Reeb dynamics inspired by Katok’s example in Finsler geometry

Abstract

Inspired by Katok's examples of Finsler metrics with a small number of closed geodesics, we present two results on Reeb flows with finitely many periodic orbits. The first result is concerned with a contact-geometric description of magnetic flows on the 2-sphere found recently by Benedetti. We give a simple interpretation of that work in terms of a quaternionic symmetry. In the second part, we use Hamiltonian circle actions on symplectic manifolds to produce compact, connected contact manifolds in dimension at least five with arbitrarily large numbers of periodic Reeb orbits. This contrasts sharply with recent work by Cristofaro-Gardiner, Hutchings and Pomerleano on Reeb flows in dimension three. With the help of Hamiltonian plugs and a surgery construction due to Laudenbach we reprove a result of Cieliebak: one can produce Hamiltonian flows in dimension at least five with any number of periodic orbits; in dimension three, with any number greater than one.