Periodic points for sphere maps preserving monopole foliations (Q2274329)

Two different foliations of \(\mathbb S^2\), each with a single singularity (pole) are described. One, called the paridianal foliation, has leaves which project from the point antipodal to the pole onto the tangent plane at the pole to a line through the pole together with circles tangent to that line at the pole. The other, the rabbit foliation, modifies the projected leaves in one half plane. Maps \(f:\mathbb S^2\to\mathbb S^2\) that are \(C^1\) near the pole and preserve the paridianal foliation are such that \(f^n\) has at least \(|\deg(f)|^n\) fixed points. \(C^1\) maps preserving the rabbit foliation have degree \(-1\), 0 or 1 and have a fixed point.