems theory in France a long tradition, with Henri Poincaré goes back to the investigation of the three-body problem, but also in the iteration of complex functions by Gaston Julia, or in the theory of fractals by Benoît Mandelbrot. In the 1950s, there arose the famous KAM theory by Kolmogorov, Vladimir Arnold and Jürgen Moser, the existence of quasi-periodic orbits around gates in phase spaces ( gates as defined by closed orbits corresponding to rational periods) describes, depending on the degree of irrationality of periods. Another pioneer was Carl Ludwig Siegel and application of topology Stephen Smale. From the 1970s, the field experienced a revival thanks to the chaos theory ( existence of " strange attractors " in the phase space to which the trajectories are asymptotically tight).

Yoccoz examined not only the qualitative behavior of such dynamical systems ( " hyperbolic " chaotic systems or quasi- regular, etc.) but also the Julia set and Mandelbrot set in the complex dynamics. He proved that the Mandelbrot set is locally connected ( for finitely - renormalizable parameter values) after Adrien Douady and John H. Hubbard previously showed that this is connected.

## Honors

1988 Yoccoz received the Salem Prize. In 1994 he became the Fields medal ( plenary lecture at the ICM in Zurich: Recent Developments in Dynamics ). In 1990 he was invited speaker at the International Congress of Mathematicians in Kyoto (optimal arithmetic conditions in some small divisor theorem ).

## Works (selection)

- Conjugaison differentiable of difféomorphismes du cercle dont le nombre de rotationally verifie une condition diophantienne. Ann. Sci. École standard. Sup (4 ) 17 ( 1984) no 3, 333-359.
- With Lennart Carleson, PW Jones: Julia and John. Bol. Soc. Brasil. Mat ( N.S. ) 25 (1994 ), no 1, 1-30.
- An introduction to small divisor problems, in Michel Waldschmidt, Claude Itzykson, Jean -Marc Luck, Pierre Moussa (Editor ): Number Theory and Physics, Les Houches 1989, Springer 1992
- Continued fraction algorithms for interval exchange maps: an introduction, in Pierre Cartier, among others Frontiers in Number Theory, Physics and Geometry, Volume 1, Springer Verlag 2006
- With Stefano Marmi, Pierre Moussa: Some properties of real and complex functions Brjuno, in Pierre Cartier, among others Frontiers in Number Theory, Physics and Geometry, Volume 1, Springer Verlag 2006