Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows

Abstract

This paper concerns connections between dynamical systems, knots and helicity of vector fields. For a divergence-free vector field on a closed $3$-manifold that generates an Anosov flow, we show that the helicity of the vector field may be recovered as the limit of appropriately weighted averages of linking numbers of periodic orbits, regarded as knots. This complements a classical result of Arnold and Vogel that, when the manifold is a real homology $3$-sphere, the helicity may be obtained as the limit of the normalised linking numbers of typical pairs of long trajectories. We also obtain results on the asymptotic distribution of weighted averages of null-homologous periodic orbits.