让-克里斯托夫·约科兹Jean-Christophe Yoccoz

让-克里斯托夫·约科兹在1975年高等师范学校和巴黎综合理工学院的入学考试中都名列第一。他于1977年以并列第一的成绩获得Agrégation de Mathématiques。

约科兹于1981至1983年在巴西服兵役。他是M Herman的学生，并于1985年提交了博士学位论文。作为Herman的学生，而Herman是动力系统领域的一位顶尖专家，约科兹本人会研究动力系统并不令人意外。然而，约科兹如此迅速地确立自己为该领域最杰出的研究者，这一点却很了不起。在他的学位论文中，约科兹改进了其导师Herman的定理，给出了更简单的证明，但也在更弱的假设下得到了相同的结果。

鉴于他杰出的工作，显然他很快就会获得职位，事实上他被任命为巴黎南大学（奥赛）教授。他成为法兰西大学研究院成员，并成为奥赛法国国家科学研究中心“拓扑学与动力系统”联合研究单位的成员。

在1994年苏黎世的国际数学家大会上，约科兹因其在动力系统方面的工作而获得了最大的荣誉，被授予了菲尔兹奖。在他自己对大会的演讲中，他首先陈述了他的主题的目标：-

广义地说，动力系统理论的目标，正如它应该的那样，是理解大多数系统的大部分动力学。

动力系统理论真正开始于儒勒·昂利·庞加莱，他正在研究太阳系的稳定性。这是动力系统理论试图做的事情的典型：它描述了一个系统在给定规则下如何随时间演化，该规则对于系统的任何特定状态，描述了接下来的状态。太阳系的例子正好显示了人们想知道的东西——给定系统的任何配置，它根据艾萨克·牛顿的定律演化，但它会保持稳定吗，或者，在许多年后，其中一颗行星会被从系统中弹出吗？

在儒勒·昂利·庞加莱之后，该理论由大量数学家发展，如弗拉基米尔·阿诺尔德、皮埃尔·法图、Herman、茹利亚、安德雷·柯尔莫哥洛夫、Palis、卡尔·西格尔、斯蒂芬·斯梅尔和约科兹。Douady在[3]中写道：-

……他们已经证明了稳定性性质——动态稳定性，如太阳系所寻求的，或结构稳定性，意味着在参数变化下系统全局性质的持久性。

约科兹在1994年苏黎世国际数学家大会上的演讲中说：-

我们能够理解的动力系统特征分为两类：双曲动力学和准周期动力学；很可能出现的情况是，特别是在保守系统中，一个系统同时表现出双曲和准周期特征。……我们寻求扩展这些概念，同时保持对动力学的合理理解，以便能够解释尽可能多的系统。那么，最大的问题就是：这些概念是否足以理解大多数系统？

Douady在[2]中概括地描述了约科兹的贡献，但在[3]中更为详细，那篇文章描述了导致约翰·查尔斯·菲尔兹奖章授予的数学工作。我们引用[2]，并建议那些寻求更多技术细节的人查阅[3]：-

他结合了极其敏锐的几何直觉、令人印象深刻的分析能力和深刻的组合意识，来玩他所擅长的棋局。他偶尔会花半天时间进行数学“实验”，用手工或计算机。“当我做这样的实验时”，他说，“我感兴趣的不仅仅是结果，还有它展开的方式，这揭示了真正发生的事情。”约科兹发展了一种对茹利亚集和曼德尔布罗集进行组合研究的方法——称为“约科兹谜题”——这允许深刻的洞察。

Jean-Christophe Yoccoz was placed first in the entrance examination for the École Normale Supérieure and also for the École Polytechnique in 1975. He received the Agrégation de Mathématiques in 1977 in joint first position.

Yoccoz did his military service in Brazil in 1981-1983. He was a student of M Herman and submitted his doctoral thesis in 1985. As a student of Herman, a leading word expert on dynamical systems, it was not surprising that Yoccoz would himself work on dynamical systems. However, it was remarkable how quickly Yoccoz was to establish himself as the most brilliant researcher in this area. In his thesis, Yoccoz improved theorems of his supervisor Herman by giving simpler proofs but also obtaining the same results under weaker hypotheses.

Given his outstanding work it was clear that he would quickly be offered appointments and indeed he was appointed as professor at the University of Paris-Sud (Orsay). He became a member of the Institut Universitaire de France and a member of the Unité Recherche Associé "Topology and Dynamics" of the Centre National de la Recherche Scientifique at Orsay.

At the International Congress of Mathematicians in Zürich in 1994, Yoccoz received his greatest honour for this work on dynamical systems when he was awarded a Fields Medal. In his own address to the Congress, he began by stating the aim of his subject:-

Broadly speaking, the goal of the theory of dynamical systems is, as it should be, to understand most of the dynamics of most systems.

The theory of dynamical systems really began with Poincaré who was studying the stability of the solar system. This is typical of what the theory of a dynamical system tries to do: it describes how a system evolves over time given a rule which, for any particular state of the system, describes the following state. The solar system example shows exactly what one wants to know - given any configuration of the system it evolves according to Newton's laws but will it remain stable or, after many years, will one of the planets be ejected from the system?

After Poincaré, the theory was developed by a large number of mathematicians such as Arnold, Fatou, Herman, Julia, Kolmogorov, Palis, Siegel, Smale and Yoccoz. Douady writes in [3]:-

... they have proved stability properties - dynamic stability, such as that sought for the solar system, or structural stability, meaning persistence under parameter changes of the global properties of the system.

Yoccoz in his own address at the International Congress of Mathematicians in Zürich in 1994 said:-

The dynamical features that we are able to understand fall into two classes, hyperbolic dynamics and quasiperiodic dynamics; it my well happen, especially in the conservative case, that a system exhibits both hyperbolic and quasiperiodic features. ... we seek to extend these concepts, keeping a reasonable understanding of the dynamics, in order to account for as many systems as we can. The big question is then: Are these concepts sufficient to understand most systems?

Douady describes Yoccoz's contributions in general terms in [2] but in more detail in [3] which was the article describing the mathematics leading to the award of the Fields Medal. We quote [2] and advise those looking for more technicalities to consult [3]:-

He combines an extremely acute geometric intuition, an impressive command of analysis, and a penetrating combinatorial sense to play the chess game at which he excels. He occasionally spends half a day on mathematical "experiments", by hand or by computer. "When I make such an experiment", he says, "it is not just the results that interest me, but the manner in which it unfolds, which sheds light on what is really going on." Yoccoz has developed a method of combinatorial study of Julia sets and Mandelbrot sets - called "Yoccoz puzzles" - which permit deep insight.