On the structure of the family of Cherry fields on the torus

Colin Boyd
1985 Ergodic Theory and Dynamical Systems  
A class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure of {t\R,°f has irrational rotation number}, where / is a certain map of the circle and R, is rotation through t. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.
doi:10.1017/s014338570000273x fatcat:eo2ziyfmnzczln54ydsmp3t7de