Periodic Points for Sphere Maps Preserving Monopole Foliations

Let S 2 be a two-dimensional sphere. We consider two types of its foliations with one singularity and maps f : S 2 → S 2 preserving these foliations, more and less regular. We prove that in both cases f has at least | deg ( f ) | fixed points, where deg ( f ) is a topological degree of f . In particular, the lower growth rate of the number of fixed points of the iterations of f is at least log | deg ( f ) | . This confirms the Shub’s conjecture in these classes of maps.