Knots and dynamics. (English) Zbl 1125.37032

Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume I: Plenary lectures and ceremonies. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 247-277 (2007).
Summary: 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 survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part is devoted to vorticity, helicity, and asymptotic cycles for flows. The second part deals with various notions of rotation and spin for surface diffeomorphisms. Finally, I 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.
For the entire collection see [Zbl 1111.00009].


37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C27 Periodic orbits of vector fields and flows
37E45 Rotation numbers and vectors
57M25 Knots and links in the \(3\)-sphere (MSC2010)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)