On the border of chaos 

Jean-Christophe Yoccoz is currently professor at Collège de France, Paris. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden. 

Jean-Christophe Yoccoz is a leading researcher of dynamical systems. Such systems can be used to study the evolution over time of natural or social phenomena, as well as that of technical systems. Diverse applications of the theory include studies of planetary orbits, financial markets, animal populations, and epidemics. Even the simplest dynamical systems can lead to surprisingly complicated mathematical structures. 

Chaotic behavior is characteristic of some of the most interesting dynamical systems. Extreme sensitivity to initial conditions is typical of chaotic systems. A common metaphor used to describe such sensitivity is called the butterfly effect: a butterfly flapping its wings in the Caribbean may cause a storm in Sweden.  

Many dynamical systems exhibit chaotic behavior. On the other hand, periodic dynamical systems evolve in a more regular fashion. Certain regular patterns can also be found in the chaotic dynamical systems. A study of such regularity in chaos is called ergodic theory. This theory was first developed over a hundred years ago by an Austrian physicist, Ludwig Boltzmann, who studied random motions of gas particles and their statistical properties. Ergodic theory is still an active area of research. 

Dynamical systems, which exhibit both chaotic and periodic features, present interesting challenges to researchers. It has been shown that one of the two patterns always appears for a certain class of models in one and two-dimensions. The goal of the project is to develop new mathematical methods for the study of low-dimensional dynamical systems and ergodic theory. Jean-Christophe Yoccoz is one of the most eminent researchers in the field.  

Photo: Renate Schmid - MFO