Proceedings of the International Congress of Mathematicians
Madrid, August 22–30, 2006

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pp: 247–277

DOI: 10.4171/022-1/11

Knots and dynamics

Étienne Ghys[1]

(1) UMPA, École Normale Supérieure de Lyon, 46, allée d'Italie, 69364, LYON CEDEX 07, FRANCE

The trajectories of a vector field in 3-space can be very entangled; the flow can swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be very complicated. In this talk, I will survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part will be devoted to vorticity, helicity, and asymptotic cycles for flows. The second part will deal with various notions of rotation and spin for surface diffeomorphisms. Finally, I will describe the important example of the geodesic flow on the modular surface, where the linking between geodesics turns out to be related to well-known arithmetical functions.

Keywords: Low-dimensional dynamical systems, periodic orbits, knots and links