Periods of Self-Maps on $${\mathbb{S}}^{2}$$ Via their Homology

Abstract

As usual, we denote a 2-dimensional sphere by \({\mathbb{S}}^{2}\). We study the periods of periodic orbits of the maps f : \({\mathbb{S}}^{2}\to {\mathbb{S}}^{2}\) that are either continuous or C¹ with all their periodic orbits being hyperbolic, or transversal, or holomorphic, or transversal holomorphic. For the first time, we summarize all known results on the periodic orbits of these distinct kinds of self-maps on \({\mathbb{S}}^{2}\) together. We note that every time when a map f : \({\mathbb{S}}^{2}\to {\mathbb{S}}^{2}\) increases its structure, the number of periodic orbits provided by its action on the homology increases.